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<!doctype html><html lang=en class=no-js> <head><meta charset=utf-8><meta name=viewport content="width=device-width,initial-scale=1"><meta name=description content="Unofficial Monero Documentation"><meta name=author content="Piotr 'Qertoip' Włodarek"><link href=https://monerodocs.org/cryptography/asymmetric/edwards25519/ rel=canonical><link rel="shortcut icon" href=../../../assets/favicon.png><meta name=generator content="mkdocs-1.1.2, mkdocs-material-6.2.5"><title>Edwards25519 Elliptic Curve - Monero Documentation</title><link rel=stylesheet href=../../../assets/stylesheets/main.15aa0b43.min.css><link rel=stylesheet href=../../../assets/stylesheets/palette.75751829.min.css><meta name=theme-color content=#ffffff><link rel=preconnect href=https://fonts.gstatic.com crossorigin><link rel=stylesheet 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type=checkbox id=__toc> <label class="md-nav__link md-nav__link--active" for=__toc> Edwards25519 curve <span class="md-nav__icon md-icon"></span> </label> <a href=./ class="md-nav__link md-nav__link--active"> Edwards25519 curve </a> <nav class="md-nav md-nav--secondary" aria-label="Table of contents"> <label class=md-nav__title for=__toc> <span class="md-nav__icon md-icon"></span> Table of contents </label> <ul class=md-nav__list data-md-scrollfix> <li class=md-nav__item> <a href=#definition class=md-nav__link> Definition </a> <nav class=md-nav aria-label=Definition> <ul class=md-nav__list> <li class=md-nav__item> <a href=#curve-equation class=md-nav__link> Curve equation </a> </li> <li class=md-nav__item> <a href=#base-point-g class=md-nav__link> Base point: G </a> </li> <li class=md-nav__item> <a href=#prime-order-of-the-base-point-l class=md-nav__link> Prime order of the base point: l </a> </li> <li class=md-nav__item> <a href=#total-number-of-points-on-the-curve class=md-nav__link> Total number of points on the curve </a> </li> </ul> </nav> </li> <li class=md-nav__item> <a href=#implementation class=md-nav__link> Implementation </a> </li> <li class=md-nav__item> <a href=#reference class=md-nav__link> Reference </a> </li> </ul> </nav> </li> <li class=md-nav__item> <a href=../key-image/ class=md-nav__link> Key image </a> </li> </ul> </nav> </li> <li class=md-nav__item> <a href=../../base58/ class=md-nav__link> Base58 </a> </li> <li class=md-nav__item> <a href=../../prng/ class=md-nav__link> PRNG </a> </li> <li class=md-nav__item> <a href=../../keccak-256/ class=md-nav__link> Keccak-256 </a> </li> </ul> </nav> </li> <li class="md-nav__item md-nav__item--nested"> <input class="md-nav__toggle md-toggle" data-md-toggle=nav-5 type=checkbox id=nav-5> <label class=md-nav__link for=nav-5> Address <span class="md-nav__icon md-icon"></span> </label> <nav class=md-nav aria-label=Address data-md-level=1> <label class=md-nav__title for=nav-5> <span class="md-nav__icon 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Mainnet, stagenet, testnet </a> </li> <li class=md-nav__item> <a href=../../../infrastructure/tor-onion-p2p-seed-nodes/ class=md-nav__link> Tor onion seed nodes </a> </li> <li class=md-nav__item> <a href=../../../infrastructure/monero-pulse/ class=md-nav__link> MoneroPulse </a> </li> </ul> </nav> </li> <li class="md-nav__item md-nav__item--nested"> <input class="md-nav__toggle md-toggle" data-md-toggle=nav-9 type=checkbox id=nav-9> <label class=md-nav__link for=nav-9> Running a Node <span class="md-nav__icon md-icon"></span> </label> <nav class=md-nav aria-label="Running a Node" data-md-level=1> <label class=md-nav__title for=nav-9> <span class="md-nav__icon md-icon"></span> Running a Node </label> <ul class=md-nav__list data-md-scrollfix> <li class=md-nav__item> <a href=../../../running-node/open-node-tor-onion/ class=md-nav__link> Open Node + Tor Onion </a> </li> </ul> </nav> </li> <li class="md-nav__item md-nav__item--nested"> <input class="md-nav__toggle md-toggle" data-md-toggle=nav-10 type=checkbox id=nav-10> <label class=md-nav__link for=nav-10> Accepting Monero <span class="md-nav__icon md-icon"></span> </label> <nav class=md-nav aria-label="Accepting Monero" data-md-level=1> <label class=md-nav__title for=nav-10> <span class="md-nav__icon md-icon"></span> Accepting Monero </label> <ul class=md-nav__list data-md-scrollfix> <li class=md-nav__item> <a href=../../../accepting-monero/overview/ class=md-nav__link> Overview </a> </li> </ul> </nav> </li> </ul> </nav> </div> </div> </div> <div class="md-sidebar md-sidebar--secondary" data-md-component=toc> <div class=md-sidebar__scrollwrap> <div