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";
var _globals = require("@jest/globals");
var _point = require("./core/point");
var _curve = require("./core/curve");
var _registry = require("./curves/registry");
var _affine = require("./core/arithmetic/affine");
var _jacobian = require("./core/arithmetic/jacobian");
function _slicedToArray(r, e) { return _arrayWithHoles(r) || _iterableToArrayLimit(r, e) || _unsupportedIterableToArray(r, e) || _nonIterableRest(); }
function _nonIterableRest() { throw new TypeError("Invalid attempt to destructure 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 _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 _iterableToArrayLimit(r, l) { var t = null == r ? null : "undefined" != typeof Symbol && r[Symbol.iterator] || r["@@iterator"]; if (null != t) { var e, n, i, u, a = [], f = !0, o = !1; try { if (i = (t = t.call(r)).next, 0 === l) { if (Object(t) !== t) return; f = !1; } else for (; !(f = (e = i.call(t)).done) && (a.push(e.value), a.length !== l); f = !0); } catch (r) { o = !0, n = r; } finally { try { if (!f && null != t["return"] && (u = t["return"](), Object(u) !== u)) return; } finally { if (o) throw n; } } return a; } }
function _arrayWithHoles(r) { if (Array.isArray(r)) return r; }
// ---------------------------------------------------------------------------
// Elliptic curve operations
//
// Tests for the Point-operator API on the short Weierstrass curve
// y² = x³ + ax + b (mod p)
//
// Covers:
// • Point addition (chord formula):
// λ = (y₂ - y₁) · (x₂ - x₁)⁻¹ (mod p)
// x₃ = λ² - x₁ - x₂ (mod p)
// y₃ = λ(x₁ - x₃) - y₁ (mod p)
//
// • Point doubling (tangent formula):
// λ = (3x₁² + a) · (2y₁)⁻¹ (mod p)
// x₃ = λ² - 2x₁ (mod p)
// y₃ = λ(x₁ - x₃) - y₁ (mod p)
//
// • Scalar multiplication k·P via double-and-add in O(log k)
// • Negation: -P = (x, -y mod p)
// • Identity element (point at infinity): P + O = P
// • Inverse: P + (-P) = O
// ---------------------------------------------------------------------------
(0, _globals.describe)('Elliptic curve operations', function () {
// secp192k1: y² = x³ + 3 over a 192-bit prime field
var curve = (0, _registry.getCurve)('secp192k1');
var affineCurve = new _curve.CurveParams({
p: curve.p,
a: curve.a,
b: curve.b,
n: curve.n,
h: curve.h,
coord: _curve.CoordinateSystem.AFFINE
});
var point1 = new _point.Point(0xf091cf6331b1747684f5d2549cd1d4b3a8bed93b94f93cb6n, 0xfd7af42e1e7565a02e6268661c5e42e603da2d98a18f2ed5n, curve);
var point2 = new _point.Point(0x6e43b7dcae2fd5e0bf2a1ba7615ca3b9065487c9a67b4583n, 0xc48dcea47ae08e84d5fedc3d09e4c19606a290f7a19a6a58n, curve);
(0, _globals.test)('affine arithmetic: addition, doubling, and scalar multiplication', function () {
var p1 = new _point.Point(point1.x, point1.y, affineCurve);
var p2 = new _point.Point(point2.x, point2.y, affineCurve);
// P₁ + P₂ (P₁ ≠ P₂) — chord formula
p1.add(p2);
// P₁ + P₁ — tangent formula (doubling)
var expectedDouble = p1.add(p1);
var calculatedDouble = p1.add(p1);
(0, _globals.expect)(calculatedDouble.x).toBe(expectedDouble.x);
(0, _globals.expect)(calculatedDouble.y).toBe(expectedDouble.y);
// 3·P₁ = 2·P₁ + P₁
var expectedProduct = p1.add(p1);
expectedProduct = new _point.Point(expectedProduct.x, expectedProduct.y, affineCurve);
expectedProduct = expectedProduct.add(p1);
var calculatedProduct = p1.mul(3n);
(0, _globals.expect)(calculatedProduct.x).toBe(expectedProduct.x);
(0, _globals.expect)(calculatedProduct.y).toBe(expectedProduct.y);
});
// P₁ + P₂ on secp192k1 — verified against known test vectors
(0, _globals.test)('point addition produces expected result in both coordinate systems', function () {
var expectedSumX = 0x3cd61e370d02ca0687c0b5f7ebf6d0373f4dd0ccccb7cc2dn;
var expectedSumY = 0x2c4befd9b02f301eb4014504f0533aa7eb19e9ea56441f78n;
// Affine: λ = (y₂-y₁)·(x₂-x₁)⁻¹, x₃ = λ²-x₁-x₂, y₃ = λ(x₁-x₃)-y₁
var p1a = new _point.Point(point1.x, point1.y, affineCurve);
var p2a = new _point.Point(point2.x, point2.y, affineCurve);
var sumAffine = p1a.add(p2a);
(0, _globals.expect)(sumAffine.x).toBe(expectedSumX);
(0, _globals.expect)(sumAffine.y).toBe(expectedSumY);
// Jacobian: same result via (X/Z², Y/Z³) representation
var sumJacobian = point1.add(point2);
(0, _globals.expect)(sumJacobian.x).toBe(expectedSumX);
(0, _globals.expect)(sumJacobian.y).toBe(expectedSumY);
});
// 2·P₁ — point doubling verified against known test vectors
(0, _globals.test)('point doubling produces expected result in both coordinate systems', function () {
var expectedDoubleX = 0xea525dd5a1353762a14e9e78b9063316d1f2d5e792f87862n;
var expectedDoubleY = 0xa936d583530982690c445427cdf2c5b0bb1c88749247b02en;
// Affine: λ = (3x₁²+a)·(2y₁)⁻¹
var p1a = new _point.Point(point1.x, point1.y, affineCurve);
var dblAffine = p1a.add(p1a);
(0, _globals.expect)(dblAffine.x).toBe(expectedDoubleX);
(0, _globals.expect)(dblAffine.y).toBe(expectedDoubleY);
// Jacobian: S = 4XY², M = 3X²+aZ⁴, X' = M²-2S
var dblJacobian = point1.add(point1);
(0, _globals.expect)(dblJacobian.x).toBe(expectedDoubleX);
(0, _globals.expect)(dblJacobian.y).toBe(expectedDoubleY);
});
// Point(x, y) must satisfy y² ≡ x³ + ax + b (mod p)
(0, _globals.test)('constructor rejects points not on the curve', function () {
