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ecdsacrack.html
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<html>
<head>
<title>Bitcoin elliptic curve calculations</title>
<meta charset=utf-8 />
<meta http-equiv=“Pragma” content=”no-cache”>
<meta http-equiv=“Expires” content=”-1″>
<meta http-equiv=“CACHE-CONTROL” content=”NO-CACHE”>
<meta name="keywords" content="bitcoin,ecdsa,example,calculations,math">
<meta name="author" content="Willem Hengeveld, itsme@xs4all.nl">
<meta name="description" content="Examples of several ECDSA calculations.">
<style>
h2 { background-color: lightgreen; }
</style>
<script src="bignum.js"></script>
<script src="gfp.js"></script>
<script src="ec.js"></script>
<script src="ecdsa.js"></script>
<script src="utils.js"></script>
<script src="bccurve.js"></script>
<script language=javascript>
'use strict';
var example = {
k: "bc614e",
x: "51c4dba2c28fc89b208550477a514c87f9d0db0354f03b7c61f08c0a0e3118a2",
px: "bb6c1de01f36618ae05f7c183c22dfa8797e779f39537752c27e2dc045b0e694",
py: "2f8af53270bf045f2258834b6dad7481ad6fca009d80f5b54697b08d104fc7b3",
r: "cabc3692f1f7ba75a8572dc5d270b35bcc00650534f6e5ecd6338e55355454d5",
s1: "f65bfc44435a91814c142a3b8ee288a9183e6a3f012b84545d1fe334ccfac25e",
m1: "9b076ad2fe6b2ce63acf4edf7fc82d5152d3c8bffb36b944da7a1cce038f544a",
s2: "9cae782a191f3e742d9d4ff8f726d097a3a256af9fbc1faf16e7ec4d9fcf6feb",
m2: "85e43d48a83d8713a0fe253bf6b1fc70b8ee780e54749dc500f2880b056c4383",
};
var B = secp256k1();
function setvalue(base, id, value)
{
var av = base.querySelector("#"+id);
av.value = value;
av.innerHTML = value;
}
function getnumber(base, id)
{
var av = base.querySelector("#"+id);
if (!av)
throw "invalid selector";
var val = av.value;
if (!val)
return B.scalar(BigInt(0));
if (!val.startsWith("0x"))
val = "0x" + val;
return B.scalar(BigInt(val));
}
function load_crack2()
{
var tab = document.getElementById("crack2");
setvalue(tab, "r", example.r);
setvalue(tab, "m1", example.m1);
setvalue(tab, "s1", example.s1);
setvalue(tab, "m2", example.m2);
setvalue(tab, "s2", example.s2);
}
function do_crack2()
{
var tab = document.getElementById("crack2");
var r = getnumber(tab, "r");
var m1 = getnumber(tab, "m1");
var m2 = getnumber(tab, "m2");
var s1 = getnumber(tab, "s1");
var s2 = getnumber(tab, "s2");
var [k, x] = B.crack2(r, m1, m2, s1, s2);
setvalue(tab, "k", k.uint().toString(16));
setvalue(tab, "x", x.uint().toString(16));
}
function load_crack1()
{
var tab = document.getElementById("crack1");
setvalue(tab, "k", example.k);
setvalue(tab, "s", example.s1);
setvalue(tab, "m", example.m1);
}
function do_crack1()
{
var tab = document.getElementById("crack1");
var k = getnumber(tab, "k");
var m = getnumber(tab, "m");
var s = getnumber(tab, "s");
var x = B.crack1(k, m, s);
setvalue(tab, "x", x.uint().toString(16));
}
function load_calcpub()
{
