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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> PLONKish Arithmetization</a></li><li class="chapter-item expanded "><a href="../concepts/chips.html"><strong aria-hidden="true">1.3.</strong> Chips</a></li><li class="chapter-item expanded "><a href="../concepts/gadgets.html"><strong aria-hidden="true">1.4.</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 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aria-hidden="true">3.1.</strong> Feature development</a></li></ol></li><li class="chapter-item expanded "><a href="../design.html"><strong aria-hidden="true">4.</strong> Design</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system.html"><strong aria-hidden="true">4.1.</strong> Proving system</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/proving-system/lookup.html"><strong aria-hidden="true">4.1.1.</strong> Lookup argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/permutation.html"><strong aria-hidden="true">4.1.2.</strong> Permutation argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/circuit-commitments.html"><strong aria-hidden="true">4.1.3.</strong> Circuit commitments</a></li><li class="chapter-item expanded "><a href="../design/proving-system/vanishing.html"><strong aria-hidden="true">4.1.4.</strong> Vanishing argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/multipoint-opening.html"><strong aria-hidden="true">4.1.5.</strong> Multipoint opening argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/inner-product.html"><strong aria-hidden="true">4.1.6.</strong> Inner product argument</a></li><li class="chapter-item expanded "><a href="../design/proving-system/comparison.html"><strong aria-hidden="true">4.1.7.</strong> Comparison to other work</a></li></ol></li><li class="chapter-item expanded "><a href="../design/protocol.html"><strong aria-hidden="true">4.2.</strong> Protocol Description</a></li><li class="chapter-item expanded "><a href="../design/implementation.html"><strong aria-hidden="true">4.3.</strong> Implementation</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/implementation/proofs.html"><strong aria-hidden="true">4.3.1.</strong> Proofs</a></li><li class="chapter-item expanded "><a href="../design/implementation/fields.html"><strong aria-hidden="true">4.3.2.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../design/implementation/selector-combining.html"><strong aria-hidden="true">4.3.3.</strong> Selector combining</a></li></ol></li><li class="chapter-item expanded "><a href="../design/gadgets.html"><strong aria-hidden="true">4.4.</strong> Gadgets</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/ecc.html"><strong aria-hidden="true">4.4.1.</strong> Elliptic curve cryptography</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../design/gadgets/ecc/witnessing-points.html"><strong aria-hidden="true">4.4.1.1.</strong> Witnessing points</a></li><li class="chapter-item expanded "><a href="../design/gadgets/ecc/addition.html"><strong aria-hidden="true">4.4.1.2.</strong> Incomplete and complete addition</a></li><li class="chapter-item expanded "><a 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"><a href="../design/gadgets/sha256/table16.html"><strong aria-hidden="true">4.4.4.1.</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">5.</strong> Background Material</a></li><li><ol class="section"><li class="chapter-item expanded "><a href="../background/fields.html"><strong aria-hidden="true">5.1.</strong> Fields</a></li><li class="chapter-item expanded "><a href="../background/polynomials.html"><strong aria-hidden="true">5.2.</strong> Polynomials</a></li><li class="chapter-item expanded "><a href="../background/groups.html"><strong aria-hidden="true">5.3.</strong> Cryptographic groups</a></li><li class="chapter-item expanded "><a href="../background/curves.html"><strong aria-hidden="true">5.4.</strong> Elliptic curves</a></li><li class="chapter-item expanded "><a href="../background/pc-ipa.html"><strong aria-hidden="true">5.5.</strong> Polynomial commitment using inner product argument</a></li><li class="chapter-item expanded "><a href="../background/recursion.html"><strong aria-hidden="true">5.6.</strong> Recursion</a></li></ol></li></ol>
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<h1 class="menu-title">The halo2 Book</h1>
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<link rel="stylesheet" href="https://cdn.jsdelivr.net/npm/katex@0.12.0/dist/katex.min.css" integrity="sha384-AfEj0r4/OFrOo5t7NnNe46zW/tFgW6x/bCJG8FqQCEo3+Aro6EYUG4+cU+KJWu/X" crossorigin="anonymous">
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<h1 id="proof-systems"><a class="header" href="#proof-systems">Proof systems</a></h1>
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<p>The aim of any <em><strong>proof system</strong></em> is to be able to prove interesting mathematical or
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cryptographic <em><strong>statements</strong></em>.</p>
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<p>Typically, in a given protocol we will want to prove families of statements that differ
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in their <em><strong>public inputs</strong></em>. The prover will also need to show that they know some
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<em><strong>private inputs</strong></em> that make the statement hold.</p>
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<p>To do this we write down a <em><strong>relation</strong></em>, <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal">R</span></span></span></span>, that specifies which
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combinations of public and private inputs are valid.</p>
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<blockquote>
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<p>The terminology above is intended to be aligned with the
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<a href="https://docs.zkproof.org/reference#latest-version">ZKProof Community Reference</a>.</p>
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</blockquote>
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<p>To be precise, we should distinguish between the relation <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathcal">R</span></span></span></span>, and its
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implementation to be used in a proof system. We call the latter a <em><strong>circuit</strong></em>.</p>
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<p>The language that we use to express circuits for a particular proof system is called an
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<em><strong>arithmetization</strong></em>. Usually, an arithmetization will define circuits in terms of
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polynomial constraints on variables over a field.</p>
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<blockquote>
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<p>The <em>process</em> of expressing a particular relation as a circuit is also sometimes called
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"arithmetization", but we'll avoid that usage.</p>
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</blockquote>
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<p>To create a proof of a statement, the prover will need to know the private inputs,
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and also intermediate values, called <em><strong>advice</strong></em> values, that are used by the circuit.</p>
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<p>We assume that we can compute advice values efficiently from the private and public inputs.
