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<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>
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<h1 id="edwards25519-elliptic-curve">Edwards25519 Elliptic Curve<a class="headerlink" href="#edwards25519-elliptic-curve" title="Permanent link">&para;</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">&para;</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">&para;</a></h3>
<div class="highlight"><pre><span></span><code>−x^2 + y^2 = 1 − (121665/121666) * x^2 * y^2
</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">&para;</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
# The hex representation of the base point
5866666666666666666666666666666666666666666666666666666666666666
</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">&para;</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
# =&gt; 7237005577332262213973186563042994240857116359379907606001950938285454250989
</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">&para;</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
</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">&para;</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">&para;</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>
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