var tab = document.getElementById("calcpub");
setvalue(tab, "x", example.x);
}
function do_calcpub()
{
var tab = document.getElementById("calcpub");
var x = getnumber(tab, "x");
var Y = B.calcpub(x);
setvalue(tab, "px", Y.x.uint().toString(16));
setvalue(tab, "py", Y.y.uint().toString(16));
}
function load_calcsig()
{
var tab = document.getElementById("calcsig");
setvalue(tab, "k", example.k);
setvalue(tab, "x", example.x);
setvalue(tab, "m", example.m1);
}
function do_calcsig()
{
var tab = document.getElementById("calcsig");
var x = getnumber(tab, "x");
var k = getnumber(tab, "k");
var m = getnumber(tab, "m");
var [r, s] = B.sign(m, x, k);
setvalue(tab, "r", r.uint().toString(16));
setvalue(tab, "s", s.uint().toString(16));
}
function load_verifysig()
{
var tab = document.getElementById("verifysig");
setvalue(tab, "m", example.m1);
setvalue(tab, "r", example.r);
setvalue(tab, "s", example.s1);
setvalue(tab, "px", example.px);
setvalue(tab, "py", example.py);
}
function do_verifysig()
{
var tab = document.getElementById("verifysig");
var r = getnumber(tab, "r");
var s = getnumber(tab, "s");
var m = getnumber(tab, "m");
var px = getnumber(tab, "px");
var py = getnumber(tab, "py");
var Y = B.ec.point(px, py);
var ok = B.verify(m, Y, r, s);
setvalue(tab, "result", ok ? "ok" : "invalid");
}
function load_findpk()
{
var tab = document.getElementById("findpk");
setvalue(tab, "m", example.m1);
setvalue(tab, "r", example.r);
setvalue(tab, "s", example.s1);
}
function do_findpk()
{
var tab = document.getElementById("findpk");
var m = getnumber(tab, "m");
var r = getnumber(tab, "r");
var s = getnumber(tab, "s");
var Y = B.findpk(m, r, s, 0);
setvalue(tab, "px0", Y.x.uint().toString(16));
setvalue(tab, "py0", Y.y.uint().toString(16));
var Y = B.findpk(m, r, s, 1);
setvalue(tab, "px1", Y.x.uint().toString(16));
setvalue(tab, "py1", Y.y.uint().toString(16));
}
function load_findk()
{
var tab = document.getElementById("findk");
setvalue(tab, "x", example.x);
setvalue(tab, "m", example.m1);
setvalue(tab, "r", example.r);
setvalue(tab, "s", example.s1);
}
function do_findk()
{
var tab = document.getElementById("findk");
var x = getnumber(tab, "x");
var m = getnumber(tab, "m");
var r = getnumber(tab, "r");
var s = getnumber(tab, "s");
var k = B.findk(m, x, r, s);
setvalue(tab, "k", k.uint().toString(16));
}
function load_add()
{
var tab = document.getElementById("addpt");
setvalue(tab, "x1", B.G.x.uint().toString(16));
setvalue(tab, "y1", B.G.y.uint().toString(16));
var BB = B.G.mul(B.scalar(BigInt("0x"+example.x)));
setvalue(tab, "x2", BB.x.uint().toString(16));
setvalue(tab, "y2", BB.y.uint().toString(16));
}
function do_add()
{
var tab = document.getElementById("addpt");
var x1 = getnumber(tab, "x1");
var y1 = getnumber(tab, "y1");
var x2 = getnumber(tab, "x2");
var y2 = getnumber(tab, "y2");
var C = B.ec.point(x1,y1).add(B.ec.point(x2,y2));