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The particular advice values will depend on how we write the circuit, not only on the
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high-level statement.</p>
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<p>The private inputs and advice values are collectively called a <em><strong>witness</strong></em>.</p>
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<blockquote>
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<p>Some authors use "witness" as just a synonym for private inputs. But in our usage,
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a witness includes advice, i.e. it includes all values that the prover supplies to
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the circuit.</p>
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</blockquote>
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<p>For example, suppose that we want to prove knowledge of a preimage <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> of a
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hash function <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span></span></span></span> for a digest <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>:</p>
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<ul>
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<li>
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<p>The private input would be the preimage <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span>.</p>
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</li>
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<li>
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<p>The public input would be the digest <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>.</p>
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</li>
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<li>
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<p>The relation would be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.1667em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">}</span></span></span></span>.</p>
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</li>
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<li>
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<p>For a particular public input <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.6833em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span></span></span></span>, the statement would be: <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">{(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">:</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.08125em;">H</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.22222em;">Y</span><span class="mclose">}</span></span></span></span>.</p>
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</li>
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<li>
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<p>The advice would be all of the intermediate values in the circuit implementing the
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hash function. The witness would be <span class="katex"><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.4306em;"></span><span class="mord mathnormal">x</span></span></span></span> and the advice.</p>
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</li>
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</ul>
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<p>A <em><strong>Non-interactive Argument</strong></em> allows a <em><strong>prover</strong></em> to create a <em><strong>proof</strong></em> for a
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given statement and witness. The proof is data that can be used to convince a <em><strong>verifier</strong></em>
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that <em>there exists</em> a witness for which the statement holds. The security property that
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such proofs cannot falsely convince a verifier is called <em><strong>soundness</strong></em>.</p>
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<p>A <em><strong>Non-interactive Argument of Knowledge</strong></em> (<em><strong>NARK</strong></em>) further convinces the verifier
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that the prover <em>knew</em> a witness for which the statement holds. This security property is
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called <em><strong>knowledge soundness</strong></em>, and it implies soundness.</p>
|
|
<p>In practice knowledge soundness is more useful for cryptographic protocols than soundness:
|
|
if we are interested in whether Alice holds a secret key in some protocol, say, we need
|
|
Alice to prove that <em>she knows</em> the key, not just that it exists.</p>
|
|
<p>Knowledge soundness is formalized by saying that an <em><strong>extractor</strong></em>, which can observe
|
|
precisely how the proof is generated, must be able to compute the witness.</p>
|
|
<blockquote>
|
|
<p>This property is subtle given that proofs can be <em><strong>malleable</strong></em>. That is, depending on the
|
|
proof system it may be possible to take an existing proof (or set of proofs) and, without
|
|
knowing the witness(es), modify it/them to produce a distinct proof of the same or a related
|
|
statement. Higher-level protocols that use malleable proof systems need to take this into
|
|
account.</p>
|
|
<p>Even without malleability, proofs can also potentially be <em><strong>replayed</strong></em>. For instance,
|
|
we would not want Alice in our example to be able to present a proof generated by someone
|
|
else, and have that be taken as a demonstration that she knew the key.</p>
|
|
</blockquote>
|
|
<p>If a proof yields no information about the witness (other than that a witness exists and was
|
|
known to the prover), then we say that the proof system is <em><strong>zero knowledge</strong></em>.</p>
|
|
<p>If a proof system produces short proofs —i.e. of length polylogarithmic in the circuit
|
|
size— then we say that it is <em><strong>succinct</strong></em>. A succinct NARK is called a <em><strong>SNARK</strong></em>
|
|
(<em><strong>Succinct Non-Interactive Argument of Knowledge</strong></em>).</p>
|
|
<blockquote>
|
|
<p>By this definition, a SNARK need not have verification time polylogarithmic in the circuit
|
|
size. Some papers use the term <em><strong>efficient</strong></em> to describe a SNARK with that property, but
|
|
we'll avoid that term since it's ambiguous for SNARKs that support amortized or recursive
|
|
verification, which we'll get to later.</p>
|
|
</blockquote>
|
|
<p>A <em><strong>zk-SNARK</strong></em> is a zero-knowledge SNARK.</p>
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