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SNARGs for Bounded Depth Computations from Sub-Exponential LWE
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For a circuit of size $S$ and depth $D$, the prover runs in time poly$(S)$, and the verifier runs in time $(D + n) \cdot S^{o(1)}$, where $n$ is the input size. We obtain this result by slightly modifying the $\mathsf{GKR}$ protocol and proving that the Fiat-Shamir heuristic is sound when applied to this modified protocol. We build on the recent works of Canetti et al. (STOC 2019) and Peikert and Shiehian (Crypto 2020), which prove the soundness of the Fiat-Shamir heuristic when applied to a specific (non-succinct) zero-knowledge protocol. As a corollary, by the work of Choudhuri et al. (STOC 2019), this implies that the complexity class $\mathsf{PPAD}$ is hard (on average) under the sub-exponential $\mathsf{LWE}$ assumption, assuming that $\mathsf{\#SAT}$ with $o(\log n \cdot \log\log n)$ variables is hard (on average)." /> <meta property="article:section" content="PROTOCOLS" /> <meta property="article:modified_time" content="2020-07-12T12:51:11+00:00" /> <meta property="article:published_time" content="2020-07-12T12:51:11+00:00" /> <meta property="article:tag" content="delegation schemes" /> <meta property="article:tag" content="non-interactive" /> <meta property="article:tag" content="Fiat-Shamir" /> <meta property="article:tag" content="sum-check" /> <meta property="article:tag" content="GKR" /> <meta property="article:tag" content="PPAD" /> </head> <body> <noscript> <h1 class="text-center">What a lovely hat</h1> <h4 class="text-center">Is it made out of <a href="https://iacr.org/tinfoil.html">tin foil</a>?</h4> </noscript> <div class="fixed-top" id="topNavbar"> 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class="col-md-7 col-lg-8 pe-md-5"> <h4>Paper 2020/860</h4> <h3 class="mb-3">SNARGs for Bounded Depth Computations from Sub-Exponential LWE</h3> <p class="fst-italic mb-3"> Yael Tauman Kalai and Rachel Zhang </p> <h5 class="mt-3">Abstract</h5> <p style="white-space: pre-wrap;">We construct a succinct non-interactive publicly-verifiable delegation scheme for any log-space uniform circuit under the sub-exponential $\mathsf{LWE}$ assumption, a standard assumption that is believed to be post-quantum secure. For a circuit of size $S$ and depth $D$, the prover runs in time poly$(S)$, and the verifier runs in time $(D + n) \cdot S^{o(1)}$, where $n$ is the input size. We obtain this result by slightly modifying the $\mathsf{GKR}$ protocol and proving that the Fiat-Shamir heuristic is sound when applied to this modified protocol. We build on the recent works of Canetti et al. (STOC 2019) and Peikert and Shiehian (Crypto 2020), which prove the soundness of the Fiat-Shamir heuristic when applied to a specific (non-succinct) zero-knowledge protocol. As a corollary, by the work of Choudhuri et al. (STOC 2019), this implies that the complexity class $\mathsf{PPAD}$ is hard (on average) under the sub-exponential $\mathsf{LWE}$ assumption, assuming that $\mathsf{\#SAT}$ with $o(\log n \cdot \log\log n)$ variables is hard (on average).</p> </div> <div id="metadata" class="col-md-5 col-lg-4 ps-md-5 mt-4 mt-md-0"> <h5>Metadata</h5> <dl> <dt> Available format(s) </dt> <dd> <a class="btn btn-sm btn-outline-dark" href="/2020/860.pdf"> <img class="icon" src="/img/file-pdf.svg">PDF</a> </dd> <dt>Category</dt> <dd><a href="/search?category=PROTOCOLS"><small class="badge category category-PROTOCOLS">Cryptographic protocols</small></a></dd> <dt>Publication info</dt> <dd>Preprint. </dd> <dt>Keywords</dt> <dd class="keywords"><a href="/search?q=delegation%20schemes" class="me-2 badge bg-secondary keyword">delegation schemes</a><a href="/search?q=non-interactive" class="me-2 badge bg-secondary keyword">non-interactive</a><a href="/search?q=Fiat-Shamir" class="me-2 badge bg-secondary keyword">Fiat-Shamir</a><a href="/search?q=sum-check" class="me-2 badge bg-secondary keyword">sum-check</a><a href="/search?q=GKR" class="me-2 badge bg-secondary keyword">GKR</a><a href="/search?q=PPAD" class="me-2 badge bg-secondary keyword">PPAD</a></dd> <dt>Contact author(s)</dt> <dd><span class="font-monospace"> yael<span class="obfuscate"> @ </span>microsoft com </span> </dd> <dt>History</dt> <dd>2020-07-12: received</dd> <dt>Short URL</dt> <dd><a href="https://ia.cr/2020/860">https://ia.cr/2020/860</a></dd> <dt>License</dt> <dd><a rel="license" target="_blank" href="https://creativecommons.org/licenses/by/4.0/"> <img class="licenseImg" src="/img/license/CC_BY.svg" alt="Creative Commons Attribution" title="Creative Commons Attribution"><br> <small>CC BY</small> </a> </dd> </dl> </div> </div> <p class="mt-4"><strong>BibTeX</strong> <button id="bibcopy" class="ms-2 btn btn-sm btn-outline-dark" aria-label="Copy to clipboard" onclick="copyBibtex()"> <img src="/img/copy-outline.svg" class="icon">Copy to clipboard</button></p> <pre id="bibtex"> @misc{cryptoeprint:2020/860, author = {Yael Tauman Kalai and Rachel Zhang}, title = {{SNARGs} for Bounded Depth Computations from Sub-Exponential {LWE}}, howpublished = {Cryptology {ePrint} Archive, Paper 2020/860}, year = {2020}, url = {https://eprint.iacr.org/2020/860} } </pre> <script> var bibcopy; function triggerTooltip() { console.log('setting tooltip'); } window.onload = triggerTooltip; function copyBibtex() { let range = document.createRange(); range.selectNode(document.getElementById('bibtex')); window.getSelection().removeAllRanges(); window.getSelection().addRange(range); document.execCommand('copy'); window.getSelection().removeAllRanges(); let bibcopy = document.getElementById('bibcopy'); let copyTooltip = new bootstrap.Tooltip(bibcopy, {trigger: 'manual', title: 'Copied!'}); copyTooltip.show(); setTimeout(function() { copyTooltip.dispose(); }, 2000); } </script> </main> <div class="container-fluid mt-auto" id="eprintFooter"> <a href="https://iacr.org/"> <img id="iacrlogo" src="/img/iacrlogo_small.png" class="img-fluid d-block mx-auto" alt="IACR Logo"> </a> <div class="colorDiv"></div> <div class="alert alert-success w-75 mx-auto"> Note: In order to protect the privacy of readers, eprint.iacr.org does not use cookies or embedded third party content. </div> </div> <script src="/css/bootstrap/js/bootstrap.bundle.min.js"></script> <script> var topNavbar = document.getElementById('topNavbar'); if (topNavbar) { document.addEventListener('scroll', function(e) { if (window.scrollY > 100) { topNavbar.classList.add('scrolled'); } else { topNavbar.classList.remove('scrolled'); } }) } </script> </body> </html>