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
"use strict";
Object.defineProperty(exports, "__esModule", {
value: true
});
exports.Point = void 0;
var _affine = require("./arithmetic/affine.js");
var _jacobian = require("./arithmetic/jacobian.js");
var _curve = require("./curve.js");
var _math = require("../utils/math.js");
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); }
function _toConsumableArray(r) { return _arrayWithoutHoles(r) || _iterableToArray(r) || _unsupportedIterableToArray(r) || _nonIterableSpread(); }
function _nonIterableSpread() { throw new TypeError("Invalid attempt to spread non-iterable instance.\nIn order to be iterable, non-array objects must have a [Symbol.iterator]() method."); }
function _unsupportedIterableToArray(r, a) { if (r) { if ("string" == typeof r) return _arrayLikeToArray(r, a); var t = {}.toString.call(r).slice(8, -1); return "Object" === t && r.constructor && (t = r.constructor.name), "Map" === t || "Set" === t ? Array.from(r) : "Arguments" === t || /^(?:Ui|I)nt(?:8|16|32)(?:Clamped)?Array$/.test(t) ? _arrayLikeToArray(r, a) : void 0; } }
function _iterableToArray(r) { if ("undefined" != typeof Symbol && null != r[Symbol.iterator] || null != r["@@iterator"]) return Array.from(r); }
function _arrayWithoutHoles(r) { if (Array.isArray(r)) return _arrayLikeToArray(r); }
function _arrayLikeToArray(r, a) { (null == a || a > r.length) && (a = r.length); for (var e = 0, n = Array(a); e < a; e++) n[e] = r[e]; return n; }
function _classCallCheck(a, n) { if (!(a instanceof n)) throw new TypeError("Cannot call a class as a function"); }
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); } /**
* The Point class — public-facing representation of a point on an elliptic curve.
*
* A point on the short Weierstrass curve y² = x³ + ax + b (mod p) belongs
* to an abelian group with the following properties:
*
* - Closure: P + Q is also on the curve.
* - Associativity: (P + Q) + R = P + (Q + R).
* - Identity: There exists a special "point at infinity" O such that P + O = P.
* - Inverse: For every P = (x, y), the inverse is -P = (x, -y mod p),
* and P + (-P) = O.
*
* Usage:
*
* const curve = new CurveParams({ p: 23n, a: 1n, b: 1n, n: 28n, h: 1n })
* const P = new Point(0n, 1n, curve)
* const Q = new Point(6n, 19n, curve)
* P.add(Q) // point addition
* P.mul(5n) // scalar multiplication
* P.neg() // point negation
*
* To switch coordinate systems, create a new CurveParams with a
* different coord field:
*
* const affineCurve = new CurveParams({
* p: 23n, a: 1n, b: 1n, n: 28n,
* coord: CoordinateSystem.AFFINE
* })
*/
var Point = exports.Point = /*#__PURE__*/function () {
/**
* Create a point on an elliptic curve.
*
* Validates that the point lies on the curve at construction time:
*
* y² ≡ x³ + ax + b (mod p)
*
* Points produced by internal arithmetic use _trusted = true to skip
* re-validation.
*
* @param {BigInt|null} x - x-coordinate, or null for identity (point at infinity).
* @param {BigInt|null} y - y-coordinate, or null for identity.
* @param {CurveParams|null} curve - Curve parameters (required for arithmetic).
* @param {boolean} _trusted - If true, skip on-curve validation.
