js-ecutils
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JavaScript Library for Elliptic Curve Cryptography: key exchanges (Diffie-Hellman, Massey-Omura), ECDSA signatures, and Koblitz encoding. Suitable for crypto education and secure systems.
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JavaScript
;
function _typeof(o) { "@babel/helpers - typeof"; return _typeof = "function" == typeof Symbol && "symbol" == typeof Symbol.iterator ? function (o) { return typeof o; } : function (o) { return o && "function" == typeof Symbol && o.constructor === Symbol && o !== Symbol.prototype ? "symbol" : typeof o; }, _typeof(o); }
Object.defineProperty(exports, "__esModule", {
value: true
});
exports.JacobianPoint = void 0;
exports.jacAdd = jacAdd;
exports.jacDouble = jacDouble;
exports.jacMul = jacMul;
exports.toAffine = toAffine;
exports.toJacobian = toJacobian;
var _math = require("../../utils/math.js");
function _defineProperties(e, r) { for (var t = 0; t < r.length; t++) { var o = r[t]; o.enumerable = o.enumerable || !1, o.configurable = !0, "value" in o && (o.writable = !0), Object.defineProperty(e, _toPropertyKey(o.key), o); } }
function _createClass(e, r, t) { return r && _defineProperties(e.prototype, r), t && _defineProperties(e, t), Object.defineProperty(e, "prototype", { writable: !1 }), e; }
function _toPropertyKey(t) { var i = _toPrimitive(t, "string"); return "symbol" == _typeof(i) ? i : i + ""; }
function _toPrimitive(t, r) { if ("object" != _typeof(t) || !t) return t; var e = t[Symbol.toPrimitive]; if (void 0 !== e) { var i = e.call(t, r || "default"); if ("object" != _typeof(i)) return i; throw new TypeError("@@toPrimitive must return a primitive value."); } return ("string" === r ? String : Number)(t); }
function _classCallCheck(a, n) { if (!(a instanceof n)) throw new TypeError("Cannot call a class as a function"); } /**
* Elliptic curve arithmetic in Jacobian (projective) coordinates.
*
* In Jacobian coordinates a point is represented as (X, Y, Z) where the
* affine equivalents are x = X/Z² and y = Y/Z³. The identity (point at
* infinity) is represented by x = null, y = null.
*
* The key advantage over affine arithmetic is that point addition and
* doubling can be performed WITHOUT modular inversions (only
* multiplications and squarings), making scalar multiplication roughly
* 3x faster for large scalars.
*
* Trade-off: each point uses three coordinates instead of two, and a
* single inversion is needed at the end to convert back to affine form.
*
* See RFC 6090, Section 4 for a detailed treatment of projective
* coordinate systems and their security considerations.
*/
/**
* A point in Jacobian projective coordinates (X, Y, Z).
*
* The affine point (x, y) corresponds to (X/Z², Y/Z³).
* The identity element is represented by x = null, y = null.
*/
var JacobianPoint = exports.JacobianPoint = /*#__PURE__*/_createClass(function JacobianPoint() {
var x = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : null;
var y = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : null;
var z = arguments.length > 2 && arguments[2] !== undefined ? arguments[2] : 1n;
_classCallCheck(this, JacobianPoint);
this.x = x;
this.y = y;
this.z = z;
Object.freeze(this);
});
/**
* Convert an affine Point to Jacobian coordinates.
*
* The affine point (x, y) maps to (x, y, 1) in Jacobian form.
* The identity point maps to JacobianPoint(null, null, 1).
*
* @param {Point} point - An affine Point.
* @returns {JacobianPoint}
*/
function toJacobian(point) {
if (point.isIdentity) {
return new JacobianPoint();
}
return new JacobianPoint(point.x, point.y, 1n);
}
/**
* Convert a JacobianPoint back to affine (x, y) coordinates.
*
* x = X · Z⁻² (mod p)
* y = Y · Z⁻³ (mod p)
*
* @param {JacobianPoint} jp
* @param {CurveParams} curve
* @returns {[BigInt|null, BigInt|null]}
*/
function toAffine(jp, curve) {
if (jp.x === null || jp.y === null || jp.z === 0n) {
return [null, null];
}
var invZ = (0, _math.modInverse)(jp.z, curve.p);
var invZ2 = invZ * invZ % curve.p;
var invZ3 = invZ2 * invZ % curve.p;
return [(0, _math.modulus)(jp.x * invZ2, curve.p), (0, _math.modulus)(jp.y * invZ3, curve.p)];
}
/**
* Double a point in Jacobian coordinates.
