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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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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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<h1><a class="header" href="#background-material" id="background-material">Background Material</a></h1>
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<p>This section covers the background material required to understand the Halo 2 proving
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system. It is targeted at an ELI15 (Explain It Like I'm 15) level; if you think anything
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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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html" class="active"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html" class="active"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/gadgets.html"><strong aria-hidden="true">3.4.</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">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../design/gadgets.html"><strong aria-hidden="true">3.4.</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">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../../design/gadgets/sha256/table16.html" class="active"><strong aria-hidden="true">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../../../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../../../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../../../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</strong> SHA-256</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../../../design/gadgets/sha256/table16.html" class="active"><strong aria-hidden="true">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html" class="active"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html" class="active"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html" class="active"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html" class="active"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html" class="active"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html" class="active"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</strong> 16-bit table chip</a></li></ol></li></ol></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/multipoint-opening.html"><strong aria-hidden="true">3.1.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/permutation.html"><strong aria-hidden="true">3.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/lookup-argument.html"><strong aria-hidden="true">3.3.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">3.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/sha256.html"><strong aria-hidden="true">3.4.1.</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">3.4.1.1.</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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