*/
function Point() {
var x = arguments.length > 0 && arguments[0] !== undefined ? arguments[0] : null;
var y = arguments.length > 1 && arguments[1] !== undefined ? arguments[1] : null;
var curve = arguments.length > 2 && arguments[2] !== undefined ? arguments[2] : null;
var _trusted = arguments.length > 3 && arguments[3] !== undefined ? arguments[3] : false;
_classCallCheck(this, Point);
this.x = x;
this.y = y;
this.curve = curve;
// Validate: y² ≡ x³ + ax + b (mod p)
if (!_trusted) {
if (x !== null && y !== null && curve !== null) {
var lhs = (0, _math.modPow)(y, 2n, curve.p);
var rhs = (0, _math.modulus)((0, _math.modPow)(x, 3n, curve.p) + curve.a * x + curve.b, curve.p);
if (lhs !== rhs) {
throw new Error("Point(".concat(x, ", ").concat(y, ") is not on the curve ") + "y\xB2 = x\xB3 + ".concat(curve.a, "x + ").concat(curve.b, " (mod ").concat(curve.p, ")."));
}
}
}
Object.freeze(this);
}
/**
* True when this point represents the point at infinity (identity element O).
*/
return _createClass(Point, [{
key: "isIdentity",
get: function get() {
return this.x === null || this.y === null;
}
/**
* Check whether this point satisfies the curve equation:
*
* y² ≡ x³ + ax + b (mod p)
*
* @returns {boolean}
*/
}, {
key: "isOnCurve",
value: function isOnCurve() {
if (this.isIdentity || this.curve === null) return false;
var lhs = (0, _math.modPow)(this.y, 2n, this.curve.p);
var rhs = (0, _math.modulus)((0, _math.modPow)(this.x, 3n, this.curve.p) + this.curve.a * this.x + this.curve.b, this.curve.p);
return lhs === rhs;
}
}, {
key: "_requireCurve",
value: function _requireCurve() {
if (this.curve === null) {
throw new Error('Cannot perform arithmetic on a Point without curve parameters. ' + "Pass a CurveParams instance via the 'curve' argument.");
}
return this.curve;
}
/**
* Ensure other carries curve params (borrow ours if needed).
*/
}, {
key: "_coerce",
value: function _coerce(other) {
if (other.curve === null && this.curve !== null) {
return new Point(other.x, other.y, this.curve, true);
}
return other;
}
}, {
key: "_wrap",
value: function _wrap(x, y) {
return new Point(x, y, this.curve, true);
}
// ----- compression -----
/**
* Compress this point to its x-coordinate and parity bit.
*
* The y-coordinate can be recovered from x using the curve equation
* and the parity bit selects which of the two square roots to use.
*
* @returns {[BigInt, BigInt]} [x, parity] where parity = y mod 2.
* @throws {Error} If this point is the identity.
*/
}, {
key: "compress",
value: function compress() {
if (this.isIdentity) {
throw new Error('Cannot compress the identity point (point at infinity).');
}
return [this.x, this.y % 2n];
}
/**
* Reconstruct a point from its compressed form.
*
* Given x and a parity bit, computes:
*
* rhs = x³ + ax + b (mod p)
* y = √rhs (mod p)
*
* and selects the root matching the parity bit.
*
* @param {BigInt} x - The x-coordinate.
* @param {BigInt} parity - 0n or 1n indicating which y to select.
* @param {CurveParams} curve
* @returns {Point}
* @throws {Error} If x does not correspond to a valid point.
*/
}, {
key: "compressSec1",
value:
// ----- SEC 1 compression (interoperable) -----
/**
* Compress this point to SEC 1 / X9.62 format.
*
* The output is a single byte prefix (0x02 for even y, 0x03
* for odd y) followed by the x-coordinate as a big-endian unsigned
* integer, zero-padded to the field size.
*
* @returns {Uint8Array} Compressed point bytes.
* @throws {Error} If this point is the identity or has no curve params.
*/
function compressSec1() {
if (this.isIdentity) {
throw new Error('Cannot compress the identity point (point at infinity).');
}
var curve = this._requireCurve();
var byteLen = Number((BigInt(curve.p.toString(2).length) + 7n) / 8n);
var prefix = this.y % 2n ? 0x03 : 0x02;
var xBytes = bigIntToBytes(this.x, byteLen);
var result = new Uint8Array(1 + byteLen);
result[0] = prefix;
result.set(xBytes, 1);
return result;
}
/**
* Serialize this point to SEC 1 / X9.62 uncompressed format.