class=md-sidebar__inner> <nav class="md-nav md-nav--secondary" aria-label="Table of contents"> <label class=md-nav__title for=__toc> <span class="md-nav__icon md-icon"></span> Table of contents </label> <ul class=md-nav__list data-md-scrollfix> <li class=md-nav__item> <a href=#definition class=md-nav__link> Definition </a> <nav class=md-nav aria-label=Definition> <ul class=md-nav__list> <li class=md-nav__item> <a href=#curve-equation class=md-nav__link> Curve equation </a> </li> <li class=md-nav__item> <a href=#base-point-g class=md-nav__link> Base point: G </a> </li> <li class=md-nav__item> <a href=#prime-order-of-the-base-point-l class=md-nav__link> Prime order of the base point: l </a> </li> <li class=md-nav__item> <a href=#total-number-of-points-on-the-curve class=md-nav__link> Total number of points on the curve </a> </li> </ul> </nav> </li> <li class=md-nav__item> <a href=#implementation class=md-nav__link> Implementation </a> </li> <li class=md-nav__item> <a href=#reference class=md-nav__link> Reference </a> </li> </ul> </nav> </div> </div> </div> <div class=md-content> <article class="md-content__inner md-typeset"> <a href=https://github.com/monerodocs/md/edit/master/docs/cryptography/asymmetric/edwards25519.md title="Edit this page" class="md-content__button md-icon"> <svg xmlns=http://www.w3.org/2000/svg viewbox="0 0 24 24"><path d="M20.71 7.04c.39-.39.39-1.04 0-1.41l-2.34-2.34c-.37-.39-1.02-.39-1.41 0l-1.84 1.83 3.75 3.75M3 17.25V21h3.75L17.81 9.93l-3.75-3.75L3 17.25z"/></svg> </a> <h1 id=edwards25519-elliptic-curve>Edwards25519 Elliptic Curve<a class=headerlink href=#edwards25519-elliptic-curve title="Permanent link">¶</a></h1> <div class="admonition note"> <p class=admonition-title>Note</p> <p>Author is nowhere close to being a cryptographer. Be sceptical on accuracy.</p> </div> <div class="admonition note"> <p class=admonition-title>Note</p> <p>This article is only about the underlying curve. Public key derivation and signing algorithm will be treated separately. </p> </div> <p>Monero employs edwards25519 elliptic curve as a basis for its key pair generation.</p> <p>The curve comes from the Ed25519 signature scheme. While Monero takes the curve unchanged, it does not exactly follow rest of the Ed25519.</p> <p>The edwards25519 curve is <a href=https://tools.ietf.org/html/rfc7748#section-4.1>birationally equivalent to Curve25519</a>.</p> <h2 id=definition>Definition<a class=headerlink href=#definition title="Permanent link">¶</a></h2> <p>This is the standard edwards25519 curve definition, no Monero specific stuff here, except the naming convention. The convention comes from the CryptoNote whitepaper and is widely used in Monero literature.</p> <h3 id=curve-equation>Curve equation<a class=headerlink href=#curve-equation title="Permanent link">¶</a></h3> <div class=highlight><pre><span></span><code>−x^2 + y^2 = 1 − (121665/121666) * x^2 * y^2
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</code></pre></div> <p>Note:</p> <ul> <li>curve is in two dimensions (nothing fancy, like all the curves is high school)</li> <li>curve is mirrored below y axis due to <code>y^2</code> part of the equation (not a polynomial)</li> </ul> <h3 id=base-point-g>Base point: <code>G</code><a class=headerlink href=#base-point-g title="Permanent link">¶</a></h3> <p>The base point is a specific point on the curve. It is used as a basis for further calculations. It is an arbitrary choice by the curve authors, just to standardize the scheme.</p> <p>Note that it is enough to specify the y value and the sign of the x value. That's because the specific x can be calculated from the curve equation.</p> <div class=highlight><pre><span></span><code>G = (x, 4/5) # take the point with the positive x
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# The hex representation of the base point
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5866666666666666666666666666666666666666666666666666666666666666
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</code></pre></div> <h3 id=prime-order-of-the-base-point-l>Prime order of the base point: <code>l</code><a class=headerlink href=#prime-order-of-the-base-point-l title="Permanent link">¶</a></h3> <p>In layment terms, the "canvas" where the curve is drawn is assumed to have a finite "resolution", so point coordinates must "wrap around" at some point. This is achieved by modulo the <code>l</code> value (lowercase L). In other words, the <code>l</code> defines the maximum scalar we can use.</p> <div class=highlight><pre><span></span><code>l = 2^252 + 27742317777372353535851937790883648493