(0, _globals.expect)(function () {
return new _point.Point(200n, 119n, curve);
}).toThrow();
});
// k·P via double-and-add: 2·P₁ should equal the known doubled value
(0, _globals.test)('scalar multiplication with known vectors', function () {
var expectedX = 0xea525dd5a1353762a14e9e78b9063316d1f2d5e792f87862n;
// Affine
var p1a = new _point.Point(point1.x, point1.y, affineCurve);
(0, _globals.expect)(p1a.mul(2n).x).toBe(expectedX);
// Large scalar
var productLarge = p1a.mul(0xea525dd5a1353762a14e9e78b9063316d1f2d5e792f87862n);
(0, _globals.expect)(productLarge.x).toBe(5095008632516147798595855149669871701227161828659032863660n);
// Jacobian
(0, _globals.expect)(point1.mul(2n).x).toBe(expectedX);
});
// y² ≡ x³ + ax + b (mod p) verification
(0, _globals.test)('isOnCurve() returns true for valid points, false for identity', function () {
(0, _globals.expect)(point1.isOnCurve()).toBe(true);
(0, _globals.expect)(new _point.Point().isOnCurve()).toBe(false);
});
// P + O = P (identity element of the group)
(0, _globals.test)('adding the identity element returns the original point', function () {
var identity = new _point.Point(null, null, curve, true);
(0, _globals.expect)(point1.add(identity).x).toBe(point1.x);
(0, _globals.expect)(identity.add(point1).x).toBe(point1.x);
});
// 0·P = O and n·P = O (group order)
(0, _globals.test)('multiplying by 0 or n yields the identity', function () {
(0, _globals.expect)(point1.mul(0n).isIdentity).toBe(true);
(0, _globals.expect)(point1.mul(curve.n).isIdentity).toBe(true);
});
// P + (-P) = O (additive inverse in affine coordinates)
(0, _globals.test)('P + (-P) = O in affine coordinates', function () {
var p1a = new _point.Point(point1.x, point1.y, affineCurve);
(0, _globals.expect)(p1a.add(p1a.neg()).isIdentity).toBe(true);
});
// P + (-P) = O (additive inverse in Jacobian coordinates)
(0, _globals.test)('P + (-P) = O in Jacobian coordinates', function () {
(0, _globals.expect)(point1.add(point1.neg()).isIdentity).toBe(true);
});
// Doubling a point with y = 0: tangent is vertical → result is O
(0, _globals.test)('doubling a point with y = 0 yields identity', function () {
var crv = new _curve.CurveParams({
p: 13n,
a: 1n,
b: 0n,
n: 4n,
h: 1n
});
var P = new _point.Point(0n, 0n, crv);
(0, _globals.expect)(P.add(P).isIdentity).toBe(true);
});
// k·O = O (scalar multiplication of identity)
(0, _globals.test)('multiplying identity by any scalar remains identity (Jacobian)', function () {
var crv = new _curve.CurveParams({
p: 13n,
a: 1n,
b: 0n,
n: 4n,
h: 1n
});
var inf = new _point.Point(null, null, crv, true);
(0, _globals.expect)(inf.mul(3n).isIdentity).toBe(true);
});
(0, _globals.test)('multiplying identity by any scalar remains identity (affine)', function () {
var crv = new _curve.CurveParams({
p: 13n,
a: 1n,
b: 0n,
n: 4n,
h: 1n,
coord: _curve.CoordinateSystem.AFFINE
});
var inf = new _point.Point(null, null, crv, true);
(0, _globals.expect)(inf.mul(3n).isIdentity).toBe(true);
});
// Affine: P + (-P) = O via affine_add detecting x₁ = x₂, y₁ ≠ y₂
(0, _globals.test)('affine addition of inverse points returns identity', function () {
var p1 = new _point.Point(point1.x, point1.y, affineCurve);
(0, _globals.expect)(p1.add(p1.neg()).isIdentity).toBe(true);
});
// Affine: doubling (0, 0) where 2y = 0 → no inverse → O
(0, _globals.test)('affine doubling with y = 0 returns identity', function () {
var crv = new _curve.CurveParams({
p: 13n,
a: 1n,
b: 0n,
n: 4n,
h: 1n,
coord: _curve.CoordinateSystem.AFFINE
});
(0, _globals.expect)(new _point.Point(0n, 0n, crv).add(new _point.Point(0n, 0n, crv)).isIdentity).toBe(true);
});
// Jacobian: jac_add detects U₁ = U₂, S₁ ≠ S₂ → identity
(0, _globals.test)('Jacobian addition of inverse points returns identity', function () {
var crv = new _curve.CurveParams({
p: 13n,
a: 1n,
b: 0n,
n: 4n,
h: 1n
});
var pJac = (0, _jacobian.toJacobian)(new _point.Point(10n, 3n, crv));
var pInvJac = (0, _jacobian.toJacobian)(new _point.Point(10n, crv.p - 3n, crv));
var result = (0, _jacobian.jacAdd)(pJac, pInvJac, crv);
(0, _globals.expect)(result.x).toBe(null);
(0, _globals.expect)(result.y).toBe(null);
});
// Jacobian: doubling with Y = 0 → identity
(0, _globals.test)('Jacobian doubling with y = 0 returns identity', function () {
var crv = new _curve.CurveParams({
p: 13n,
a: 1n,
b: 0n,
n: 4n,
h: 1n
});
var result = (0, _jacobian.jacDouble)((0, _jacobian.toJacobian)(new _point.Point(0n, 0n, crv)), crv);
(0, _globals.expect)(result.x).toBe(null);
(0, _globals.expect)(result.y).toBe(null);
});
// O + O = O
(0, _globals.test)('doubling identity returns identity', function () {
var identity = new _point.Point(null, null, curve, true);
(0, _globals.expect)(identity.add(identity).isIdentity).toBe(true);
});
// jac_add(P, O) = P
(0, _globals.test)('Jacobian add with identity returns original point', function () {
var crv = new _curve.CurveParams({
p: 13n,
a: 1n,
b: 0n,
n: 4n,
h: 1n
});
var pJac = (0, _jacobian.toJacobian)(new _point.Point(10n, 3n, crv));
var result = (0, _jacobian.jacAdd)(pJac, new _jacobian.JacobianPoint(), crv);
(0, _globals.expect)(result.x).toBe(pJac.x);
(0, _globals.expect)(result.y).toBe(pJac.y);