*
* The output is 0x04 || x || y, where x and y are big-endian
* unsigned integers zero-padded to the field size.
*
* @returns {Uint8Array} Uncompressed point bytes.
* @throws {Error} If this point is the identity or has no curve params.
*/
}, {
key: "toUncompressedSec1",
value: function toUncompressedSec1() {
if (this.isIdentity) {
throw new Error('Cannot serialize the identity point (point at infinity).');
}
var curve = this._requireCurve();
var byteLen = Number((BigInt(curve.p.toString(2).length) + 7n) / 8n);
var xBytes = bigIntToBytes(this.x, byteLen);
var yBytes = bigIntToBytes(this.y, byteLen);
var result = new Uint8Array(1 + 2 * byteLen);
result[0] = 0x04;
result.set(xBytes, 1);
result.set(yBytes, 1 + byteLen);
return result;
}
/**
* Deserialize a point from SEC 1 / X9.62 format.
*
* Supports both compressed (0x02/0x03 prefix) and
* uncompressed (0x04 prefix) encodings.
*
* @param {Uint8Array} data - The SEC 1 encoded point bytes.
* @param {CurveParams} curve
* @returns {Point}
* @throws {Error} If the data is malformed or the point is invalid.
*/
}, {
key: "neg",
value:
// ----- operators -----
/**
* Return the additive inverse: -P = (x, -y mod p).
*
* @returns {Point}
*/
function neg() {
if (this.isIdentity) return this;
var curve = this._requireCurve();
return new Point(this.x, (0, _math.modulus)(-this.y, curve.p), curve, true);
}
/**
* Add two points on the same curve using the group law.
*
* Delegates to affine or Jacobian arithmetic depending on
* curve.coord. The chord-and-tangent formulas are:
*
* Addition (P ≠ Q):
* λ = (y₂ - y₁) · (x₂ - x₁)⁻¹
* x₃ = λ² - x₁ - x₂
* y₃ = λ(x₁ - x₃) - y₁
*
* Doubling (P = Q):
* λ = (3x₁² + a) · (2y₁)⁻¹
* x₃ = λ² - 2x₁
* y₃ = λ(x₁ - x₃) - y₁
*
* @param {Point} other
* @returns {Point}
*/
}, {
key: "add",
value: function add(other) {
var curve = this._requireCurve();
other = this._coerce(other);
if (curve.coord === _curve.CoordinateSystem.JACOBIAN) {
var jp1 = (0, _jacobian.toJacobian)(this);
var jp2 = (0, _jacobian.toJacobian)(other);
var jp3 = (0, _jacobian.jacAdd)(jp1, jp2, curve);
return this._wrap.apply(this, _toConsumableArray((0, _jacobian.toAffine)(jp3, curve)));
}
return this._wrap.apply(this, _toConsumableArray((0, _affine.affineAdd)(this.x, this.y, other.x, other.y, curve)));
}
/**
* Subtract: P - Q = P + (-Q).
*
* @param {Point} other
* @returns {Point}
*/
}, {
key: "sub",
value: function sub(other) {
return this.add(other.neg ? other.neg() : new Point(other.x, (0, _math.modulus)(-other.y, this._requireCurve().p), other.curve || this.curve, true));
}
/**
* Scalar multiplication: k · P via double-and-add in O(log k).
*
* The scalar k is reduced modulo the group order n before
* multiplication, so that n·P = O (the identity).
*
* @param {BigInt} k - Scalar multiplier.
