"use strict"; Object.defineProperty(exports, "__esModule", { value: true }); exports.PrimeEdwardsPoint = void 0; exports.edwards = edwards; exports.eddsa = eddsa; exports.twistedEdwards = twistedEdwards; /** * Twisted Edwards curve. The formula is: ax² + y² = 1 + dx²y². * For design rationale of types / exports, see weierstrass module documentation. * Untwisted Edwards curves exist, but they aren't used in real-world protocols. * @module */ /*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */ const utils_ts_1 = require("../utils.js"); const curve_ts_1 = require("./curve.js"); const modular_ts_1 = require("./modular.js"); // Be friendly to bad ECMAScript parsers by not using bigint literals // prettier-ignore const _0n = BigInt(0), _1n = BigInt(1), _2n = BigInt(2), _8n = BigInt(8); function isEdValidXY(Fp, CURVE, x, y) { const x2 = Fp.sqr(x); const y2 = Fp.sqr(y); const left = Fp.add(Fp.mul(CURVE.a, x2), y2); const right = Fp.add(Fp.ONE, Fp.mul(CURVE.d, Fp.mul(x2, y2))); return Fp.eql(left, right); } function edwards(params, extraOpts = {}) { const validated = (0, curve_ts_1._createCurveFields)('edwards', params, extraOpts, extraOpts.FpFnLE); const { Fp, Fn } = validated; let CURVE = validated.CURVE; const { h: cofactor } = CURVE; (0, utils_ts_1._validateObject)(extraOpts, {}, { uvRatio: 'function' }); // Important: // There are some places where Fp.BYTES is used instead of nByteLength. // So far, everything has been tested with curves of Fp.BYTES == nByteLength. // TODO: test and find curves which behave otherwise. const MASK = _2n << (BigInt(Fn.BYTES * 8) - _1n); const modP = (n) => Fp.create(n); // Function overrides // sqrt(u/v) const uvRatio = extraOpts.uvRatio || ((u, v) => { try { return { isValid: true, value: Fp.sqrt(Fp.div(u, v)) }; } catch (e) { return { isValid: false, value: _0n }; } }); // Validate whether the passed curve params are valid. // equation ax² + y² = 1 + dx²y² should work for generator point. if (!isEdValidXY(Fp, CURVE, CURVE.Gx, CURVE.Gy)) throw new Error('bad curve params: generator point'); /** * Asserts coordinate is valid: 0 <= n < MASK. * Coordinates >= Fp.ORDER are allowed for zip215. */ function acoord(title, n, banZero = false) { const min = banZero ? _1n : _0n; (0, utils_ts_1.aInRange)('coordinate ' + title, n, min, MASK); return n; } function aextpoint(other) { if (!(other instanceof Point)) throw new Error('ExtendedPoint expected'); } // Converts Extended point to default (x, y) coordinates. // Can accept precomputed Z^-1 - for example, from invertBatch. const toAffineMemo = (0, utils_ts_1.memoized)((p, iz) => { const { X, Y, Z } = p; const is0 = p.is0(); if (iz == null) iz = is0 ? _8n : Fp.inv(Z); // 8 was chosen arbitrarily const x = modP(X * iz); const y = modP(Y * iz); const zz = Fp.mul(Z, iz); if (is0) return { x: _0n, y: _1n }; if (zz !== _1n) throw new Error('invZ was invalid'); return { x, y }; }); const assertValidMemo = (0, utils_ts_1.memoized)((p) => { const { a, d } = CURVE; if (p.is0()) throw new Error('bad point: ZERO'); // TODO: optimize, with vars below? // Equation in affine coordinates: ax² + y² = 1 + dx²y² // Equation in projective coordinates (X/Z, Y/Z, Z): (aX² + Y²)Z² = Z⁴ + dX²Y² const { X, Y, Z, T } = p; const X2 = modP(X * X); // X² const Y2 = modP(Y * Y); // Y² const Z2 = modP(Z * Z); // Z² const Z4 = modP(Z2 * Z2); // Z⁴ const aX2 = modP(X2 * a); // aX² const left = modP(Z2 * modP(aX2 + Y2)); // (aX² + Y²)Z² const right = modP(Z4 + modP(d * modP(X2 * Y2))); // Z⁴ + dX²Y² if (left !== right) throw new Error('bad point: equation left != right (1)'); // In Extended coordinates we also have T, which is x*y=T/Z: check X*Y == Z*T const XY = modP(X * Y); const ZT = modP(Z * T); if (XY !