setvalue(tab, "x3", C.x.uint().toString(16));
setvalue(tab, "y3", C.y.uint().toString(16));
}
function load_mul()
{
var tab = document.getElementById("mulpt");
setvalue(tab, "x1", B.G.x.uint().toString(16));
setvalue(tab, "y1", B.G.y.uint().toString(16));
setvalue(tab, "a", example.x);
}
function do_mul()
{
var tab = document.getElementById("mulpt");
var x1 = getnumber(tab, "x1");
var y1 = getnumber(tab, "y1");
var a = getnumber(tab, "a");
var C = B.ec.point(x1,y1).mul(B.scalar(a));
setvalue(tab, "x2", C.x.uint().toString(16));
setvalue(tab, "y2", C.y.uint().toString(16));
}
function load_div()
{
var tab = document.getElementById("divpt");
setvalue(tab, "x1", B.G.x.uint().toString(16));
setvalue(tab, "y1", B.G.y.uint().toString(16));
setvalue(tab, "a", example.x);
}
function do_div()
{
var tab = document.getElementById("divpt");
var x1 = getnumber(tab, "x1");
var y1 = getnumber(tab, "y1");
var a = getnumber(tab, "a");
var C = B.ec.point(x1,y1).div(B.scalar(a));
setvalue(tab, "x2", C.x.uint().toString(16));
setvalue(tab, "y2", C.y.uint().toString(16));
}
function load_decompx()
{
var tab = document.getElementById("decompx");
setvalue(tab, "x", B.G.x.uint().toString(16));
}
function do_decompx()
{
var tab = document.getElementById("decompx");
var x = getnumber(tab, "x");
var X0 = B.ec.decompress(B.ec.coord(x), 0);
var X1 = B.ec.decompress(B.ec.coord(x), 1);
setvalue(tab, "y0", X0.y.uint().toString(16));
setvalue(tab, "y1", X1.y.uint().toString(16));
}
function load_decompy()
{
var tab = document.getElementById("decompy");
setvalue(tab, "y", B.G.y.uint().toString(16));
}
function do_decompy()
{
var tab = document.getElementById("decompy");
var y = getnumber(tab, "y");
y = B.ec.coord(y);
var X0 = B.ec.ydecompress(B.ec.coord(y), 0);
var X1 = B.ec.ydecompress(B.ec.coord(y), 1);
var X2 = B.ec.ydecompress(B.ec.coord(y), 2);
setvalue(tab, "x0", X0.x.uint().toString(16));
setvalue(tab, "x1", X1.x.uint().toString(16));
setvalue(tab, "x2", X2.x.uint().toString(16));
}
function load_validatexy()
{
var tab = document.getElementById("validatexy");
setvalue(tab, "px", example.px);
setvalue(tab, "py", example.py);
}
function do_validatexy()
{
var tab = document.getElementById("validatexy");
var px = getnumber(tab, "px");
var py = getnumber(tab, "py");
var Y = B.ec.point(px, py);
var ok = Y.isoncurve();
setvalue(tab, "result", ok ? "ok" : "invalid");
}
function start()
{
}
</script>
</head>
<body onLoad="start()">
Menu:
<a href="ecdsacrack.html">crack demo</a>
<a href="linearequations.html">using linear algebra</a>
<a href="calculator.html">curve calculator</a>
<a href="curve.html">curve demo</a>
<a href="transaction.html">bitcoin transaction</a>
<a href="unittest.html">unittest</a><br>
Author: Willem Hengeveld, <a href="mailto:itsme@xs4all.nl">itsme@xs4all.nl</a>,
Source: <a href="https://github.com/nlitsme/bitcoinexplainer">on github</a>.
