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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html" class="active"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html" class="active"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html" class="active"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html" class="active"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html" class="active"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html" class="active"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html" class="active"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html" class="active"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html" class="active"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html" class="active"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html" class="active"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html" class="active"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html" class="active"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html" class="active"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html" class="active"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html" class="active"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html" class="active"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html" class="active"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html" class="active"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html" class="active"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html" class="active"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html" class="active"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html" class="active"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html" class="active"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html" class="active"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html" class="active"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html" class="active"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html" class="active"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html" class="active"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html" class="active"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html" class="active"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html" class="active"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256.html" class="active"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256.html" class="active"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html" class="active"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html" class="active"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html" class="active"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html" class="active"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
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<h1><a class="header" href="#proving-system" id="proving-system">Proving system</a></h1>
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<p>The Halo 2 proving system can be broken down into five stages:</p>
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||||
<ol>
|
||||
<li>Commit to polynomials encoding the main components of the circuit:
|
||||
<ul>
|
||||
<li>Cell assignments.</li>
|
||||
<li>Permuted values and products for each lookup argument.</li>
|
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<li>Equality constraint permutations.</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li>Construct the vanishing argument to constrain all circuit relations to zero:
|
||||
<ul>
|
||||
<li>Standard and custom gates.</li>
|
||||
<li>Lookup argument rules.</li>
|
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<li>Equality constraint permutation rules.</li>
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</li>
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<li>Evaluate the above polynomials at all necessary points:
|
||||
<ul>
|
||||
<li>All relative rotations used by custom gates across all columns.</li>
|
||||
<li>Vanishing argument pieces.</li>
|
||||
</ul>
|
||||
</li>
|
||||
<li>Construct the multipoint opening argument to check that all evaluations are consistent
|
||||
with their respective commitments.</li>
|
||||
<li>Run the inner product argument to create a polynomial commitment opening proof for the
|
||||
multipoint opening argument polynomial.</li>
|
||||
</ol>
|
||||
<p>These stages are presented in turn across this section of the book.</p>
|
||||
<h2><a class="header" href="#example" id="example">Example</a></h2>
|
||||
<p>To aid our explanations, we will at times refer to the following example constraint
|
||||
system:</p>
|
||||
<ul>
|
||||
<li>Four advice columns <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">a</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">b</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">c</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal">d</span></span></span></span>.</li>
|
||||
<li>One fixed column <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span></span></span></span>.</li>
|
||||
<li>Three custom gates:
|
||||
<ul>
|
||||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.44445em;vertical-align:0em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">b</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.791661em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathnormal">c</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></li>
|
||||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.902771em;vertical-align:-0.208331em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.10764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></li>
|
||||
<li><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">d</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">a</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span></li>
|
||||
</ul>
|
||||
</li>
|
||||
</ul>
|
||||
<h2><a class="header" href="#tldr" id="tldr">tl;dr</a></h2>
|
||||
<p>The table below provides a (probably too) succinct description of the Halo 2 protocol.