(0, _globals.expect)(result.z).toBe(pJac.z);
});
// jac_double(O) = O
(0, _globals.test)('Jacobian doubling identity returns identity', function () {
var crv = new _curve.CurveParams({
p: 13n,
a: 1n,
b: 0n,
n: 4n,
h: 1n
});
var result = (0, _jacobian.jacDouble)(new _jacobian.JacobianPoint(), crv);
(0, _globals.expect)(result.x).toBe(null);
(0, _globals.expect)(result.y).toBe(null);
});
// toJacobian(O) = JacobianPoint(null, null, 1)
(0, _globals.test)('converting affine identity to Jacobian gives Jacobian identity', function () {
var result = (0, _jacobian.toJacobian)(new _point.Point());
(0, _globals.expect)(result.x).toBe(null);
(0, _globals.expect)(result.y).toBe(null);
});
// -P = (x, -y mod p)
(0, _globals.test)('negation computes (x, p - y)', function () {
var negPoint = point1.neg();
(0, _globals.expect)(negPoint.x).toBe(point1.x);
(0, _globals.expect)(negPoint.y).toBe((-point1.y % curve.p + curve.p) % curve.p);
});
// P₁ - P₂ = P₁ + (-P₂)
(0, _globals.test)('subtraction equals addition of negation', function () {
var resultSub = point1.sub(point2);
var resultAddNeg = point1.add(point2.neg());
(0, _globals.expect)(resultSub.x).toBe(resultAddNeg.x);
(0, _globals.expect)(resultSub.y).toBe(resultAddNeg.y);
});
// affine_double(null, null) = (null, null)
(0, _globals.test)('affine_double of identity returns identity', function () {
(0, _globals.expect)((0, _affine.affineDouble)(null, null, affineCurve)).toEqual([null, null]);
});
// affine_add(P, O) = P
(0, _globals.test)('affine_add with identity as second operand returns first point', function () {
(0, _globals.expect)((0, _affine.affineAdd)(point1.x, point1.y, null, null, affineCurve)).toEqual([point1.x, point1.y]);
});
// Arithmetic requires curve parameters
(0, _globals.test)('arithmetic without curve parameters throws', function () {
var p = new _point.Point(1n, 2n);
(0, _globals.expect)(function () {
return p.add(p);
}).toThrow();
});
// _coerce assigns curve params to a point that lacks them
(0, _globals.test)('_coerce borrows curve from the other operand', function () {
var pWithoutCurve = new _point.Point(point2.x, point2.y);
var coerced = point1._coerce(pWithoutCurve);
(0, _globals.expect)(coerced.curve).toBe(curve);
(0, _globals.expect)(point1.add(pWithoutCurve).isIdentity).toBe(false);
});
// -O = O
(0, _globals.test)('negation of identity returns identity', function () {
var identity = new _point.Point(null, null, curve, true);
(0, _globals.expect)(identity.neg().isIdentity).toBe(true);
});
// toString representation
(0, _globals.test)('toString of identity displays Point(∞)', function () {
(0, _globals.expect)(new _point.Point().toString()).toBe('Point(∞)');
});
(0, _globals.test)('toString of non-identity displays coordinates', function () {
(0, _globals.expect)(point1.toString()).toContain('Point(x=');
(0, _globals.expect)(point1.toString()).toContain(point1.x.toString());
});
// --- SEC 1 compression / decompression ---
(0, _globals.test)('compressSec1 produces correct prefix and length', function () {
var G = (0, _registry.getGenerator)('secp256k1');
var compressed = G.compressSec1();
// 256-bit curve → 32-byte x + 1-byte prefix = 33 bytes
(0, _globals.expect)(compressed.length).toBe(33);
(0, _globals.expect)([0x02, 0x03]).toContain(compressed[0]);
});
(0, _globals.test)('compressSec1 throws for identity', function () {
var identity = new _point.Point(null, null, curve, true);
(0, _globals.expect)(function () {
return identity.compressSec1();
}).toThrow('identity');
});
(0, _globals.test)('toUncompressedSec1 produces correct prefix and length', function () {
var G = (0, _registry.getGenerator)('secp256k1');
var uncompressed = G.toUncompressedSec1();
// 256-bit curve → 32-byte x + 32-byte y + 1-byte prefix = 65 bytes
(0, _globals.expect)(uncompressed.length).toBe(65);
(0, _globals.expect)(uncompressed[0]).toBe(0x04);
});
(0, _globals.test)('toUncompressedSec1 throws for identity', function () {
var identity = new _point.Point(null, null, curve, true);
(0, _globals.expect)(function () {
return identity.toUncompressedSec1();
}).toThrow('identity');
});
(0, _globals.test)('fromSec1 compressed roundtrip', function () {
var G = (0, _registry.getGenerator)('secp256k1');
var sec1Curve = (0, _registry.getCurve)('secp256k1');
var compressed = G.compressSec1();
var recovered = _point.Point.fromSec1(compressed, sec1Curve);
(0, _globals.expect)(recovered.x).toBe(G.x);
(0, _globals.expect)(recovered.y).toBe(G.y);
});
(0, _globals.test)('fromSec1 uncompressed roundtrip', function () {
var G = (0, _registry.getGenerator)('secp256k1');
var sec1Curve = (0, _registry.getCurve)('secp256k1');
var uncompressed = G.toUncompressedSec1();
var recovered = _point.Point.fromSec1(uncompressed, sec1Curve);
(0, _globals.expect)(recovered.x).toBe(G.x);
(0, _globals.expect)(recovered.y).toBe(G.y);
});
(0, _globals.test)('fromSec1 throws for data too short', function () {
var sec1Curve = (0, _registry.getCurve)('secp256k1');
(0, _globals.expect)(function () {
return _point.Point.fromSec1(new Uint8Array([0x02]), sec1Curve);
}).toThrow();
});
(0, _globals.test)('fromSec1 throws for unknown prefix', function () {
var sec1Curve = (0, _registry.getCurve)('secp256k1');
var bad = new Uint8Array(33);