== ZT) throw new Error('bad point: equation left != right (2)'); return true; }); // Extended Point works in extended coordinates: (X, Y, Z, T) ∋ (x=X/Z, y=Y/Z, T=xy). // https://en.wikipedia.org/wiki/Twisted_Edwards_curve#Extended_coordinates class Point { constructor(X, Y, Z, T) { this.X = acoord('x', X); this.Y = acoord('y', Y); this.Z = acoord('z', Z, true); this.T = acoord('t', T); Object.freeze(this); } static CURVE() { return CURVE; } static fromAffine(p) { if (p instanceof Point) throw new Error('extended point not allowed'); const { x, y } = p || {}; acoord('x', x); acoord('y', y); return new Point(x, y, _1n, modP(x * y)); } // Uses algo from RFC8032 5.1.3. static fromBytes(bytes, zip215 = false) { const len = Fp.BYTES; const { a, d } = CURVE; bytes = (0, utils_ts_1.copyBytes)((0, utils_ts_1._abytes2)(bytes, len, 'point')); (0, utils_ts_1._abool2)(zip215, 'zip215'); const normed = (0, utils_ts_1.copyBytes)(bytes); // copy again, we'll manipulate it const lastByte = bytes[len - 1]; // select last byte normed[len - 1] = lastByte & ~0x80; // clear last bit const y = (0, utils_ts_1.bytesToNumberLE)(normed); // zip215=true is good for consensus-critical apps. =false follows RFC8032 / NIST186-5. // RFC8032 prohibits >= p, but ZIP215 doesn't // zip215=true: 0 <= y < MASK (2^256 for ed25519) // zip215=false: 0 <= y < P (2^255-19 for ed25519) const max = zip215 ? MASK : Fp.ORDER; (0, utils_ts_1.aInRange)('point.y', y, _0n, max); // Ed25519: x² = (y²-1)/(dy²+1) mod p. Ed448: x² = (y²-1)/(dy²-1) mod p. Generic case: // ax²+y²=1+dx²y² => y²-1=dx²y²-ax² => y²-1=x²(dy²-a) => x²=(y²-1)/(dy²-a) const y2 = modP(y * y); // denominator is always non-0 mod p. const u = modP(y2 - _1n); // u = y² - 1 const v = modP(d * y2 - a); // v = d y² + 1. let { isValid, value: x } = uvRatio(u, v); // √(u/v) if (!isValid) throw new Error('bad point: invalid y coordinate'); const isXOdd = (x & _1n) === _1n; // There are 2 square roots. Use x_0 bit to select proper const isLastByteOdd = (lastByte & 0x80) !== 0; // x_0, last bit if (!zip215 && x === _0n && isLastByteOdd) // if x=0 and x_0 = 1, fail throw new Error('bad point: x=0 and x_0=1'); if (isLastByteOdd !== isXOdd) x = modP(-x); // if x_0 != x mod 2, set x = p-x return Point.fromAffine({ x, y }); } static fromHex(bytes, zip215 = false) { return Point.fromBytes((0, utils_ts_1.ensureBytes)('point', bytes), zip215); } get x() { return this.toAffine().x; } get y() { return this.toAffine().y; } precompute(windowSize = 8, isLazy = true) { wnaf.createCache(this, windowSize); if (!isLazy) this.multiply(_2n); // random number return this; } // Useful in fromAffine() - not for fromBytes(), which always created valid points. assertValidity() { assertValidMemo(this); } // Compare one point to another. equals(other) { aextpoint(other); const { X: X1, Y: Y1, Z: Z1 } = this; const { X: X2, Y: Y2, Z: Z2 } = other; const X1Z2 = modP(X1 * Z2); const X2Z1 = modP(X2 * Z1); const Y1Z2 = modP(Y1 * Z2); const Y2Z1 = modP(Y2 * Z1); return X1Z2 === X2Z1 && Y1Z2 === Y2Z1; } is0() { return this.equals(Point.ZERO); } negate() { // Flips point sign to a negative one (-x, y in affine coords) return new Point(modP(-this.X), this.Y, this.Z, modP(-this.T)); } // Fast algo for doubling Extended Point. // https://hyperelliptic.org/EFD/g1p/auto-twisted-extended.html#doubling-dbl-2008-hwcd // Cost: 4M + 4S + 1*a + 6add + 1*2. double() { const { a } = CURVE; const { X: X1, Y: Y1, Z: Z1 } = this; const A = modP(X1 * X1); // A = X12 const B = modP(Y1 * Y1); // B = Y12 const C = modP(_2n * modP(Z1 * Z1)); // C = 2*Z12 const D = modP(a * A); // D = a*A const x1y1 = X1 + Y1; const E = modP(modP(x1y1 * x1y1) - A - B); // E = (X1+Y1)2-A-B const G = D + B; // G = D+B const F = G - C; // F = G-C const H = D - B; // H = D-B const X3 = modP(E * F); // X3 = E*F const Y3 = modP(G * H); // Y3 = G*H const T3 = modP(E * H); // T3 = E*H const Z3 = modP(F * G); // Z3 = F*G return new Point(X3, Y3, Z3, T3); } // Fast algo for adding 2 Extended Points. // https://hyperelliptic.org/EFD/g1p/auto-twisted-extended.html#addition-add-2008-hwcd // Cost: 9M + 1*a + 1*d + 7add. add(other) { aextpoint(other); const { a, d } = CURVE; const { X: X1, Y: Y1, Z: Z1, T: T1 } = this; const { X: X2, Y: Y2, Z: Z2, T: T2 } = other; const A = modP(X1 * X2); // A = X1*X2 const B = modP(Y1 * Y2); // B = Y1*Y2 const C = modP(T1 * d * T2); // C = T1*d*T2 const D = modP(Z1 * Z2); // D = Z1*Z2 const E = modP((X1 + Y1) * (X2 + Y2) - A - B); // E = (X1+Y1)*(X2+Y2)-A-B const F = D - C; // F = D-C const G = D + C; // G = D+C const H = modP(B - a * A); // H = B-a*A const X3 = modP(E * F); // X3 = E*F const Y3 = modP(G * H); // Y3 = G*H const T3 = modP(E * H); // T3 = E*H const Z3 = modP(F * G); // Z3 = F*G return new Point(X3, Y3, Z3, T3); } subtract(other) { return this.add(other.negate()); } // Constant-time multiplication. multiply(scalar) { // 1 <= scalar < L if (!Fn.isValidNot0(scalar)) throw new Error('invalid scalar: expected 1 <= sc < curve.n'); const { p, f } = wnaf.cached(this, scalar, (p) => (0, curve_ts_1.normalizeZ)(Point, p)); return (0, curve_ts_1.normalizeZ)(Point, [p, f])[0]; } // Non-constant-time multiplication. Uses double-and-add algorithm. // It's faster, but should only be used when you don't care about // an exposed private key e.g. sig verification. // Does NOT allow scalars higher than CURVE.n. // Accepts optional accumulator to merge with multiply (important for sparse scalars) multiplyUnsafe(scalar, acc = Point.ZERO) { // 0 <= scalar < L if (!Fn.isValid(scalar)) throw new Error('invalid scalar: expected 0 <= sc < curve.n'); if (scalar === _0n) return Point.ZERO; if (this.is0() || scalar === _1n) return this; return wnaf.unsafe(this, scalar, (p) => (0, curve_ts_1.normalizeZ)(Point, p), acc); } // Checks if point is of small order. // If you add something to small order point, you will have "dirty" // point with torsion component. // Multiplies point by cofactor and checks if the result is 0. isSmallOrder() { return this.multiplyUnsafe(cofactor).is0(); } // Multiplies point by curve order and checks if the result is 0. // Returns `false` is the point is dirty. isTorsionFree() { return wnaf.unsafe(this, CURVE.n).is0(); } // Converts Extended point to default (x, y) coordinates. // Can accept precomputed Z^-1 - for example, from invertBatch. toAffine(invertedZ) { return toAffineMemo(this, invertedZ); } clearCofactor() { if (cofactor === _1n) return this; return this.multiplyUnsafe(cofactor); } toBytes() { const { x, y } = this.toAffine(); // Fp.toBytes() allows non-canonical encoding of y (>= p). const bytes = Fp.toBytes(y); // Each y has 2 valid points: (x, y), (x,-y). // When compressing, it's enough to store y and use the last byte to encode sign of x bytes[bytes.length - 1] |= x & _1n ? 0x80 : 0; return bytes; } toHex() { return (0, utils_ts_1.bytesToHex)(this.toBytes()); } toString() { return ``; } // TODO: remove get ex() { return this.X; } get ey() { return this.Y; } get ez() { return this.Z; } get et() { return this.T; } static normalizeZ(points) { return (0, curve_ts_1.normalizeZ)(Point, points); } static msm(points, scalars) { return (0, curve_ts_1.pippenger)(Point, Fn, points, scalars); } _setWindowSize(windowSize) { this.precompute(windowSize); } toRawBytes() { return this.toBytes(); } } // base / generator point Point.BASE = new Point(CURVE.Gx, CURVE.Gy, _1n, modP(CURVE.Gx * CURVE.Gy)); // zero / infinity / identity point Point.ZERO = new Point(_0n, _1n, _1n, _0n); // 0, 1, 1, 0 // math field Point.Fp = Fp; // scalar field Point.Fn = Fn; const wnaf = new curve_ts_1.wNAF(Point, Fn.BITS); Point.BASE.precompute(8); // Enable precomputes. Slows down first publicKey computation by 20ms. return Point; } /** * Base class for prime-order points like Ristretto255 and Decaf448. * These points eliminate cofactor issues by representing equivalence classes * of Edwards curve points. */ class PrimeEdwardsPoint { constructor(ep) { this.ep = ep; } // Static methods that must be implemented by subclasses static fromBytes(_bytes) { (0, utils_ts_1.notImplemented)(); } static fromHex(_hex) { (0, utils_ts_1.notImplemented)(); } get x() { return this.toAffine().x; } get y() { return this.toAffine().y; } // Common implementations clearCofactor() { // no-op for prime-order groups return this; } assertValidity() { this.ep.assertValidity(); } toAffine(invertedZ) { return this.ep.toAffine(invertedZ); } toHex() { return (0, utils_ts_1.bytesToHex)(this.toBytes()); } toString() { return this.toHex(); } isTorsionFree() { return true; } isSmallOrder() { return false; } add(other) { this.assertSame(other); return this.init(this.ep.add(other.ep)); } subtract(other) { this.assertSame(other); return this.init(this.ep.subtract(other.ep)); } multiply(scalar) { return this.init(this.ep.multiply(scalar)); } multiplyUnsafe(scalar) { return this.init(this.ep.multiplyUnsafe(scalar)); } double() { return this.init(this.ep.double()); } negate() { return this.init(this.ep.negate()); } precompute(windowSize, isLazy) { return this.init(this.ep.precompute(windowSize, isLazy)); } /** @deprecated use `toBytes` */ toRawBytes() { return this.toBytes(); } } exports.PrimeEdwardsPoint = PrimeEdwardsPoint; /** * Initializes EdDSA signatures over given Edwards curve. */ function eddsa(Point, cHash, eddsaOpts = {}) { if (typeof cHash !== 'function') throw new Error('"hash" function param is required'); (0, utils_ts_1._validateObject)(eddsaOpts, {}, { adjustScalarBytes: 'function', randomBytes: 'function', domain: 'function', prehash: 'function', mapToCurve: 'function', }); const { prehash } = eddsaOpts; const { BASE, Fp, Fn } = Point; const randomBytes = eddsaOpts.randomBytes || utils_ts_1.randomBytes; const adjustScalarBytes = eddsaOpts.adjustScalarBytes || ((bytes) => bytes); const domain = eddsaOpts.domain || ((data, ctx, phflag) => { (0, utils_ts_1._abool2)(phflag, 'phflag'); if (ctx.length || phflag) throw new Error('Contexts/pre-hash are not supported'); return data; }); // NOOP // Little-endian SHA512 with modulo n function modN_LE(hash) { return Fn.create((0, utils_ts_1.bytesToNumberLE)(hash)); // Not Fn.fromBytes: it has length limit } // Get the hashed private scalar per RFC8032 5.1.5 function getPrivateScalar(key) { const len = lengths.secretKey; key = (0, utils_ts_1.ensureBytes)('private key', key, len); // Hash private key with curve's hash function to produce uniformingly random input // Check byte lengths: ensure(64, h(ensure(32, key))) const hashed = (0, utils_ts_1.ensureBytes)('hashed private key', cHash(key), 2 * len); const head = adjustScalarBytes(hashed.slice(0, len)); // clear first half bits, produce FE const prefix = hashed.slice(len, 2 * len); // second half is called key prefix (5.1.6) const scalar = modN_LE(head); // The actual private scalar return { head, prefix, scalar }; } /** Convenience method that creates public key from scalar. RFC8032 5.1.5 */ function getExtendedPublicKey(secretKey) { const { head, prefix, scalar } = getPrivateScalar(secretKey); const point = BASE.multiply(scalar); // Point on Edwards curve aka