|
||||
This description will likely be replaced by the Halo 2 paper and security proof, but for
|
||||
now serves as a summary of the following sub-sections.</p>
|
||||
<table><thead><tr><th>Prover</th><th></th><th>Verifier</th></tr></thead><tbody>
|
||||
<tr><td></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">←</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">t</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">n</span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord">1</span><span class="mclose">)</span></span></span></span></td></tr>
|
||||
<tr><td></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">←</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></td></tr>
|
||||
<tr><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">A</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span></td><td></td></tr>
|
||||
<tr><td></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">←</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.02778em;">θ</span></span></span></span></td></tr>
|
||||
<tr><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">L</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.058331em;vertical-align:-0.30643899999999996em;"></span><span class="mopen">[</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.4518920000000004em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.4518920000000004em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.24810799999999997em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.451892em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.30643899999999996em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.05764em;">S</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-2.451892em;margin-left:-0.05764em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.30643899999999996em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mclose">]</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span></td><td></td></tr>
|
||||
<tr><td></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">←</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.05278em;">β</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.05556em;">γ</span></span></span></span></td></tr>
|
||||
<tr><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">P</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.13889em;">P</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.13889em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span></td><td></td></tr>
|
||||
<tr><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">Z</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07153em;">Z</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.301108em;"><span style="top:-2.5500000000000003em;margin-left:-0.07153em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">m</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span></td><td></td></tr>
|
||||
<tr><td></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">←</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span></td></tr>
|
||||
<tr><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.647568em;vertical-align:-0.52em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.127568em;"><span style="top:-2.655em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">t</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mclose mtight">)</span></span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.496108em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">gate</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.19444571428571428em;"><span style="top:-2.2341314285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26586857142857145em;"><span></span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mclose mtight">)</span><span class="mbin mtight">+</span><span class="minner mtight">⋯</span><span class="mbin mtight">+</span><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.03588em;">y</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9020857142857143em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span></span></span></span></span><span class="mbin mtight">⋅</span><span class="mord mtight"><span class="mord text mtight"><span class="mord mtight">gate</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.20521714285714282em;"><span style="top:-2.2341314285714287em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mathnormal mtight">i</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.26586857142857145em;"><span></span></span></span></span></span></span><span class="mopen mtight">(</span><span class="mord mathnormal mtight" style="margin-right:0.07847em;">X</span><span class="mclose mtight">)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.52em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></td><td></td><td></td></tr>
|
||||
<tr><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.66666em;vertical-align:-0.08333em;"></span><span class="minner">⋯</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.138em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8879999999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">n</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">d</span><span class="mbin mtight">−</span><span class="mord mtight">1</span><span class="mclose mtight">)</span></span></span></span></span></span></span></span></span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span></td><td></td><td></td></tr>
|
||||
<tr><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68611em;vertical-align:0em;"></span><span class="mord"><span class="mord mathbf">H</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mclose">]</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span></td><td></td></tr>
|
||||
<tr><td></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">←</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span></td></tr>
|
||||
<tr><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.03588em;">v</span><span class="mord mathnormal">a</span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">s</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">A</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.3361079999999999em;"><span style="top:-2.5500000000000003em;margin-left:-0.08125em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">d</span><span class="mbin mtight">−</span><span class="mord mtight">1</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.208331em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mclose">]</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span></td><td></td></tr>
|
||||
<tr><td></td><td></td><td>Checks <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span></td></tr>
|
||||
<tr><td></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">←</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">2</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td></tr>
|
||||
<tr><td>Constructs <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> multipoint opening poly</td><td></td><td></td></tr>
|
||||
<tr><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.001892em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">Commit</span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">h</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.751892em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">′</span></span></span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mclose">)</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span></td><td></td></tr>
|
||||
<tr><td></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">←</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td></tr>
|
||||
<tr><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63888em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">q</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">evals</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">[</span><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">0</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">Q</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="minner">…</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mclose">]</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span></td><td></td></tr>
|
||||
<tr><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">eval</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span><span class="mopen">(</span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mclose">)</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">→</span></span></span></span></td><td></td></tr>
|
||||
<tr><td></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">←</span></span></span></span></td><td><span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span></td></tr>
|
||||
</tbody></table>
|
||||
<p>Then the prover and verifier:</p>
|
||||
<ul>
|
||||
