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Analysis of Multivariate Encryption Schemes: Application to Dob and C*
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In this work we study the effectiveness of these modifications, by deriving estimates for the number of degree fall polynomials. After developing the necessary tools, we focus on encryption schemes using the $C^*$ and Dobbertin central maps, with the internal perturbation (ip), and $Q_+$ modifications. For these constructions we are able to accurately predict the number of degree fall polynomials produced in a Gröbner basis attack, up to and including degree five for the Dob encryption scheme and four for $C^*$. The predictions remain accurate even when fixing variables. Based on this new theory we design a novel attack on Dob, which completely recovers the secret key for the parameters suggested by its designers. Due to the generality of the presented techniques, we also believe that they are of interest to the analysis of other big field schemes." /> <meta property="article:section" content="ATTACKS" /> <meta property="article:modified_time" content="2024-04-16T12:00:32+00:00" /> <meta property="article:published_time" content="2024-04-16T12:00:32+00:00" /> </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"> <nav class="navbar navbar-custom navbar-expand-lg"> <div class="container px-0 justify-content-between justify-content-lg-evenly"> <div class="order-0 align-items-center d-flex"> <button class="navbar-toggler btnNoOutline" type="button" data-bs-toggle="collapse" data-bs-target="#navbarContent" aria-controls="navbarContent" aria-expanded="false"> <span class="icon-bar top-bar"></span> <span class="icon-bar 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class="authorName">Håvard Raddum</span><span class="affiliation">, Simula UiB</span></div> <h5 class="mt-3">Abstract</h5> <p style="white-space: pre-wrap;">A common strategy for constructing multivariate encryption schemes is to use a central map that is easy to invert over an extension field, along with a small number of modifications to thwart potential attacks. In this work we study the effectiveness of these modifications, by deriving estimates for the number of degree fall polynomials. After developing the necessary tools, we focus on encryption schemes using the $C^*$ and Dobbertin central maps, with the internal perturbation (ip), and $Q_+$ modifications. For these constructions we are able to accurately predict the number of degree fall polynomials produced in a Gröbner basis attack, up to and including degree five for the Dob encryption scheme and four for $C^*$. The predictions remain accurate even when fixing variables. Based on this new theory we design a novel attack on Dob, which completely recovers the secret key for the parameters suggested by its designers. Due to the generality of the presented techniques, we also believe that they are of interest to the analysis of other big field schemes.</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="/2024/595.pdf"> <img class="icon" src="/img/file-pdf.svg">PDF</a> </dd> <dt>Category</dt> <dd><a href="/search?category=ATTACKS"><small class="badge category category-ATTACKS">Attacks and cryptanalysis</small></a></dd> <dt>Publication info</dt> <dd>Published by the IACR in JOC 2024</dd> <dt>DOI</dt> <dd><a href="https://doi.org/10.1007/s00145-024-09501-w" target="_blank" class="ms-1">10.1007/s00145-024-09501-w <img src="/img/open-outline.svg" style="height:1.25rem;"></a></dd> <dt>Contact author(s)</dt> <dd><span class="font-monospace"> morten oygarden<span class="obfuscate"> @ </span>simula no<br>patrick felke<span class="obfuscate"> @ </span>hs-emden-leer de<br>haavardr<span class="obfuscate"> @ </span>simula no </span> </dd> <dt>History</dt> <dd>2024-04-18: approved</dd> <dd>2024-04-16: received</dd> <dd><a rel="nofollow" href="/archive/versions/2024/595">See all versions</a></dd> <dt>Short URL</dt> <dd><a href="https://ia.cr/2024/595">https://ia.cr/2024/595</a></dd> <dt>License</dt> <dd><a rel="license" target="_blank" href="https://creativecommons.org/publicdomain/zero/1.0/"> <img class="licenseImg" src="/img/license/CC0.svg" alt="No rights reserved" title="No rights reserved"><br> <small>CC0</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:2024/595, author = {Morten Øygarden and Patrick Felke and Håvard Raddum}, title = {Analysis of Multivariate Encryption Schemes: Application to Dob and C*}, howpublished = {Cryptology {ePrint} Archive, Paper 2024/595}, year = {2024}, doi = {10.1007/s00145-024-09501-w}, url = {https://eprint.iacr.org/2024/595} } </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>