bad[0] = 0x05;
(0, _globals.expect)(function () {
return _point.Point.fromSec1(bad, sec1Curve);
}).toThrow('Unknown SEC 1 prefix');
});
(0, _globals.test)('fromSec1 throws for wrong compressed length', function () {
var sec1Curve = (0, _registry.getCurve)('secp256k1');
var bad = new Uint8Array(20);
bad[0] = 0x02;
(0, _globals.expect)(function () {
return _point.Point.fromSec1(bad, sec1Curve);
}).toThrow();
});
(0, _globals.test)('fromSec1 throws for wrong uncompressed length', function () {
var sec1Curve = (0, _registry.getCurve)('secp256k1');
var bad = new Uint8Array(20);
bad[0] = 0x04;
(0, _globals.expect)(function () {
return _point.Point.fromSec1(bad, sec1Curve);
}).toThrow();
});
(0, _globals.test)('compress throws for identity', function () {
var identity = new _point.Point(null, null, curve, true);
(0, _globals.expect)(function () {
return identity.compress();
}).toThrow('identity');
});
(0, _globals.test)('decompress with parity flip selects correct root', function () {
var toyCurve = new _curve.CurveParams({
p: 23n,
a: 1n,
b: 1n,
n: 28n,
h: 1n
});
var P = new _point.Point(0n, 1n, toyCurve);
var _P$compress = P.compress(),
_P$compress2 = _slicedToArray(_P$compress, 2),
x = _P$compress2[0],
parity = _P$compress2[1];
// Decompress with opposite parity → y = p - original_y
var flipped = _point.Point.decompress(x, parity === 0n ? 1n : 0n, toyCurve);
(0, _globals.expect)(flipped.y).toBe(toyCurve.p - P.y);
});
(0, _globals.test)('decompress throws for invalid x', function () {
// Use a curve where x=999 has no valid point
var toyCurve = new _curve.CurveParams({
p: 23n,
a: 1n,
b: 1n,
n: 28n,
h: 1n
});
// x=2: rhs = 8 + 2 + 1 = 11. 11 is not a QR mod 23.
(0, _globals.expect)(function () {
return _point.Point.decompress(2n, 0n, toyCurve);
}).toThrow('does not correspond');
});
(0, _globals.test)('SEC 1 roundtrip on secp521r1 (odd-byte field size)', function () {
var G = (0, _registry.getGenerator)('secp521r1');
var c = (0, _registry.getCurve)('secp521r1');
var comp = G.compressSec1();
var uncomp = G.toUncompressedSec1();
(0, _globals.expect)(_point.Point.fromSec1(comp, c).x).toBe(G.x);
(0, _globals.expect)(_point.Point.fromSec1(uncomp, c).x).toBe(G.x);
});
});
// ---------------------------------------------------------------------------
// Group law properties on E: y² = x³ + x + 1 over F₂₃ (n = 28)
//
// An elliptic curve over a finite field forms an abelian group:
// 1. Identity: P + O = P
// 2. Inverse: P + (-P) = O
// 3. Associativity: (P + Q) + R = P + (Q + R)
// 4. Commutativity: P + Q = Q + P
// 5. Distributivity: (a + b)·P = a·P + b·P
// ---------------------------------------------------------------------------
(0, _globals.describe)('Group law properties (toy curve E/F₂₃)', function () {
var curve = new _curve.CurveParams({
p: 23n,
a: 1n,
b: 1n,
n: 28n,
h: 1n
});
var P = new _point.Point(0n, 1n, curve);
var Q = new _point.Point(6n, 19n, curve);
var inf = new _point.Point(null, null, curve, true);
(0, _globals.test)('P + O = P (right identity)', function () {
var result = P.add(inf);
(0, _globals.expect)(result.x).toBe(P.x);
(0, _globals.expect)(result.y).toBe(P.y);
});
(0, _globals.test)('O + P = P (left identity)', function () {
var result = inf.add(P);
(0, _globals.expect)(result.x).toBe(P.x);
(0, _globals.expect)(result.y).toBe(P.y);
});
(0, _globals.test)('P + (-P) = O (inverse)', function () {
(0, _globals.expect)(P.add(P.neg()).isIdentity).toBe(true);
});
(0, _globals.test)('n·P = O (group order)', function () {
(0, _globals.expect)(P.mul(curve.n).isIdentity).toBe(true);
});
(0, _globals.test)('0·P = O', function () {
(0, _globals.expect)(P.mul(0n).isIdentity).toBe(true);
});
(0, _globals.test)('1·P = P', function () {
var result = P.mul(1n);
(0, _globals.expect)(result.x).toBe(P.x);
(0, _globals.expect)(result.y).toBe(P.y);
});
(0, _globals.test)('(P + Q) + R = P + (Q + R) (associativity)', function () {
var R = P.mul(3n);
var lhs = P.add(Q).add(R);
var rhs = P.add(Q.add(R));
(0, _globals.expect)(lhs.x).toBe(rhs.x);
(0, _globals.expect)(lhs.y).toBe(rhs.y);
});
(0, _globals.test)('P + Q = Q + P (commutativity)', function () {
var pq = P.add(Q);
var qp = Q.add(P);
(0, _globals.expect)(pq.x).toBe(qp.x);
(0, _globals.expect)(pq.y).toBe(qp.y);
});
(0, _globals.test)('(a + b)·P = a·P + b·P (distributivity)', function () {
var a = 7n,
b = 13n;
var lhs = P.mul(a + b);
var rhs = P.mul(a).add(P.mul(b));
(0, _globals.expect)(lhs.x).toBe(rhs.x);
(0, _globals.expect)(lhs.y).toBe(rhs.y);
});
(0, _globals.test)('P + P = 2·P (doubling via addition equals scalar multiplication)', function () {
var pp = P.add(P);
var twoP = P.mul(2n);
(0, _globals.expect)(pp.x).toBe(twoP.x);
(0, _globals.expect)(pp.y).toBe(twoP.y);
});
(0, _globals.test)('k·P computed via mul is consistent', function () {
var r1 = P.mul(5n);
var r2 = P.mul(5n);
(0, _globals.expect)(r1.x).toBe(r2.x);
(0, _globals.expect)(r1.y).toBe(r2.y);
});
});
// ---------------------------------------------------------------------------
// Jacobian ↔ Affine consistency
//
// Jacobian coordinates (X, Y, Z) map to affine (x, y) via:
// x = X / Z², y = Y / Z³
//
// Both systems must produce identical results for all scalar multiples.