* @returns {Point}
*/
}, {
key: "mul",
value: function mul(k) {
var curve = this._requireCurve();
k = (0, _math.modulus)(k, curve.n);
if (curve.coord === _curve.CoordinateSystem.JACOBIAN) {
var jp = (0, _jacobian.toJacobian)(this);
var jpResult = (0, _jacobian.jacMul)(k, jp, curve);
return this._wrap.apply(this, _toConsumableArray((0, _jacobian.toAffine)(jpResult, curve)));
}
return this._wrap.apply(this, _toConsumableArray((0, _affine.affineMul)(k, this.x, this.y, curve)));
}
}, {
key: "toString",
value: function toString() {
if (this.isIdentity) return 'Point(∞)';
return "Point(x=".concat(this.x, ", y=").concat(this.y, ")");
}
}], [{
key: "decompress",
value: function decompress(x, parity, curve) {
var rhs = (0, _math.modulus)((0, _math.modPow)(x, 3n, curve.p) + curve.a * x + curve.b, curve.p);
var y = (0, _math.modularSqrt)(rhs, curve.p);
if (y === null) {
throw new Error("x=".concat(x, " does not correspond to a valid point on the curve ") + "y\xB2 = x\xB3 + ".concat(curve.a, "x + ").concat(curve.b, " (mod ").concat(curve.p, ")."));
}
if (y % 2n !== parity) {
y = curve.p - y;
}
return new Point(x, y, curve);
}
}, {
key: "fromSec1",
value: function fromSec1(data, curve) {
if (data.length < 2) {
throw new Error('SEC 1 data too short.');
}
var byteLen = Number((BigInt(curve.p.toString(2).length) + 7n) / 8n);
var prefix = data[0];
if (prefix === 0x02 || prefix === 0x03) {
if (data.length !== 1 + byteLen) {
throw new Error("Compressed SEC 1 data must be ".concat(1 + byteLen, " bytes, got ").concat(data.length, "."));
}
var x = bytesToBigInt(data.slice(1));
var parity = BigInt(prefix - 0x02); // 0 for even, 1 for odd
return Point.decompress(x, parity, curve);
}
if (prefix === 0x04) {
if (data.length !== 1 + 2 * byteLen) {
throw new Error("Uncompressed SEC 1 data must be ".concat(1 + 2 * byteLen, " bytes, got ").concat(data.length, "."));
}
var _x = bytesToBigInt(data.slice(1, 1 + byteLen));
var y = bytesToBigInt(data.slice(1 + byteLen));
return new Point(_x, y, curve);
}
throw new Error("Unknown SEC 1 prefix: 0x".concat(prefix.toString(16).padStart(2, '0'), ". Expected 0x02, 0x03, or 0x04."));
}
}]);
}(); // ----- helper functions for byte ↔ BigInt conversion -----
function bigIntToBytes(value, length) {
var hex = value.toString(16).padStart(length * 2, '0');
var bytes = new Uint8Array(length);
for (var i = 0; i < length; i++) {
bytes[i] = parseInt(hex.slice(i * 2, i * 2 + 2), 16);
}
return bytes;
}
function bytesToBigInt(bytes) {
var hex = '';
for (var i = 0; i < bytes.length; i++) {
hex += bytes[i].toString(16).padStart(2, '0');
}
return BigInt('0x' + hex);
}
//# sourceMappingURL=data:application/json;charset=utf-8;base64,{"version":3,"names":["_affine","require","_jacobian","_curve","_math","_typeof","o","Symbol","iterator","constructor","prototype","_toConsumableArray","r","_arrayWithoutHoles","_iterableToArray","_unsupportedIterableToArray","_nonIterableSpread","TypeError","a","_arrayLikeToArray","t","toString","call","slice","name","Array","from","test","isArray","length","e","n","_classCallCheck","_defineProperties","enumerable","configurable","writable","Object","defineProperty","_toPropertyKey","key","_createClass","i","_toPrimitive","toPrimitive","String","Number","Point","exports","x","arguments","undefined","y","curve","_trusted","lhs","modPow","p","rhs","modulus","b","Error","concat","freeze","get","value","isOnCurve","isIdentity","_requireCurve","_coerce","other","_wrap","compress","compressSec1","byteLen","BigInt","prefix","xBytes","bigIntToBytes","result","Uint8Array","set","toUncompressedSec1","yBytes","neg","add","coord","CoordinateSystem","JACOBIAN","jp1","toJacobian","jp2","jp3","jacAdd","apply","toAffine","affineAdd","sub","mul","k","jp","jpResult","jacMul","affineMul","decompress","parity","modularSqrt","fromSec1","data","bytesToBigInt","padStart","hex","bytes","parseInt"],"sources":["../../../src/core/point.js"],"sourcesContent":["/**\n * The Point class — public-facing representation of a point on an elliptic curve.