<li>Construct <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">finalPoly</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span></span></span></span> as a linear combination of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8805499999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathbf">Q</span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.10903em;">U</span></span></span></span> using
|
||||
powers of <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">4</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>;</li>
|
||||
<li>Construct <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord text"><span class="mord">finalPolyEval</span></span></span></span></span> as the equivalent linear combination of
|
||||
<span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.63888em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord"><span class="mord mathbf">q</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">evals</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.58056em;vertical-align:-0.15em;"></span><span class="mord"><span class="mord mathnormal">u</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.33610799999999996em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">eval</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span></span></span></span>; and</li>
|
||||
<li>Perform <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">InnerProduct</span></span><span class="mopen">(</span><span class="mord text"><span class="mord">finalPoly</span></span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.07847em;">X</span><span class="mclose">)</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">x</span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.30110799999999993em;"><span style="top:-2.5500000000000003em;margin-left:0em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight">3</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord text"><span class="mord">finalPolyEval</span></span><span class="mclose">)</span><span class="mord">.</span></span></span></span></li>
|
||||
</ul>
|
||||
<blockquote>
|
||||
<p>TODO: Write up protocol components that provide zero-knowledge.</p>
|
||||
</blockquote>
|
||||
|
||||
</main>
|
||||
|
||||
|
@ -177,7 +258,7 @@
|
|||
|
||||
|
||||
|
||||
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<i class="fa fa-angle-right"></i>
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</a>
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@ -195,7 +276,7 @@
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<a rel="next" href="../design/proving-system/multipoint-opening.html" class="nav-chapters next" title="Next chapter" aria-label="Next chapter" aria-keyshortcuts="Right">
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/comparison.html" class="active"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
|
||||
<ol class="chapter"><li class="chapter-item expanded affix "><a href="../../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/comparison.html" class="active"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/inner-product.html" class="active"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<h1><a class="header" href="#inner-product-argument" id="inner-product-argument">Inner product argument</a></h1>
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<p>Halo 2 uses a polynomial commitment scheme for which we can create polynomial commitment
|
||||
opening proofs, based around the Inner Product Argument.</p>
|
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<blockquote>
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<p>TODO: Explain Halo 2's variant of the IPA.</p>
|
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<p>It is very similar to <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord text"><span class="mord">PC</span></span><span class="msupsub"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.32833099999999993em;"><span style="top:-2.5500000000000003em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord text mtight"><span class="mord mtight">DL</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.15em;"><span></span></span></span></span></span></span><span class="mord">.</span><span class="mord text"><span class="mord">Open</span></span></span></span></span> from Appendix A.2 of <a href="https://eprint.iacr.org/2020/499">BCMS20</a>.
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See <a href="comparison.html#bcms20-appendix-a2">this comparison</a> for details.</p>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/lookup-argument.html" class="active"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/proving-system/lookup.html" class="active"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html" class="active">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html" class="active">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html" class="active"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="index.html">halo2</a></li><li class="chapter-item expanded "><a href="concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="user.html" class="active"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html" class="active"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html" class="active"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html" class="active"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html" class="active"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html" class="active"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html" class="active"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html" class="active"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/lookup-argument.html"><strong aria-hidden="true">3.1.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.4.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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<ol class="chapter"><li class="chapter-item expanded affix "><a href="../index.html">halo2</a></li><li class="chapter-item expanded "><a href="../concepts.html"><strong aria-hidden="true">1.</strong> Concepts</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../concepts/proofs.html"><strong aria-hidden="true">1.1.</strong> Proof systems</a></li><li class="chapter-item expanded "><a href="../concepts/arithmetization.html"><strong aria-hidden="true">1.2.</strong> UltraPLONK Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/cores.html"><strong aria-hidden="true">1.3.</strong> Cores</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.4.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.5.</strong> Gadgets</a></li></ol></li><li class="chapter-item expanded "><a href="../user.html"><strong aria-hidden="true">2.</strong> User Documentation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../user/simple-example.html"><strong aria-hidden="true">2.1.</strong> A simple example</a></li><li class="chapter-item expanded "><a href="../user/lookup-tables.html"><strong aria-hidden="true">2.2.</strong> Lookup tables</a></li><li class="chapter-item expanded "><a href="../user/gadgets.html"><strong aria-hidden="true">2.3.</strong> Gadgets</a></li><li class="chapter-item expanded "><a href="../user/tips-and-tricks.html" class="active"><strong aria-hidden="true">2.4.</strong> Tips and tricks</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">3.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">3.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">3.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">3.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">3.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">3.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">3.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">3.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">3.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">3.2.</strong> Implementation</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.3.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.3.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">3.3.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></li></ol></li><li class="chapter-item expanded "><a href="../background.html"><strong aria-hidden="true">4.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">4.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">4.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">4.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">4.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/upa.html"><strong aria-hidden="true">4.5.</strong> UltraPLONK arithmetisation</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">4.6.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">4.7.</strong> Recursion</a></li></ol></li></ol>
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