// ---------------------------------------------------------------------------
(0, _globals.describe)('Jacobian ↔ Affine consistency (E/F₂₃)', function () {
var curveJac = new _curve.CurveParams({
p: 23n,
a: 1n,
b: 1n,
n: 28n,
h: 1n,
coord: _curve.CoordinateSystem.JACOBIAN
});
var curveAff = new _curve.CurveParams({
p: 23n,
a: 1n,
b: 1n,
n: 28n,
h: 1n,
coord: _curve.CoordinateSystem.AFFINE
});
var Pjac = new _point.Point(0n, 1n, curveJac);
var Paff = new _point.Point(0n, 1n, curveAff);
(0, _globals.test)('k·P matches for k = 1, 2, ..., n-1', function () {
for (var k = 1n; k < curveJac.n; k++) {
var rj = Pjac.mul(k);
var ra = Paff.mul(k);
(0, _globals.expect)(rj.x).toBe(ra.x);
(0, _globals.expect)(rj.y).toBe(ra.y);
}
});
(0, _globals.test)('n·P = O in both coordinate systems', function () {
(0, _globals.expect)(Pjac.mul(curveJac.n).isIdentity).toBe(true);
(0, _globals.expect)(Paff.mul(curveAff.n).isIdentity).toBe(true);
});
(0, _globals.test)('P + Q matches in both coordinate systems', function () {
var Qjac = new _point.Point(6n, 19n, curveJac);
var Qaff = new _point.Point(6n, 19n, curveAff);
var rj = Pjac.add(Qjac);
var ra = Paff.add(Qaff);
(0, _globals.expect)(rj.x).toBe(ra.x);
(0, _globals.expect)(rj.y).toBe(ra.y);
});
});
// ---------------------------------------------------------------------------
// Group law properties on secp256k1 (production curve)
//
// secp256k1: y² = x³ + 7 over a 256-bit prime field
// Used by Bitcoin and many other cryptocurrencies.
// Order n ≈ 2²⁵⁶ — ensures ECDLP security.
// ---------------------------------------------------------------------------
(0, _globals.describe)('Group law properties (secp256k1)', function () {
var curve = (0, _registry.getCurve)('secp256k1');
var G = (0, _registry.getGenerator)('secp256k1');
(0, _globals.test)('generator G lies on the curve', function () {
(0, _globals.expect)(G.isOnCurve()).toBe(true);
});
(0, _globals.test)('n·G = O (generator order)', function () {
(0, _globals.expect)(G.mul(curve.n).isIdentity).toBe(true);
});
(0, _globals.test)('(P + Q) + R = P + (Q + R) (associativity)', function () {
var P = G.mul(7n);
var Q = G.mul(13n);
var R = G.mul(42n);
var lhs = P.add(Q).add(R);
var rhs = P.add(Q.add(R));
(0, _globals.expect)(lhs.x).toBe(rhs.x);
(0, _globals.expect)(lhs.y).toBe(rhs.y);
});
(0, _globals.test)('P + Q = Q + P (commutativity)', function () {
var P = G.mul(7n);
var Q = G.mul(13n);
var pq = P.add(Q);
var qp = Q.add(P);
(0, _globals.expect)(pq.x).toBe(qp.x);
(0, _globals.expect)(pq.y).toBe(qp.y);
});
(0, _globals.test)('(a + b)·G = a·G + b·G (distributivity)', function () {
var a = 123n,
b = 456n;
var lhs = G.mul(a + b);
var rhs = G.mul(a).add(G.mul(b));
(0, _globals.expect)(lhs.x).toBe(rhs.x);
(0, _globals.expect)(lhs.y).toBe(rhs.y);
});
(0, _globals.test)('P + (-P) = O (inverse)', function () {
var P = G.mul(42n);
(0, _globals.expect)(P.add(P.neg()).isIdentity).toBe(true);
});
});
//# sourceMappingURL=data:application/json;charset=utf-8;base64,{"version":3,"names":["_globals","require","_point","_curve","_registry","_affine","_jacobian","_slicedToArray","r","e","_arrayWithHoles","_iterableToArrayLimit","_unsupportedIterableToArray","_nonIterableRest","TypeError","a","_arrayLikeToArray","t","toString","call","slice","constructor","name","Array","from","test","length","n","l","Symbol","iterator","i","u","f","o","next","Object","done","push","value","isArray","describe","curve","getCurve","affineCurve","CurveParams","p","b","h","coord","CoordinateSystem","AFFINE","point1","Point","point2","p1","x","y","p2","add","expectedDouble","calculatedDouble","expect","toBe","expectedProduct","calculatedProduct","mul","expectedSumX","expectedSumY","p1a","p2a","sumAffine","sumJacobian","expectedDoubleX","expectedDoubleY","dblAffine","dblJacobian","toThrow","expectedX","productLarge","isOnCurve","identity","isIdentity","neg","crv","P","inf","pJac","toJacobian","pInvJac","result","jacAdd","jacDouble","JacobianPoint","z","negPoint","resultSub","sub","resultAddNeg","affineDouble","toEqual","affineAdd","pWithoutCurve","coerced","_coerce","toContain","G","getGenerator","compressed","compressSec1","uncompressed","toUncompressedSec1","sec1Curve","recovered","fromSec1","Uint8Array","bad","compress","toyCurve","_P$compress","_P$compress2","parity","flipped","decompress","c","comp","uncomp","Q","R","lhs","rhs","pq","qp","pp","twoP","r1","r2","curveJac","JACOBIAN","curveAff","Pjac","Paff","k","rj","ra","Qjac","Qaff"],"sources":["../../src/core.test.js"],"sourcesContent":["import { test, expect, describe } from '@jest/globals'\nimport { Point } from './core/point'\nimport { CurveParams, CoordinateSystem } from './core/curve'\nimport { getCurve, getGenerator } from './curves/registry'\nimport { affineDouble, affineAdd } from './core/arithmetic/affine'\nimport {\n  JacobianPoint,\n  toJacobian,\n  jacAdd,\n  jacDouble,\n} from './core/arithmetic/jacobian'\n\n// ---------------------------------------------------------------------------\n// Elliptic curve operations\n//\n// Tests for the Point-operator API on the short Weierstrass curve\n//   y² = x³ + ax + b  (mod