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McEliece in the world of Escher
<!DOCTYPE html> <html lang="en"> <head> <meta charset="utf-8"> <meta name="viewport" content="width=device-width, initial-scale=1, shrink-to-fit=no"> <link href="/css/dist/css/bootstrap.min.css" rel="stylesheet"> <title>McEliece in the world of Escher</title> <link rel="stylesheet" href="/css/eprint.css?v=10"> <link rel="shortcut icon" href="/favicon.ico" type="image/x-icon" /> <link rel="apple-touch-icon" href="/img/apple-touch-icon-180x180.png" /> <style> a.toggle-open:after { content:' -'; font-weight: 800; } a.toggle-closed:after { content: " ›"; font-weight: 800; } .paper-abstract { white-space: pre-wrap; } #metadata dt { margin-top: 1rem; } #metadata dt + dd { /* gap between dt and first dd */ margin-top: .75rem; } #metadata dd { margin-left: 2rem; } #metadata dd.keywords { padding-bottom: .5rem; } span.authorName { margin-top: .5rem; font-style: italic; } </style> <script> MathJax = { tex: { inlineMath: [['$', '$'], ['\\(', '\\)']], displayMath: [ ['$$','$$'], ["\\[","\\]"] ], processEnvironments: false }, loader: { load: [ "ui/safe", "ui/lazy", ], }, options: { safeOptions: { allow: { URLs: "none", classes: "safe", cssIDs: "safe", styles: "safe", }, }, } }; </script> <script id="MathJax-script" async src="/js/mathjax/tex-chtml.js"></script> <meta name="citation_title" content="McEliece in the world of Escher"> <meta name="citation_author" content="Danilo Gligoroski"> <meta name="citation_author" content="Simona Samardjiska"> <meta name="citation_author" content="Håkon Jacobsen"> <meta name="citation_author" content="Sergey Bezzateev"> <meta name="citation_journal_title" content="Cryptology ePrint Archive"> <meta name="citation_publication_date" content="2014"> <meta name="citation_pdf_url" content="https://eprint.iacr.org/2014/360.pdf"> <meta property="og:image" content="https://eprint.iacr.org/img/iacrlogo.png"/> <meta property="og:image:alt" content="IACR logo"/> <meta property="og:url" content="https://eprint.iacr.org/2014/360"> <meta property="og:site_name" content="IACR Cryptology ePrint Archive" /> <meta property="og:type" content="article" /> <meta property="og:title" content="McEliece in the world of Escher" /> <meta property="og:description" content="We present a new family of linear binary codes of length n and dimension k accompanied with a fast list decoding algorithm that can correct up to n/2 errors in a bounded channel with an error density $\rho$. The decisional problem of decoding random codes using these generalized error sets is NP-complete. Next we use the properties of these codes to design both an encryption scheme and a signature scheme. Although in the open literature there have been several proposals how to produce digital signatures from the McEliece public key scheme, as far as we know, this is the first public key scheme based on codes where signatures are produced in a straightforward manner from the decryption procedure of the scheme. The security analysis of our scheme have four parts: 1. An extensive list of attacks using the Information Set Decoding techniques adopted for our codes; 2. An analysis of the cost of a distinguishing attack based on rank attacks on the generator matrix of the code or on its dual code; 3. An analysis of the cost of cheap distinguishing attacks on the generator matrix of the code or on its dual code that have expensive list-decoding properties; 4. We interpret our scheme as multivariate quadratic system and discuss difficulties of solving that system using algebraic approaches such as Gröbner bases. Based on this security analysis we suggest some concrete parameters for the security levels in the range of $2^{80} - 2^{128}$. An additional feature of the decryption process is that it admits massive and trivial parallelization that could potentially make our scheme in hardware as fast as the symmetric crypto primitives." /> <meta property="article:section" content="" /> <meta property="article:modified_time" content="2014-09-24T11:18:11+00:00" /> <meta property="article:published_time" content="2014-05-23T07:53:54+00:00" /> <meta property="article:tag" content="Public Key" /> <meta property="article:tag" content="Cryptography" /> <meta property="article:tag" content="McEliece PKC" /> <meta property="article:tag" content="Error Correcting Codes" /> <meta property="article:tag" content="List Decoding" /> </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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class="fst-italic mb-3"> Danilo Gligoroski, Simona Samardjiska, Håkon Jacobsen, and Sergey Bezzateev </p> <h5 class="mt-3">Abstract</h5> <p style="white-space: pre-wrap;">We present a new family of linear binary codes of length n and dimension k accompanied with a fast list decoding algorithm that can correct up to n/2 errors in a bounded channel with an error density $\rho$. The decisional problem of decoding random codes using these generalized error sets is NP-complete. Next we use the properties of these codes to design both an encryption scheme and a signature scheme. Although in the open literature there have been several proposals how to produce digital signatures from the McEliece public key scheme, as far as we know, this is the first public key scheme based on codes where signatures are produced in a straightforward manner from the decryption procedure of the scheme. The security analysis of our scheme have four parts: 1. An extensive list of attacks using the Information Set Decoding techniques adopted for our codes; 2. An analysis of the cost of a distinguishing attack based on rank attacks on the generator matrix of the code or on its dual code; 3. An analysis of the cost of cheap distinguishing attacks on the generator matrix of the code or on its dual code that have expensive list-decoding properties; 4. We interpret our scheme as multivariate quadratic system and discuss difficulties of solving that system using algebraic approaches such as Gröbner bases. Based on this security analysis we suggest some concrete parameters for the security levels in the range of $2^{80} - 2^{128}$. An additional feature of the decryption process is that it admits massive and trivial parallelization that could potentially make our scheme in hardware as fast as the symmetric crypto primitives.</p> <p><strong>Note:</strong> The same material as previously, but restructured in a form of monograph. The cheap distinguisher attack section is updated to cover signatures.</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="/2014/360.pdf"> <img class="icon" src="/img/file-pdf.svg">PDF</a> </dd> <dt>Publication info</dt> <dd>Preprint. MINOR revision.</dd> <dt>Keywords</dt> <dd class="keywords"><a href="/search?q=Public%20Key" class="me-2 badge bg-secondary keyword">Public Key</a><a href="/search?q=Cryptography" class="me-2 badge bg-secondary keyword">Cryptography</a><a href="/search?q=McEliece%20PKC" class="me-2 badge bg-secondary keyword">McEliece PKC</a><a href="/search?q=Error%20Correcting%20Codes" class="me-2 badge bg-secondary keyword">Error Correcting Codes</a><a href="/search?q=List%20Decoding" class="me-2 badge bg-secondary keyword">List Decoding</a></dd> <dt>Contact author(s)</dt> <dd><span class="font-monospace"> danilog<span class="obfuscate"> @ </span>item ntnu no<br>simonas<span class="obfuscate"> @ </span>item ntnu no<br>hakoja<span class="obfuscate"> @ </span>item ntnu no<br>bsv<span class="obfuscate"> @ </span>aanet ru </span> </dd> <dt>History</dt> <dd>2014-09-24: last of 4 revisions</dd> <dd>2014-05-23: received</dd> <dd><a rel="nofollow" href="/archive/versions/2014/360">See all versions</a></dd> <dt>Short URL</dt> <dd><a href="https://ia.cr/2014/360">https://ia.cr/2014/360</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:2014/360, author = {Danilo Gligoroski and Simona Samardjiska and Håkon Jacobsen and Sergey Bezzateev}, title = {{McEliece} in the world of Escher}, howpublished = {Cryptology {ePrint} Archive, Paper 2014/360}, year = {2014}, url = {https://eprint.iacr.org/2014/360} } </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>