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# => 7237005577332262213973186563042994240857116359379907606001950938285454250989
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</code></pre></div> <p>The <code>l</code> is a prime number specified by the curve authors.</p> <p>In practice this is the private key's strength.</p> <h3 id=total-number-of-points-on-the-curve>Total number of points on the curve<a class=headerlink href=#total-number-of-points-on-the-curve title="Permanent link">¶</a></h3> <p>The total number of points on the curve is also a prime number:</p> <div class=highlight><pre><span></span><code>q = 2^255 - 19
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</code></pre></div> <p>In practice not all points are "useful" and so the private key strength is limited to <code>l</code> describe above.</p> <h2 id=implementation>Implementation<a class=headerlink href=#implementation title="Permanent link">¶</a></h2> <p>Monero uses (apparently modified) Ref10 implementation by Daniel J. Bernstein.</p> <h2 id=reference>Reference<a class=headerlink href=#reference title="Permanent link">¶</a></h2> <ul> <li><a href=https://blog.cloudflare.com/a-relatively-easy-to-understand-primer-on-elliptic-curve-cryptography/ >A (Relatively Easy To Understand) Primer on Elliptic Curve Cryptography</a></li> <li><a href=https://tools.ietf.org/html/rfc8032>RFC 8032 defining EdDSA</a></li> <li><a href=https://steemit.com/monero/@luigi1111/understanding-monero-cryptography-privacy-introduction>Understanding Monero Cryptography</a> - excellent writeup by Luigi</li> <li><a href=https://monero.stackexchange.com/questions/2290/why-how-does-monero-generate-public-ed25519-keys-without-using-the-standard-publ>StackOverflow answer</a></li> <li><a href=https://github.com/monero-project/mininero/blob/master/ed25519.py>Python implementation</a> - not the reference one but easier to understand</li> <li><a href=https://monero.stackexchange.com/questions/6050/what-is-the-base-point-g-from-the-whitepaper-and-how-is-it-represented-as-a>Encoding point to hex</a></li> <li><a href=https://en.wikipedia.org/wiki/EdDSA>EdDSA on Wikipedia</a></li> </ul> </article> </div> </div> </main> <footer class=md-footer> <div class=md-footer-nav> <nav class="md-footer-nav__inner md-grid" aria-label=Footer> <a href=../public-key/ class="md-footer-nav__link md-footer-nav__link--prev" rel=prev> <div class="md-footer-nav__button md-icon"> <svg xmlns=http://www.w3.org/2000/svg viewbox="0 0 24 24"><path d="M20 11v2H8l5.5 5.5-1.42 1.42L4.16 12l7.92-7.92L13.5 5.5 8 11h12z"/></svg> </div> <div class=md-footer-nav__title> <div class=md-ellipsis> <span class=md-footer-nav__direction> Previous </span> Public keys </div> </div> </a> <a href=../key-image/ class="md-footer-nav__link md-footer-nav__link--next" rel=next> <div class=md-footer-nav__title> <div class=md-ellipsis> <span class=md-footer-nav__direction> Next </span> Key image </div> </div> <div class="md-footer-nav__button md-icon"> <svg xmlns=http://www.w3.org/2000/svg viewbox="0 0 24 24"><path d="M4 11v2h12l-5.5 5.5 1.42 1.42L19.84 12l-7.92-7.92L10.5 5.5 16 11H4z"/></svg> </div> </a> </nav> </div> <div class="md-footer-meta md-typeset"> <div class="md-footer-meta__inner md-grid"> <div class=md-footer-copyright style="margin: auto;"> <a href=http://bumz4sduzxzlobbgzy5fiufdflg3mo2jyecdwdx5rphrqldms7wlmiid.onion/ >Tor onion version</a> | <a href=https://qertoip.com/ >contact</a> | © 2021 MoneroDocs under <a href=https://opensource.org/licenses/MIT>MIT</a> | built w/ <a href=https://www.mkdocs.org/ >mkdocs</a> and <a href=https://squidfunk.github.io/mkdocs-material/ >squidfunk/material</a> </div> </div> </div> </footer> </div> <script src=../../../assets/javascripts/vendor.93c04032.min.js></script> <script src=../../../assets/javascripts/bundle.83e5331e.min.js></script><script id=__lang type=application/json>{"clipboard.copy": "Copy to clipboard", "clipboard.copied": "Copied to clipboard", "search.config.lang": "en", "search.config.pipeline": "trimmer, stopWordFilter", "search.config.separator": "[\\s\\-]+", "search.placeholder": "Search", "search.result.placeholder": "Type to start searching", "search.result.none": "No matching documents", "search.result.one": "1 matching document", "search.result.other": "# matching documents", "search.result.more.one": "1 more on this page", "search.result.more.other": "# more on this page", "search.result.term.missing": "Missing"}</script> <script>
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