<p>
Several example calculations with the bitcoin parameters. In these calculations the following parameters are used:
<ul>
<li>p - the curve base prime field, this is used for calculations involving coordinates.</li>
<li>G - the fixed generator point</li>
<li>n - the curve group order, this is the total number of points.</li>
<li>x - the secret key, a value between 0 and the group order</li>
<li>k - the signing secret, a value between 0 and the group order</li>
<li>px, py - the coordinates of the public key point, values between 0 and the coordinate order</li>
<li>r - the first part of the signature, a value between 0 and the coordinate order</li>
<li>s - the first part of the signature, a value between 0 and the group order</li>
<li>m - the message, a value between 0 and the group order</li>
<li>Y - the public key</li>
</ul>
<em>Notes</em>
<ul>
<li>All numbers are hexadecimal</li>
<li>Note that in bitcoin calculations the message is the hash of the prepared transaction.</li>
<li>Also note that in bitcoin the s value is required to be between 0 and half the group order, you have to take the negative when it is larger.</li>
</ul>
<p>
<h2>Example, sign a message with a secret key</h2>
A ecdsa signature is calculated like this:
<pre>r = xcoord(G*k), s = (m+x*r)/k</pre>
<table id=calcsig>
<tr>
<td>x: <input size=64 id='x'></td>
<td>k: <input size=64 id='k'></td>
</tr>
<tr>
<td>m: <input size=64 id='m'></td>
<td><button onclick='do_calcsig()'>Calc sig</button>
<button onclick='load_calcsig()'>Load example</button>
</td>
</tr>
<tr>
<td>r: <span id='r'></span></td>
<td>s: <span id='s'></span></td>
</tr>
</table>
<p>
<h2>Example, verify a message signature.</h2>
A ECDSA signature is verified using this calculation:
<pre>G*m+Y*r==R*s</pre>
<table id=verifysig>
<tr>
<td>px: <input size=64 id='px'></td>
<td>py: <input size=64 id='py'></td>
</tr>
<tr>
<td>r: <input size=64 id='r'></td>
<td>s: <input size=64 id='s'></td>
</tr>
<tr>
<td>m: <input size=64 id='m'></td>
<td><button onclick='do_verifysig()'>Verify sig</button>
<button onclick='load_verifysig()'>Load example</button>
</td>
</tr>
<tr>
<td>result: <span id='result'></span></td>
</tr>
</table>
<p>
<h2>Example, cracking a key using secret-reuse</h2>
When a signing secret was used to sign two different messages, you can recover the signing secret with this
calculation: <pre> k = (m1-m2)/(s1-s2) </pre>
And then calculate x in the same way as in the next example.
<p>
<table id=crack2>
<tr>
<td>r: <input size=64 id='r'></td>
<td><button onclick='do_crack2()'>Do Crack</button>
<button onclick='load_crack2()'>Load example</button>
</td>
</tr>
<tr>
<td>m1: <input size=64 id='m1'></td>
<td>s1: <input size=64 id='s1'></td>
</tr>
<tr>
<td>m2: <input size=64 id='m2'></td>
<td>s2: <input size=64 id='s2'></td>
</tr>
<tr>
<td>k: <span id='k'></span></td>
<td>x: <span id='x'></span></td>
</tr>
</table>
<p>
<h2>Example, cracking a key using known secret</h2>
When you have cracked or otherwise guessed a signing secret for a signature, the private key
is calculated like this:
<pre>x = (s*k-m)/r</pre>
<table id=crack1>
<tr>
<td>k: <input size=64 id='k'></td>
<td><button onclick='do_crack1()'>Do Crack</button>
<button onclick='load_crack1()'>Load example</button>
</td>
</tr>
<tr>
<td>s: <input size=64 id='s'></td>
<td>m: <input size=64 id='m'></td>
</tr>
<tr>
<td>x: <span id='x'></span></td>
</tr>
</table>
<p>
<h2>Example, calculate a public key</h2>
Given the private key, you can calculate the public key like this:
<pre>Y = G * x</pre>
<table id=calcpub>
<tr>
<td>x: <input size=64 id='x'></td>
<td><button onclick='do_calcpub()'>Calc pub</button>
