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Algebraic Cryptanalysis of Small-Scale Variants of Stream Cipher E0
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By systematically reducing the size of the linear feedback shift registers (LFSRs) while preserving the cipher’s core structure, we investigate the relationship between the number of unknowns and the number of consecutive keystream bits required to recover the internal states of the LFSRs. Our work demonstrates an approximately linear relationship between the number of consecutive keystream bits and the size of small-scale E0 variants, as indicated by our experimental results. To this end, we utilize two approaches: the computation of Gröbner bases using Magma’s F4 algorithm and the application of CryptoMiniSat’s SAT solver. Our experimental results show that increasing the number of keystream bits significantly improves computational efficiency, with the F4 algorithm achieving a speedup of up to 733× when additional equations are supplied. Furthermore, we verify the non-existence of equations of degree four or lower for up to seven consecutive keystream bits, and the non-existence of equations of degree three or lower for up to eight consecutive keystream bits, extending prior results on the algebraic properties of E0." /> <meta property="article:section" content="ATTACKS" /> <meta property="article:modified_time" content="2025-03-12T09:52:12+00:00" /> <meta property="article:published_time" content="2025-03-12T09:52:12+00:00" /> <meta property="article:tag" content="E0" /> <meta property="article:tag" content="small-scale variants" /> <meta property="article:tag" content="stream cipher" /> <meta property="article:tag" content="algebraic cryptanalysis" /> <meta property="article:tag" content="Gröbner bases" /> <meta property="article:tag" content="SAT" /> </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 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<div class="row mt-4"> <div class="col-md-7 col-lg-8 pe-md-5"> <h4>Paper 2025/466</h4> <h3 class="mb-3">Algebraic Cryptanalysis of Small-Scale Variants of Stream Cipher E0</h3> <div class="author"><span class="authorName">Jan Dolejš</span><a class="ms-1" target="_blank" href="https://orcid.org/0009-0003-4396-3026"><img class="align-baseline orcidIcon" src="/img/orcid.svg"></a><span class="affiliation">, Czech Technical University in Prague</span></div> <div class="author"><span class="authorName">Martin Jureček</span><a class="ms-1" target="_blank" href="https://orcid.org/0000-0002-6546-8953"><img class="align-baseline orcidIcon" src="/img/orcid.svg"></a><span class="affiliation">, Czech Technical University in Prague</span></div> <h5 class="mt-3">Abstract</h5> <p style="white-space: pre-wrap;">This study explores the algebraic cryptanalysis of small-scale variants of the E0 stream cipher, a legacy cipher used in the Bluetooth protocol. By systematically reducing the size of the linear feedback shift registers (LFSRs) while preserving the cipher’s core structure, we investigate the relationship between the number of unknowns and the number of consecutive keystream bits required to recover the internal states of the LFSRs. Our work demonstrates an approximately linear relationship between the number of consecutive keystream bits and the size of small-scale E0 variants, as indicated by our experimental results. To this end, we utilize two approaches: the computation of Gröbner bases using Magma’s F4 algorithm and the application of CryptoMiniSat’s SAT solver. Our experimental results show that increasing the number of keystream bits significantly improves computational efficiency, with the F4 algorithm achieving a speedup of up to 733× when additional equations are supplied. Furthermore, we verify the non-existence of equations of degree four or lower for up to seven consecutive keystream bits, and the non-existence of equations of degree three or lower for up to eight consecutive keystream bits, extending prior results on the algebraic properties of E0.</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="/2025/466.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>Preprint. </dd> <dt>Keywords</dt> <dd class="keywords"><a href="/search?q=E0" class="me-2 badge bg-secondary keyword">E0</a><a href="/search?q=small-scale%20variants" class="me-2 badge bg-secondary keyword">small-scale variants</a><a href="/search?q=stream%20cipher" class="me-2 badge bg-secondary keyword">stream cipher</a><a href="/search?q=algebraic%20cryptanalysis" class="me-2 badge bg-secondary keyword">algebraic cryptanalysis</a><a href="/search?q=Gr%C3%B6bner%20bases" class="me-2 badge bg-secondary keyword">Gröbner bases</a><a href="/search?q=SAT" class="me-2 badge bg-secondary keyword">SAT</a></dd> <dt>Contact author(s)</dt> <dd><span class="font-monospace"> dolejj13<span class="obfuscate"> @ </span>fit cvut cz<br>martin jurecek<span class="obfuscate"> @ </span>fit cvut cz </span> </dd> <dt>History</dt> <dd>2025-03-13: approved</dd> <dd>2025-03-12: received</dd> <dd><a rel="nofollow" href="/archive/versions/2025/466">See all versions</a></dd> <dt>Short URL</dt> <dd><a href="https://ia.cr/2025/466">https://ia.cr/2025/466</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:2025/466, author = {Jan Dolejš and Martin Jureček}, title = {Algebraic Cryptanalysis of Small-Scale Variants of Stream Cipher E0}, howpublished = {Cryptology {ePrint} Archive, Paper 2025/466}, year = {2025}, url = {https://eprint.iacr.org/2025/466} } </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>