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Optimized Frobenius and Cyclotomic Cubing for Enhanced Pairing Computation
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A small embedding degree is crucial because part of the arithmetic for pairing computation occurs in $\mathbb{F}_{{p}^k}$ and includes operations such as squaring, multiplication, and Frobenius operations. In this paper, we present a fast and efficient method for computing the Frobenius endomorphism and its complexity. Additionally, we introduce an improvement in the efficiency of cyclotomic cubing operations for several pairing-friendly elliptic curves, which are essential for the calculation of Tate pairing and its derivatives." /> <meta property="article:section" content="FOUNDATIONS" /> <meta property="article:modified_time" content="2025-03-12T10:38:47+00:00" /> <meta property="article:published_time" content="2025-03-12T10:38:47+00:00" /> <meta property="article:tag" content="Optimal Ate Pairing" /> <meta property="article:tag" content="Frobenius maps" /> <meta property="article:tag" content="Kronecker products" /> <meta property="article:tag" content="Finite fields Cyclotomic cubing" /> </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 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Enhanced Pairing Computation</h3> <div class="author"><span class="authorName">Leila Ben Abdelghani</span><span class="affiliation">, Faculty of Sciences of Monastir</span></div> <div class="author"><span class="authorName">Nadia El Mrabet</span><a class="ms-1" target="_blank" href="https://orcid.org/0000-0003-3840-584X"><img class="align-baseline orcidIcon" src="/img/orcid.svg"></a><span class="affiliation">, Ecole des Mines de Saint Etienne</span></div> <div class="author"><span class="authorName">Loubna Ghammam</span><a class="ms-1" target="_blank" href="https://orcid.org/0000-0003-3438-1860"><img class="align-baseline orcidIcon" src="/img/orcid.svg"></a><span class="affiliation">, ITK Engineering GmbH</span></div> <div class="author"><span class="authorName">Lina Mortajine</span><span class="affiliation">, ITK Engineering GmbH</span></div> <h5 class="mt-3">Abstract</h5> <p style="white-space: pre-wrap;">Efficient implementation of a pairing-based cryptosystem relies on high-performance arithmetic in finite fields $\mathbb{F}_{p}$ and their extensions $\mathbb{F}_{p^k}$, where $k$ is the embedding degree. A small embedding degree is crucial because part of the arithmetic for pairing computation occurs in $\mathbb{F}_{{p}^k}$ and includes operations such as squaring, multiplication, and Frobenius operations. In this paper, we present a fast and efficient method for computing the Frobenius endomorphism and its complexity. Additionally, we introduce an improvement in the efficiency of cyclotomic cubing operations for several pairing-friendly elliptic curves, which are essential for the calculation of Tate pairing and its derivatives.</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="/2025/468.pdf"> <img class="icon" src="/img/file-pdf.svg">PDF</a> </dd> <dt>Category</dt> <dd><a href="/search?category=FOUNDATIONS"><small class="badge category category-FOUNDATIONS">Foundations</small></a></dd> <dt>Publication info</dt> <dd>Preprint. </dd> <dt>Keywords</dt> <dd class="keywords"><a href="/search?q=Optimal%20Ate%20Pairing" class="me-2 badge bg-secondary keyword">Optimal Ate Pairing</a><a href="/search?q=Frobenius%20maps" class="me-2 badge bg-secondary keyword">Frobenius maps</a><a href="/search?q=Kronecker%20products" class="me-2 badge bg-secondary keyword">Kronecker products</a><a href="/search?q=Finite%20fields%20%20Cyclotomic%20cubing" class="me-2 badge bg-secondary keyword">Finite fields Cyclotomic cubing</a></dd> <dt>Contact author(s)</dt> <dd><span class="font-monospace"> leila benabdelghani<span class="obfuscate"> @ </span>gmail com<br>nadia el-mrabet<span class="obfuscate"> @ </span>emse fr<br>loubna ghammam<span class="obfuscate"> @ </span>itk-engineering de<br>lina mortajine<span class="obfuscate"> @ </span>itk-engineering de </span> </dd> <dt>History</dt> <dd>2025-03-13: approved</dd> <dd>2025-03-12: received</dd> <dd><a rel="nofollow" href="/archive/versions/2025/468">See all versions</a></dd> <dt>Short URL</dt> <dd><a href="https://ia.cr/2025/468">https://ia.cr/2025/468</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:2025/468, author = {Leila Ben Abdelghani and Nadia El Mrabet and Loubna Ghammam and Lina Mortajine}, title = {Optimized Frobenius and Cyclotomic Cubing for Enhanced Pairing Computation}, howpublished = {Cryptology {ePrint} Archive, Paper 2025/468}, year = {2025}, url = {https://eprint.iacr.org/2025/468} } </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>