\n *\n * A point on the short Weierstrass curve y² = x³ + ax + b (mod p) belongs\n * to an abelian group with the following properties:\n *\n * - Closure:       P + Q is also on the curve.\n * - Associativity: (P + Q) + R = P + (Q + R).\n * - Identity:      There exists a special \"point at infinity\" O such that P + O = P.\n * - Inverse:       For every P = (x, y), the inverse is -P = (x, -y mod p),\n *                  and P + (-P) = O.\n *\n * Usage:\n *\n *     const curve = new CurveParams({ p: 23n, a: 1n, b: 1n, n: 28n, h: 1n })\n *     const P = new Point(0n, 1n, curve)\n *     const Q = new Point(6n, 19n, curve)\n *     P.add(Q)          // point addition\n *     P.mul(5n)         // scalar multiplication\n *     P.neg()           // point negation\n *\n * To switch coordinate systems, create a new CurveParams with a\n * different coord field:\n *\n *     const affineCurve = new CurveParams({\n *       p: 23n, a: 1n, b: 1n, n: 28n,\n *       coord: CoordinateSystem.AFFINE\n *     })\n */\n\nimport { affineAdd, affineMul } from './arithmetic/affine.js'\nimport { jacAdd, jacMul, toAffine, toJacobian } from './arithmetic/jacobian.js'\nimport { CoordinateSystem } from './curve.js'\nimport { modularSqrt, modulus, modPow } from '../utils/math.js'\n\nexport class Point {\n  /**\n   * Create a point on an elliptic curve.\n   *\n   * Validates that the point lies on the curve at construction time:\n   *\n   *     y² ≡ x³ + ax + b  (mod p)\n   *\n   * Points produced by internal arithmetic use _trusted = true to skip\n   * re-validation.\n   *\n   * @param {BigInt|null} x          - x-coordinate, or null for identity (point at infinity).\n   * @param {BigInt|null} y          - y-coordinate, or null for identity.\n   * @param {CurveParams|null} curve - Curve parameters (required for arithmetic).\n   * @param {boolean} _trusted       - If true, skip on-curve validation.\n   */\n  constructor(x = null, y = null, curve = null, _trusted = false) {\n    this.x = x\n    this.y = y\n    this.curve = curve\n\n    // Validate: y² ≡ x³ + ax + b (mod p)\n    if (!_trusted) {\n      if (x !== null && y !== null && curve !== null) {\n        const lhs = modPow(y, 2n, curve.p)\n        const rhs = modulus(\n          modPow(x, 3n, curve.p) + curve.a * x + curve.b,\n          curve.p,\n        )\n        if (lhs !== rhs) {\n          throw new Error(\n            `Point(${x}, ${y}) is not on the curve ` +\n              `y² = x³ + ${curve.a}x + ${curve.b} (mod ${curve.p}).`,\n          )\n        }\n      }\n    }\n\n    Object.freeze(this)\n  }\n\n  /**\n   * True when this point represents the point at infinity (identity element O).\n   */\n  get isIdentity() {\n    return this.x === null || this.y === null\n  }\n\n  /**\n   * Check whether this point satisfies the curve equation:\n   *\n   *     y² ≡ x³ + ax + b  (mod p)\n   *\n   * @returns {boolean}\n   */\n  isOnCurve() {\n    if (this.isIdentity || this.curve === null) return false\n    const lhs = modPow(this.y, 2n, this.curve.p)\n    const rhs = modulus(\n      modPow(this.x, 3n, this.curve.p) + this.curve.a * this.x + this.curve.b,\n      this.curve.p,\n    )\n    return lhs === rhs\n  }\n\n  _requireCurve() {\n    if (this.curve === null) {\n      throw new Error(\n        'Cannot perform arithmetic on a Point without curve parameters. ' +\n          \"Pass a CurveParams instance via the 'curve' argument.