p)\n//\n// Covers:\n//   • Point addition (chord formula):\n//       λ  = (y₂ - y₁) · (x₂ - x₁)⁻¹  (mod p)\n//       x₃ = λ² - x₁ - x₂              (mod p)\n//       y₃ = λ(x₁ - x₃) - y₁           (mod p)\n//\n//   • Point doubling (tangent formula):\n//       λ  = (3x₁² + a) · (2y₁)⁻¹      (mod p)\n//       x₃ = λ² - 2x₁                  (mod p)\n//       y₃ = λ(x₁ - x₃) - y₁           (mod p)\n//\n//   • Scalar multiplication k·P via double-and-add in O(log k)\n//   • Negation: -P = (x, -y mod p)\n//   • Identity element (point at infinity): P + O = P\n//   • Inverse: P + (-P) = O\n// ---------------------------------------------------------------------------\n\ndescribe('Elliptic curve operations', () => {\n  // secp192k1: y² = x³ + 3 over a 192-bit prime field\n  const curve = getCurve('secp192k1')\n  const affineCurve = new CurveParams({\n    p: curve.p,\n    a: curve.a,\n    b: curve.b,\n    n: curve.n,\n    h: curve.h,\n    coord: CoordinateSystem.AFFINE,\n  })\n\n  const point1 = new Point(\n    0xf091cf6331b1747684f5d2549cd1d4b3a8bed93b94f93cb6n,\n    0xfd7af42e1e7565a02e6268661c5e42e603da2d98a18f2ed5n,\n    curve,\n  )\n  const point2 = new Point(\n    0x6e43b7dcae2fd5e0bf2a1ba7615ca3b9065487c9a67b4583n,\n    0xc48dcea47ae08e84d5fedc3d09e4c19606a290f7a19a6a58n,\n    curve,\n  )\n\n  test('affine arithmetic: addition, doubling, and scalar multiplication', () => {\n    const p1 = new Point(point1.x, point1.y, affineCurve)\n    const p2 = new Point(point2.x, point2.y, affineCurve)\n\n    // P₁ + P₂ (P₁ ≠ P₂) — chord formula\n    p1.add(p2)\n\n    // P₁ + P₁ — tangent formula (doubling)\n    const expectedDouble = p1.add(p1)\n    const calculatedDouble = p1.add(p1)\n    expect(calculatedDouble.x).toBe(expectedDouble.x)\n    expect(calculatedDouble.y).toBe(expectedDouble.y)\n\n    // 3·P₁ = 2·P₁ + P₁\n    let expectedProduct = p1.add(p1)\n    expectedProduct = new Point(\n      expectedProduct.x,\n      expectedProduct.y,\n      affineCurve,\n    )\n    expectedProduct = expectedProduct.add(p1)\n    const calculatedProduct = p1.mul(3n)\n    expect(calculatedProduct.x).toBe(expectedProduct.x)\n    expect(calculatedProduct.y).toBe(expectedProduct.y)\n  })\n\n  // P₁ + P₂ on secp192k1 — verified against known test vectors\n  test('point addition produces expected result in both coordinate systems', () => {\n    const expectedSumX = 0x3cd61e370d02ca0687c0b5f7ebf6d0373f4dd0ccccb7cc2dn\n    const expectedSumY = 0x2c4befd9b02f301eb4014504f0533aa7eb19e9ea56441f78n\n\n    // Affine: λ = (y₂-y₁)·(x₂-x₁)⁻¹, x₃ = λ²-x₁-x₂, y₃ = λ(x₁-x₃)-y₁\n    const p1a = new Point(point1.x, point1.y, affineCurve)\n    const p2a = new Point(point2.x, point2.y, affineCurve)\n    const sumAffine = p1a.add(p2a)\n    expect(sumAffine.x).toBe(expectedSumX)\n    expect(sumAffine.y).toBe(expectedSumY)\n\n    // Jacobian: same result via (X/Z², Y/Z³) representation\n    const sumJacobian = point1.add(point2)\n    expect(sumJacobian.x).toBe(expectedSumX)\n    expect(sumJacobian.y).toBe(expectedSumY)\n  })\n\n  // 2·P₁ — point doubling verified against known test vectors\n  test('point doubling produces expected result in both coordinate systems', () => {\n    const expectedDoubleX = 0xea525dd5a1353762a14e9e78b9063316d1f2d5e792f87862n\n    const expectedDoubleY = 0xa936d583530982690c445427cdf2c5b0bb1c88749247b02en\n\n    // Affine: λ = (3x₁²+a)·(2y₁)⁻¹\n    const p1a = new Point(point1.x, point1.y, affineCurve)\n    const dblAffine = p1a.add(p1a)\n    expect(dblAffine.x).toBe(expectedDoubleX)\n    expect(dblAffine.y).toBe(expectedDoubleY)\n\n    // Jacobian: S = 4XY², M = 3X²+aZ⁴, X' = M²-2S\n    const dblJacobian = point1.add(point1)\n    expect(dblJacobian.x).toBe(expectedDoubleX)\n    expect(dblJacobian.y).toBe(expectedDoubleY)\n  })\n\n  // Point(x, y) must satisfy y² ≡ x³ + ax + b (mod p)\n  test('constructor rejects points not on the curve', () => {\n    expect(() => new Point(200n, 119n, curve)).toThrow()\n  })\n\n  // k·P via double-and-add: 2·P₁ should equal the known doubled value\n  test('scalar multiplication with known vectors', () => {\n    const expectedX = 0xea525dd5a1353762a14e9e78b9063316d1f2d5e792f87862n\n\n    // Affine\n    const p1a = new Point(point1.x, point1.y, affineCurve)\n    expect(p1a.mul(2n).x).toBe(expectedX)\n\n    // Large scalar\n    const productLarge =\n      p1a.mul(0xea525dd5a1353762a14e9e78b9063316d1f2d5e792f87862n)\n    expect(productLarge.x).toBe(\n      5095008632516147798595855149669871701227161828659032863660n,\n    )\n\n    // Jacobian\n    expect(point1.mul(2n).x).toBe(expectedX)\n  })\n\n  // y² ≡ x³ + ax + b (mod p) verification\n  test('isOnCurve() returns true for valid points, false for identity', () => {\n    expect(point1.isOnCurve()).toBe(true)\n    expect(new Point().isOnCurve()).toBe(false)\n  })\n\n  // P + O = P  (identity element of the group)\n  test('adding the identity element returns the original point', () => {\n    const identity = new Point(null, null, curve, true)\n    expect(point1.add(identity).x).toBe(point1.x)\n    expect(identity.add(point1).x).toBe(point1.x)\n  })\n\n  // 0·P = O  and  