<button onclick='load_calcpub()'>Load example</button></td>
</tr>
<tr>
<td>px: <span id='px'></span></td>
<td>py: <span id='py'></span></td>
</tr>
</table>
<p>
<h2>Example, find pubkey.</h2>
Given a signature and messagehash, you can calculate the public key:
<pre>Y = (R*s-G*m)/r</pre>
<table id=findpk>
<tr>
<td>r: <input size=64 id='r'></td>
<td>s: <input size=64 id='s'></td>
</tr>
<tr>
<td>m: <input size=64 id='m'></td>
<td><button onclick='do_findpk()'>Find Pubkey</button>
<button onclick='load_findpk()'>Load example</button>
</td>
</tr>
<tr>
<td>px: <span id='px0'></span></td>
<td>py: <span id='py0'></span></td>
</tr>
<tr>
<td>px: <span id='px1'></span></td>
<td>py: <span id='py1'></span></td>
</tr>
</table>
<p>
<h2>Example, find signing secret.</h2>
given a privatekey, message and signature, you can calculate the signing secret which was used:
<pre>k = (m+x*r)/s</pre>
<table id=findk>
<tr>
<td>r: <input size=64 id='r'></td>
<td>s: <input size=64 id='s'></td>
</tr>
<tr>
<td>x: <input size=64 id='x'></td>
</tr>
<tr>
<td>m: <input size=64 id='m'></td>
<td><button onclick='do_findk()'>Find secret</button>
<button onclick='load_findk()'>Load example</button>
</td>
</tr>
<tr>
<td>k: <span id='k'></span></td>
</tr>
</table>
<p>
<h2>Example, add points.</h2>
<table id=addpt>
<tr>
<td>x: <input size=64 id='x1'></td>
<td>y: <input size=64 id='y1'></td>
</tr>
<tr>
<td>x: <input size=64 id='x2'></td>
<td>y: <input size=64 id='y2'></td>
</tr>
<tr>
<td><button onclick='do_add()'>add</button>
<button onclick='load_add()'>Load example</button>
</td>
</tr>
<tr>
<td>x: <span id='x3' /></td>
<td>y: <span id='y3' /></td>
</tr>
</table>
<p>
<h2>Example, multiply point by a number.</h2>
<table id=mulpt>
<tr>
<td>x: <input size=64 id='x1'></td>
<td>y: <input size=64 id='y1'></td>
</tr>
<tr>
<td>a: <input size=64 id='a'></td>
<td><button onclick='do_mul()'>multiply</button>
<button onclick='load_mul()'>Load example</button>
</td>
</tr>
<tr>
<td>x: <span id='x2'/></td>
<td>y: <span id='y2'/></td>
</tr>
</table>
<p>
<h2>Example, divide point by a number.</h2>
<pre>multiply by the modular inverse in GFn</pre>
<table id=divpt>
<tr>
<td>x: <input size=64 id='x1'></td>
<td>y: <input size=64 id='y1'></td>
</tr>
<tr>
<td>a: <input size=64 id='a'></td>
<td><button onclick='do_div()'>divide</button>
<button onclick='load_div()'>Load example</button>
</td>
</tr>
<tr>
<td>x: <span id='x2'/></td>
<td>y: <span id='y2'/></td>
</tr>
</table>
<p>
<h2>Decompress a point</h2>
<pre> y = sqrt(x^3+a*x+b)</pre>
<table id=decompx>
<tr>
<td>x: <input size=64 id='x'></td>
<td><button onclick='do_decompx()'>decompress</button>
<button onclick='load_decompx()'>Load example</button>
</td>
</tr>
<tr>
<td>even</td>
<td>y: <span id='y0'/></td>
</tr>
<tr>
<td>odd</td>
<td>y: <span id='y1'/></td>
</tr>
</table>
<p>
<h2>You can decompress from `y` as well.</h2>
<pre> x = cuberoot(y^2-7)</pre>
This works, because for the secp256k1 curve, 'a' is zero.
<table id=decompy>
<tr>
<td><button onclick='do_decompy()'>decompress</button>
<button onclick='load_decompy()'>Load example</button>
</td>
<td>y: <input size=64 id='y'></td>
</tr>
<tr> <td>first</td> <td>x: <span id='x0'/></td></tr>
<tr> <td>second</td><td>x: <span id='x1'/></td></tr>
<tr> <td>third</td> <td>x: <span id='x2'/></td></tr>
</table>
<p>
<h2>Validate point.</h2>
checks that: <pre>y<sup>2</sup> == x<sup>3</sup>+a*x+b</pre>
<table id=validatexy>
<tr>
<td><button onclick='do_validatexy()'>validate</button>
<button onclick='load_validatexy()'>Load example</button>
</td>
<td>result: <span id='result'></span></td>
</tr>
<tr>
<td>x: <input size=64 id='px'></td>
<td>y: <input size=64 id='py'></td>
</tr>
</table>
</body>
</html>