\",\n      )\n    }\n    return this.curve\n  }\n\n  /**\n   * Ensure other carries curve params (borrow ours if needed).\n   */\n  _coerce(other) {\n    if (other.curve === null && this.curve !== null) {\n      return new Point(other.x, other.y, this.curve, true)\n    }\n    return other\n  }\n\n  _wrap(x, y) {\n    return new Point(x, y, this.curve, true)\n  }\n\n  // ----- compression -----\n\n  /**\n   * Compress this point to its x-coordinate and parity bit.\n   *\n   * The y-coordinate can be recovered from x using the curve equation\n   * and the parity bit selects which of the two square roots to use.\n   *\n   * @returns {[BigInt, BigInt]} [x, parity] where parity = y mod 2.\n   * @throws {Error} If this point is the identity.\n   */\n  compress() {\n    if (this.isIdentity) {\n      throw new Error('Cannot compress the identity point (point at infinity).')\n    }\n    return [this.x, this.y % 2n]\n  }\n\n  /**\n   * Reconstruct a point from its compressed form.\n   *\n   * Given x and a parity bit, computes:\n   *\n   *     rhs = x³ + ax + b  (mod p)\n   *     y   = √rhs         (mod p)\n   *\n   * and selects the root matching the parity bit.\n   *\n   * @param {BigInt} x       - The x-coordinate.\n   * @param {BigInt} parity  - 0n or 1n indicating which y to select.\n   * @param {CurveParams} curve\n   * @returns {Point}\n   * @throws {Error} If x does not correspond to a valid point.\n   */\n  static decompress(x, parity, curve) {\n    const rhs = modulus(modPow(x, 3n, curve.p) + curve.a * x + curve.b, curve.p)\n    let y = modularSqrt(rhs, curve.p)\n    if (y === null) {\n      throw new Error(\n        `x=${x} does not correspond to a valid point on the curve ` +\n          `y² = x³ + ${curve.a}x + ${curve.b} (mod ${curve.p}).`,\n      )\n    }\n    if (y % 2n !== parity) {\n      y = curve.p - y\n    }\n    return new Point(x, y, curve)\n  }\n\n  // ----- SEC 1 compression (interoperable) -----\n\n  /**\n   * Compress this point to SEC 1 / X9.62 format.\n   *\n   * The output is a single byte prefix (0x02 for even y, 0x03\n   * for odd y) followed by the x-coordinate as a big-endian unsigned\n   * integer, zero-padded to the field size.\n   *\n   * @returns {Uint8Array} Compressed point bytes.\n   * @throws {Error} If this point is the identity or has no curve params.\n   */\n  compressSec1() {\n    if (this.isIdentity) {\n      throw new Error('Cannot compress the identity point (point at infinity).')\n    }\n    const curve = this._requireCurve()\n    const byteLen = Number((BigInt(curve.p.toString(2).length) + 7n) / 8n)\n    const prefix = this.y % 2n ? 0x03 : 0x02\n    const xBytes = bigIntToBytes(this.x, byteLen)\n    const result = new Uint8Array(1 + byteLen)\n    result[0] = prefix\n    result.set(xBytes, 1)\n    return result\n  }\n\n  /**\n   * Serialize this point to SEC 1 / X9.62 uncompressed format.\n   *\n   * The output is 0x04 || x || y, where x and y are big-endian\n   * unsigned integers zero-padded to the field size.\n   *\n   * @returns {Uint8Array} Uncompressed point bytes.\n   * @throws {Error} If this point is the identity or has no curve params.