n·P = O  (group order)\n  test('multiplying by 0 or n yields the identity', () => {\n    expect(point1.mul(0n).isIdentity).toBe(true)\n    expect(point1.mul(curve.n).isIdentity).toBe(true)\n  })\n\n  // P + (-P) = O  (additive inverse in affine coordinates)\n  test('P + (-P) = O in affine coordinates', () => {\n    const p1a = new Point(point1.x, point1.y, affineCurve)\n    expect(p1a.add(p1a.neg()).isIdentity).toBe(true)\n  })\n\n  // P + (-P) = O  (additive inverse in Jacobian coordinates)\n  test('P + (-P) = O in Jacobian coordinates', () => {\n    expect(point1.add(point1.neg()).isIdentity).toBe(true)\n  })\n\n  // Doubling a point with y = 0: tangent is vertical → result is O\n  test('doubling a point with y = 0 yields identity', () => {\n    const crv = new CurveParams({ p: 13n, a: 1n, b: 0n, n: 4n, h: 1n })\n    const P = new Point(0n, 0n, crv)\n    expect(P.add(P).isIdentity).toBe(true)\n  })\n\n  // k·O = O  (scalar multiplication of identity)\n  test('multiplying identity by any scalar remains identity (Jacobian)', () => {\n    const crv = new CurveParams({ p: 13n, a: 1n, b: 0n, n: 4n, h: 1n })\n    const inf = new Point(null, null, crv, true)\n    expect(inf.mul(3n).isIdentity).toBe(true)\n  })\n\n  test('multiplying identity by any scalar remains identity (affine)', () => {\n    const crv = new CurveParams({\n      p: 13n,\n      a: 1n,\n      b: 0n,\n      n: 4n,\n      h: 1n,\n      coord: CoordinateSystem.AFFINE,\n    })\n    const inf = new Point(null, null, crv, true)\n    expect(inf.mul(3n).isIdentity).toBe(true)\n  })\n\n  // Affine: P + (-P) = O via affine_add detecting x₁ = x₂, y₁ ≠ y₂\n  test('affine addition of inverse points returns identity', () => {\n    const p1 = new Point(point1.x, point1.y, affineCurve)\n    expect(p1.add(p1.neg()).isIdentity).toBe(true)\n  })\n\n  // Affine: doubling (0, 0) where 2y = 0 → no inverse → O\n  test('affine doubling with y = 0 returns identity', () => {\n    const crv = new CurveParams({\n      p: 13n,\n      a: 1n,\n      b: 0n,\n      n: 4n,\n      h: 1n,\n      coord: CoordinateSystem.AFFINE,\n    })\n    expect(new Point(0n, 0n, crv).add(new Point(0n, 0n, crv)).isIdentity).toBe(\n      true,\n    )\n  })\n\n  // Jacobian: jac_add detects U₁ = U₂, S₁ ≠ S₂ → identity\n  test('Jacobian addition of inverse points returns identity', () => {\n    const crv = new CurveParams({ p: 13n, a: 1n, b: 0n, n: 4n, h: 1n })\n    const pJac = toJacobian(new Point(10n, 3n, crv))\n    const pInvJac = toJacobian(new Point(10n, crv.p - 3n, crv))\n    const result = jacAdd(pJac, pInvJac, crv)\n    expect(result.x).toBe(null)\n    expect(result.y).toBe(null)\n  })\n\n  // Jacobian: doubling with Y = 0 → identity\n  test('Jacobian doubling with y = 0 returns identity', () => {\n    const crv = new CurveParams({ p: 13n, a: 1n, b: 0n, n: 4n, h: 1n })\n    const result = jacDouble(toJacobian(new Point(0n, 0n, crv)), crv)\n    expect(result.x).toBe(null)\n    expect(result.y).toBe(null)\n  })\n\n  // O + O = O\n  test('doubling identity returns identity', () => {\n    const identity = new Point(null, null, curve, true)\n    expect(identity.add(identity).isIdentity).toBe(true)\n  })\n\n  // jac_add(P, O) = P\n  test('Jacobian add with identity returns original point', () => {\n    const crv = new CurveParams({ p: 13n, a: 1n, b: 0n, n: 4n, h: 1n })\n    const pJac = toJacobian(new Point(10n, 3n, crv))\n    const result = jacAdd(pJac, new JacobianPoint(), crv)\n    expect(result.x).toBe(pJac.x)\n    expect(result.y).toBe(pJac.y)\n    expect(result.z).toBe(pJac.z)\n  })\n\n  // jac_double(O) = O\n  test('Jacobian doubling identity returns identity', () => {\n    const crv = new CurveParams({ p: 13n, a: 1n, b: 0n, n: 4n, h: 1n })\n    const result = jacDouble(new JacobianPoint(), crv)\n    expect(result.x).toBe(null)\n    expect(result.y).toBe(null)\n  })\n\n  // toJacobian(O) = JacobianPoint(null, null, 1)\n  test('converting affine identity to Jacobian gives Jacobian identity', () => {\n    const result = toJacobian(new Point())\n    expect(result.x).toBe(null)\n    expect(result.y).toBe(null)\n  })\n\n  // -P = (x, -y mod p)\n  test('negation computes (x, p - y)', () => {\n    const negPoint = point1.neg()\n    expect(negPoint.x).toBe(point1.x)\n    expect(negPoint.y).toBe(((-point1.y % curve.p) + curve.p) % curve.p)\n  })\n\n  // P₁ - P₂ = P₁ + (-P₂)\n  test('subtraction equals addition of negation', () => {\n    const resultSub = point1.sub(point2)\n    const resultAddNeg = point1.add(point2.neg())\n    expect(resultSub.x).toBe(resultAddNeg.x)\n    expect(resultSub.y).toBe(resultAddNeg.y)\n  })\n\n  // affine_double(null, null) = (null, null)\n  test('affine_double of identity returns identity', () => {\n    expect(affineDouble(null, null, affineCurve)).toEqual([null, null])\n  })\n\n  // affine_add(P, O) = P\n  test('affine_add with identity as second operand returns first point', () => {\n    expect(affineAdd(point1.x, point1.y, null, null, affineCurve)).toEqual([\n      point1.x,\n      point1.y,\n    ])\n  })\n\n  // Arithmetic requires curve parameters\n  test('arithmetic without