*
* Uses the standard Jacobian doubling formulas (see RFC 6090 Section 4):
*
* S = 4 · X · Y²
* M = 3 · X² + a · Z⁴
* X' = M² - 2·S
* Y' = M · (S - X') - 8·Y⁴
* Z' = 2 · Y · Z
*
* Cost: 1S + 4M (no field inversions).
*
* @param {JacobianPoint} jp
* @param {CurveParams} curve
* @returns {JacobianPoint}
*/
function jacDouble(jp, curve) {
if (jp.x === null || jp.y === null || jp.y === 0n) {
return new JacobianPoint();
}
var p = curve.p;
var ysq = (0, _math.modulus)(jp.y * jp.y, p);
var zsqr = (0, _math.modulus)(jp.z * jp.z, p);
var s = (0, _math.modulus)(4n * jp.x * ysq, p);
var m = (0, _math.modulus)(3n * jp.x * jp.x + curve.a * zsqr * zsqr, p);
var nx = (0, _math.modulus)(m * m - 2n * s, p);
var ny = (0, _math.modulus)(m * (s - nx) - 8n * ysq * ysq, p);
var nz = (0, _math.modulus)(2n * jp.y * jp.z, p);
return new JacobianPoint(nx, ny, nz);
}
/**
* Add two points in Jacobian coordinates.
*
* Uses the standard Jacobian addition formulas (see RFC 6090 Section 4):
*
* U₁ = X₁ · Z₂², U₂ = X₂ · Z₁²
* S₁ = Y₁ · Z₂³, S₂ = Y₂ · Z₁³
* H = U₂ - U₁, R = 2 · (S₂ - S₁)
* X' = R² - H³ - 2 · U₁ · H²
* Y' = R · (U₁ · H² - X') - 2 · S₁ · H³
* Z' = ((Z₁ + Z₂)² - Z₁² - Z₂²) · H
*
* If U₁ = U₂ and S₁ ≠ S₂ the points are inverses → identity.
* If U₁ = U₂ and S₁ = S₂ the points are equal → delegates to jacDouble.
*
* @param {JacobianPoint} jp1
* @param {JacobianPoint} jp2
* @param {CurveParams} curve
* @returns {JacobianPoint}
*/
function jacAdd(jp1, jp2, curve) {
if (jp1.x === null || jp1.y === null) return jp2;
if (jp2.x === null || jp2.y === null) return jp1;
var p = curve.p;
var z1z1 = (0, _math.modulus)(jp1.z * jp1.z, p);
var z2z2 = (0, _math.modulus)(jp2.z * jp2.z, p);
var u1 = (0, _math.modulus)(jp1.x * z2z2, p);
var u2 = (0, _math.modulus)(jp2.x * z1z1, p);
var s1 = (0, _math.modulus)(jp1.y * jp2.z * z2z2, p);
var s2 = (0, _math.modulus)(jp2.y * jp1.z * z1z1, p);
if (u1 === u2) {
if (s1 !== s2) return new JacobianPoint();
return jacDouble(jp1, curve);
}
var h = u2 - u1;
var i = (0, _math.modulus)(2n * h * (2n * h), p);
var j = (0, _math.modulus)(h * i, p);
var r = (0, _math.modulus)(2n * (s2 - s1), p);
var v = (0, _math.modulus)(u1 * i, p);
var x = (0, _math.modulus)(r * r - j - 2n * v, p);
var y = (0, _math.modulus)(r * (v - x) - 2n * s1 * j, p);
var z = (0, _math.modulus)(((jp1.z + jp2.z) * (jp1.z + jp2.z) - z1z1 - z2z2) * h, p);
return new JacobianPoint(x, y, z);
}
/**
* Scalar multiplication in Jacobian coordinates (double-and-add).
*
* Computes k·P using the binary expansion of k. Runs in O(log k)
* doublings and at most O(log k) additions, all without field inversions
* until the final conversion back to affine.
*
* @param {BigInt} k - Scalar multiplier.
* @param {JacobianPoint} jp - Base point in Jacobian coordinates.