\n   */\n  toUncompressedSec1() {\n    if (this.isIdentity) {\n      throw new Error(\n        'Cannot serialize the identity point (point at infinity).',\n      )\n    }\n    const curve = this._requireCurve()\n    const byteLen = Number((BigInt(curve.p.toString(2).length) + 7n) / 8n)\n    const xBytes = bigIntToBytes(this.x, byteLen)\n    const yBytes = bigIntToBytes(this.y, byteLen)\n    const result = new Uint8Array(1 + 2 * byteLen)\n    result[0] = 0x04\n    result.set(xBytes, 1)\n    result.set(yBytes, 1 + byteLen)\n    return result\n  }\n\n  /**\n   * Deserialize a point from SEC 1 / X9.62 format.\n   *\n   * Supports both compressed (0x02/0x03 prefix) and\n   * uncompressed (0x04 prefix) encodings.\n   *\n   * @param {Uint8Array} data  - The SEC 1 encoded point bytes.\n   * @param {CurveParams} curve\n   * @returns {Point}\n   * @throws {Error} If the data is malformed or the point is invalid.\n   */\n  static fromSec1(data, curve) {\n    if (data.length < 2) {\n      throw new Error('SEC 1 data too short.')\n    }\n    const byteLen = Number((BigInt(curve.p.toString(2).length) + 7n) / 8n)\n    const prefix = data[0]\n\n    if (prefix === 0x02 || prefix === 0x03) {\n      if (data.length !== 1 + byteLen) {\n        throw new Error(\n          `Compressed SEC 1 data must be ${1 + byteLen} bytes, got ${data.length}.`,\n        )\n      }\n      const x = bytesToBigInt(data.slice(1))\n      const parity = BigInt(prefix - 0x02) // 0 for even, 1 for odd\n      return Point.decompress(x, parity, curve)\n    }\n\n    if (prefix === 0x04) {\n      if (data.length !== 1 + 2 * byteLen) {\n        throw new Error(\n          `Uncompressed SEC 1 data must be ${1 + 2 * byteLen} bytes, got ${data.length}.`,\n        )\n      }\n      const x = bytesToBigInt(data.slice(1, 1 + byteLen))\n      const y = bytesToBigInt(data.slice(1 + byteLen))\n      return new Point(x, y, curve)\n    }\n\n    throw new Error(\n      `Unknown SEC 1 prefix: 0x${prefix.toString(16).padStart(2, '0')}. Expected 0x02, 0x03, or 0x04.`,\n    )\n  }\n\n  // ----- operators -----\n\n  /**\n   * Return the additive inverse: -P = (x, -y mod p).\n   *\n   * @returns {Point}\n   */\n  neg() {\n    if (this.isIdentity) return this\n    const curve = this._requireCurve()\n    return new Point(this.x, modulus(-this.y, curve.p), curve, true)\n  }\n\n  /**\n   * Add two points on the same curve using the group law.\n   *\n   * Delegates to affine or Jacobian arithmetic depending on\n   * curve.coord.  The chord-and-tangent formulas are:\n   *\n   * Addition (P ≠ Q):\n   *     λ  = (y₂ - y₁) · (x₂ - x₁)⁻¹\n   *     x₃ = λ² - x₁ - x₂\n   *     y₃ = λ(x₁ - x₃) - y₁\n   *\n   * Doubling (P = Q):\n   *     λ  = (3x₁² + a) · (2y₁)⁻¹\n   *     x₃ = λ² - 2x₁\n   *     y₃ = λ(x₁ - x₃) - y₁\n   *\n   * @param {Point} other\n   * @returns {Point}\n   */\n  add(other) {\n    const curve = this._requireCurve()\n    other = this._coerce(other)\n\n    if (curve.coord === CoordinateSystem.JACOBIAN) {\n      const jp1 = toJacobian(this)\n      const jp2 = toJacobian(other)\n      const jp3 = jacAdd(jp1, jp2, curve)\n      return this._wrap(...toAffine(jp3, curve))\n    }\n\n    return this._wrap(...affineAdd(this.x, this.y, other.x, other.y, curve))\n  }\n\n  /**\n   * Subtract: P - Q = P + (-Q).\n   *\n   * @param {Point} other\n   * @returns {Point}\n   */\n  sub(other) {\n    return this.add(\n      other.neg\n        ? other.neg()\n        : new Point(\n            other.x,\n            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