curve parameters throws', () => {\n    const p = new Point(1n, 2n)\n    expect(() => p.add(p)).toThrow()\n  })\n\n  // _coerce assigns curve params to a point that lacks them\n  test('_coerce borrows curve from the other operand', () => {\n    const pWithoutCurve = new Point(point2.x, point2.y)\n    const coerced = point1._coerce(pWithoutCurve)\n    expect(coerced.curve).toBe(curve)\n    expect(point1.add(pWithoutCurve).isIdentity).toBe(false)\n  })\n\n  // -O = O\n  test('negation of identity returns identity', () => {\n    const identity = new Point(null, null, curve, true)\n    expect(identity.neg().isIdentity).toBe(true)\n  })\n\n  // toString representation\n  test('toString of identity displays Point(∞)', () => {\n    expect(new Point().toString()).toBe('Point(∞)')\n  })\n\n  test('toString of non-identity displays coordinates', () => {\n    expect(point1.toString()).toContain('Point(x=')\n    expect(point1.toString()).toContain(point1.x.toString())\n  })\n\n  // --- SEC 1 compression / decompression ---\n\n  test('compressSec1 produces correct prefix and length', () => {\n    const G = getGenerator('secp256k1')\n    const compressed = G.compressSec1()\n    // 256-bit curve → 32-byte x + 1-byte prefix = 33 bytes\n    expect(compressed.length).toBe(33)\n    expect([0x02, 0x03]).toContain(compressed[0])\n  })\n\n  test('compressSec1 throws for identity', () => {\n    const identity = new Point(null, null, curve, true)\n    expect(() => identity.compressSec1()).toThrow('identity')\n  })\n\n  test('toUncompressedSec1 produces correct prefix and length', () => {\n    const G = getGenerator('secp256k1')\n    const uncompressed = G.toUncompressedSec1()\n    // 256-bit curve → 32-byte x + 32-byte y + 1-byte prefix = 65 bytes\n    expect(uncompressed.length).toBe(65)\n    expect(uncompressed[0]).toBe(0x04)\n  })\n\n  test('toUncompressedSec1 throws for identity', () => {\n    const identity = new Point(null, null, curve, true)\n    expect(() => identity.toUncompressedSec1()).toThrow('identity')\n  })\n\n  test('fromSec1 compressed roundtrip', () => {\n    const G = getGenerator('secp256k1')\n    const sec1Curve = getCurve('secp256k1')\n    const compressed = G.compressSec1()\n    const recovered = Point.fromSec1(compressed, sec1Curve)\n    expect(recovered.x).toBe(G.x)\n    expect(recovered.y).toBe(G.y)\n  })\n\n  test('fromSec1 uncompressed roundtrip', () => {\n    const G = getGenerator('secp256k1')\n    const sec1Curve = getCurve('secp256k1')\n    const uncompressed = G.toUncompressedSec1()\n    const recovered = Point.fromSec1(uncompressed, sec1Curve)\n    expect(recovered.x).toBe(G.x)\n    expect(recovered.y).toBe(G.y)\n  })\n\n  test('fromSec1 throws for data too short', () => {\n    const sec1Curve = getCurve('secp256k1')\n    expect(() => Point.fromSec1(new Uint8Array([0x02]), sec1Curve)).toThrow()\n  })\n\n  test('fromSec1 throws for unknown prefix', () => {\n    const sec1Curve = getCurve('secp256k1')\n    const bad = new Uint8Array(33)\n    bad[0] = 0x05\n    expect(() => Point.fromSec1(bad, sec1Curve)).toThrow('Unknown SEC 1 prefix')\n  })\n\n  test('fromSec1 throws for wrong compressed length', () => {\n    const sec1Curve = getCurve('secp256k1')\n    const bad = new Uint8Array(20)\n    bad[0] = 0x02\n    expect(() => Point.fromSec1(bad, sec1Curve)).toThrow()\n  })\n\n  test('fromSec1 throws for wrong uncompressed length', () => {\n    const sec1Curve = getCurve('secp256k1')\n    const bad = new Uint8Array(20)\n    bad[0] = 0x04\n    expect(() => Point.fromSec1(bad, sec1Curve)).toThrow()\n  })\n\n  test('compress throws for identity', () => {\n    const identity = new Point(null, null, curve, true)\n    expect(() => identity.compress()).toThrow('identity')\n  })\n\n  test('decompress with parity flip selects correct root', () => {\n    const toyCurve = new CurveParams({ p: 23n, a: 1n, b: 1n, n: 28n, h: 1n })\n    const P = new Point(0n, 1n, toyCurve)\n    const [x, parity] = P.compress()\n    // Decompress with opposite parity → y = p - original_y\n    const flipped = Point.decompress(x, parity === 0n ? 1n : 0n, toyCurve)\n    expect(flipped.y).toBe(toyCurve.p - P.y)\n  })\n\n  test('decompress throws for invalid x', () => {\n    // Use a curve where x=999 has no valid point\n    const toyCurve = new CurveParams({ p: 23n, a: 1n, b: 1n, n: 28n, h: 1n })\n    // x=2: rhs = 8 + 2 + 1 = 11. 11 is not a QR mod 23.\n    expect(() => Point.decompress(2n, 0n, toyCurve)).toThrow(\n      'does not correspond',\n    )\n  })\n\n  test('SEC 1 roundtrip on secp521r1 (odd-byte field size)', () => {\n    const G = getGenerator('secp521r1')\n    const c = getCurve('secp521r1')\n    const comp = G.compressSec1()\n    const uncomp = G.toUncompressedSec1()\n    expect(Point.fromSec1(comp, c).x).toBe(G.x)\n    expect(Point.fromSec1(uncomp, c).x).toBe(G.x)\n  })\n})\n\n// ---------------------------------------------------------------------------\n// Group law properties on E: y² = x³ + x + 1 over F₂₃  (n = 28)\n//\n// An elliptic curve over a finite field forms an abelian group:\n//   1. Identity:       P + O = P\n//   2. Inverse:      