* @param {CurveParams} curve
* @returns {JacobianPoint}
*/
function jacMul(k, jp, curve) {
if (k === 0n || jp.x === null || jp.y === null) {
return new JacobianPoint();
}
var result = new JacobianPoint();
var bits = k.toString(2);
for (var idx = 0; idx < bits.length; idx++) {
if (bits[bits.length - idx - 1] === '1') {
result = jacAdd(result, jp, curve);
}
jp = jacDouble(jp, curve);
}
return result;
}
//# sourceMappingURL=data:application/json;charset=utf-8;base64,{"version":3,"names":["_math","require","_defineProperties","e","r","t","length","o","enumerable","configurable","writable","Object","defineProperty","_toPropertyKey","key","_createClass","prototype","i","_toPrimitive","_typeof","Symbol","toPrimitive","call","TypeError","String","Number","_classCallCheck","a","n","JacobianPoint","exports","x","arguments","undefined","y","z","freeze","toJacobian","point","isIdentity","toAffine","jp","curve","invZ","modInverse","p","invZ2","invZ3","modulus","jacDouble","ysq","zsqr","s","m","nx","ny","nz","jacAdd","jp1","jp2","z1z1","z2z2","u1","u2","s1","s2","h","j","v","jacMul","k","result","bits","toString","idx"],"sources":["../../../../src/core/arithmetic/jacobian.js"],"sourcesContent":["/**\n * Elliptic curve arithmetic in Jacobian (projective) coordinates.\n *\n * In Jacobian coordinates a point is represented as (X, Y, Z) where the\n * affine equivalents are x = X/Z² and y = Y/Z³.  The identity (point at\n * infinity) is represented by x = null, y = null.\n *\n * The key advantage over affine arithmetic is that point addition and\n * doubling can be performed WITHOUT modular inversions (only\n * multiplications and squarings), making scalar multiplication roughly\n * 3x faster for large scalars.\n *\n * Trade-off: each point uses three coordinates instead of two, and a\n * single inversion is needed at the end to convert back to affine form.\n *\n * See RFC 6090, Section 4 for a detailed treatment of projective\n * coordinate systems and their security considerations.\n */\n\nimport { modulus, modInverse } from '../../utils/math.js'\n\n/**\n * A point in Jacobian projective coordinates (X, Y, Z).\n *\n * The affine point (x, y) corresponds to (X/Z², Y/Z³).\n * The identity element is represented by x = null, y = null.\n */\nexport class JacobianPoint {\n  constructor(x = null, y = null, z = 1n) {\n    this.x = x\n    this.y = y\n    this.z = z\n    Object.freeze(this)\n  }\n}\n\n/**\n * Convert an affine Point to Jacobian coordinates.\n *\n * The affine point (x, y) maps to (x, y, 1) in Jacobian form.\n * The identity point maps to JacobianPoint(null, null, 1).\n *\n * @param {Point} point - An affine Point.\n * @returns {JacobianPoint}\n */\nexport function toJacobian(point) {\n  if (point.isIdentity) {\n    return new JacobianPoint()\n  }\n  return new JacobianPoint(point.x, point.y, 1n)\n}\n\n/**\n * Convert a JacobianPoint back to affine (x, y) coordinates.\n *\n *     x = X · Z⁻²  (mod p)\n *     y = Y · Z⁻³  (mod p)\n *\n * @param {JacobianPoint} jp\n * @param {CurveParams} curve\n * @returns {[BigInt|null, BigInt|null]}\n */\nexport function toAffine(jp, curve) {\n  if (jp.x === null || jp.y === null || jp.z === 0n) {\n    return [null, null]\n  }\n  const invZ = modInverse(jp.z, curve.p)\n  const invZ2 = (invZ * invZ) % curve.p\n  const invZ3 = (invZ2 * invZ) % curve.p\n  return [modulus(jp.x * invZ2, curve.p), modulus(jp.y * invZ3, curve.p)]\n}\n\n/**\n * Double a point in Jacobian coordinates.\n *\n * Uses the standard Jacobian doubling formulas (see RFC 6090 Section 4):\n *\n *     S  = 4 · X · Y²\n *     M  = 3 · X² + a · Z⁴\n *     X' = M² - 2·S\n *     Y' = M · (S - X') - 8·Y⁴\n *     Z' = 2 · Y · Z\n *\n * Cost: 1S + 4M (no field inversions).\n *\n * @param {JacobianPoint} jp\n * @param {CurveParams} curve\n * @returns {JacobianPoint}\n */\nexport function jacDouble(jp, curve) {\n  if (jp.x === null || jp.y === null || jp.y === 0n) {\n    return new JacobianPoint()\n  }\n  const p = curve.p\n  const ysq = modulus(jp.y * jp.y, p)\n  const zsqr = modulus(jp.z * jp.z, p)\n  const s = modulus(4n * jp.x * ysq, p)\n  const m = modulus(3n * jp.x * jp.x + curve.a * zsqr * zsqr, p)\n  const nx = modulus(m * m - 2n * s, p)\n  const ny = modulus(m * (s - nx) - 8n * ysq * ysq, p)\n  const nz = modulus(2n * jp.y * jp.z, p)\n  return new JacobianPoint(nx, ny, nz)\n}\n\n/**\n * Add two points in Jacobian coordinates.\n *\n * Uses the standard Jacobian addition formulas (see RFC 6090 Section 4):\n *\n *     U₁ = X₁ · Z₂²,   U₂ = X₂ · Z₁²\n *     S₁ = Y₁ · Z₂³,   S₂ = Y₂ · Z₁³\n *     H  = U₂ - U₁,     R  = 2 · (S₂ - S₁)\n *     X' = R² - H³ - 2 · U₁ · H²\n *     Y' = R · (U₁ · H² - X') - 2 · S₁ · H³\n *     Z' = ((Z₁ + Z₂)² - Z₁² - Z₂²) · H\n *\n * If U₁ = U₂ and S₁ ≠ S₂ the points are inverses → identity.\n * If U₁ = U₂ and S₁ = S₂ the points are equal → delegates to jacDouble.\n *\n * @param {JacobianPoint} jp1\n * @param {JacobianPoint} jp2\n * @param {CurveParams} curve\n * @returns {JacobianPoint}\n */\nexport function jacAdd(jp1, jp2, curve) {\n  if (jp1.x === null || jp1.y === null) return jp2\n  if (jp2.x === null || jp2.y === null) return jp1\n  const p = curve.p\n  const z1z1 = modulus(jp1.z * jp1.z, p)\n  const z2z2 = modulus(jp2.z * jp2.z, p)\n  const u1 = modulus(jp1.x * z2z2, p)\n  const u2 = modulus(jp2.x * z1z1, p)\n  const s1 = modulus(jp1.y * jp2.z * z2z2, p)\n  const s2 = modulus(jp2.y * jp1.z * z1z1, p)\n  if (u1 === u2) {\n    if (s1 !== s2) return new JacobianPoint()\n    return jacDouble(jp1, curve)\n  }\n  const h = u2 - u1\n  const i = modulus(2n * h * (2n * h), p)\n  const j = modulus(h * i, p)\n  const r = modulus(2n * (s2 - s1), p)\n  const v = modulus(u1 * i, p)\n  const x = modulus(r * r - j - 2n * v, p)\n  const y = modulus(r * (v - x) - 2n * s1 * j, p)\n  const z = modulus(((jp1.z + jp2.z) * (jp1.z + jp2.z) - z1z1 - z2z2) * h, p)\n  return new JacobianPoint(x, y, z)\n}\n\n/**\n * Scalar multiplication in Jacobian coordinates (double-and-add).\n *\n * Computes k·P using the binary expansion of k.  Runs in O(log k)\n * doublings and at most O(log k) additions, all without field inversions\n * until the final conversion back to affine.\n *\n * @param {BigInt} k   - Scalar multiplier.\n * @param {JacobianPoint} jp - Base point in Jacobian coordinates.\n * @param {CurveParams} curve\n * @returns {JacobianPoint}\n */\nexport function jacMul(k, jp, curve) {\n  if (k === 0n || jp.x === null || jp.y === null) {\n    return new JacobianPoint()\n  }\n  let result = new JacobianPoint()\n  const bits = k.toString(2)\n  for (let idx = 0; idx < bits.length; idx++) {\n    if (bits[bits.length - idx - 1] === '1') {\n      result = jacAdd(result, jp, curve)\n    }\n    jp = jacDouble(jp, curve)\n  }\n  return 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