public key const pointBytes = point.toBytes(); return { head, prefix, scalar, point, pointBytes }; } /** Calculates EdDSA pub key. RFC8032 5.1.5. */ function getPublicKey(secretKey) { return getExtendedPublicKey(secretKey).pointBytes; } // int('LE', SHA512(dom2(F, C) || msgs)) mod N function hashDomainToScalar(context = Uint8Array.of(), ...msgs) { const msg = (0, utils_ts_1.concatBytes)(...msgs); return modN_LE(cHash(domain(msg, (0, utils_ts_1.ensureBytes)('context', context), !!prehash))); } /** Signs message with privateKey. RFC8032 5.1.6 */ function sign(msg, secretKey, options = {}) { msg = (0, utils_ts_1.ensureBytes)('message', msg); if (prehash) msg = prehash(msg); // for ed25519ph etc. const { prefix, scalar, pointBytes } = getExtendedPublicKey(secretKey); const r = hashDomainToScalar(options.context, prefix, msg); // r = dom2(F, C) || prefix || PH(M) const R = BASE.multiply(r).toBytes(); // R = rG const k = hashDomainToScalar(options.context, R, pointBytes, msg); // R || A || PH(M) const s = Fn.create(r + k * scalar); // S = (r + k * s) mod L if (!Fn.isValid(s)) throw new Error('sign failed: invalid s'); // 0 <= s < L const rs = (0, utils_ts_1.concatBytes)(R, Fn.toBytes(s)); return (0, utils_ts_1._abytes2)(rs, lengths.signature, 'result'); } // verification rule is either zip215 or rfc8032 / nist186-5. Consult fromHex: const verifyOpts = { zip215: true }; /** * Verifies EdDSA signature against message and public key. RFC8032 5.1.7. * An extended group equation is checked. */ function verify(sig, msg, publicKey, options = verifyOpts) { const { context, zip215 } = options; const len = lengths.signature; sig = (0, utils_ts_1.ensureBytes)('signature', sig, len); msg = (0, utils_ts_1.ensureBytes)('message', msg); publicKey = (0, utils_ts_1.ensureBytes)('publicKey', publicKey, lengths.publicKey); if (zip215 !== undefined) (0, utils_ts_1._abool2)(zip215, 'zip215'); if (prehash) msg = prehash(msg); // for ed25519ph, etc const mid = len / 2; const r = sig.subarray(0, mid); const s = (0, utils_ts_1.bytesToNumberLE)(sig.subarray(mid, len)); let A, R, SB; try { // zip215=true is good for consensus-critical apps. =false follows RFC8032 / NIST186-5. // zip215=true: 0 <= y < MASK (2^256 for ed25519) // zip215=false: 0 <= y < P (2^255-19 for ed25519) A = Point.fromBytes(publicKey, zip215); R = Point.fromBytes(r, zip215); SB = BASE.multiplyUnsafe(s); // 0 <= s < l is done inside } catch (error) { return false; } if (!zip215 && A.isSmallOrder()) return false; // zip215 allows public keys of small order const k = hashDomainToScalar(context, R.toBytes(), A.toBytes(), msg); const RkA = R.add(A.multiplyUnsafe(k)); // Extended group equation // [8][S]B = [8]R + [8][k]A' return RkA.subtract(SB).clearCofactor().is0(); } const _size = Fp.BYTES; // 32 for ed25519, 57 for ed448 const lengths = { secretKey: _size, publicKey: _size, signature: 2 * _size, seed: _size, }; function randomSecretKey(seed = randomBytes(lengths.seed)) { return (0, utils_ts_1._abytes2)(seed, lengths.seed, 'seed'); } function keygen(seed) { const secretKey = utils.randomSecretKey(seed); return { secretKey, publicKey: getPublicKey(secretKey) }; } function isValidSecretKey(key) { return (0, utils_ts_1.isBytes)(key) && key.length === Fn.BYTES; } function isValidPublicKey(key, zip215) { try { return !!Point.fromBytes(key, zip215); } catch (error) { return false; } } const utils = { getExtendedPublicKey, randomSecretKey, isValidSecretKey, isValidPublicKey, /** * Converts ed public key to x public key. Uses formula: * - ed25519: * - `(u, v) = ((1+y)/(1-y), sqrt(-486664)*u/x)` * - `(x, y) = (sqrt(-486664)*u/v, (u-1)/(u+1))` * - ed448: * - `(u, v) = ((y-1)/(y+1), sqrt(156324)*u/x)` * - `(x, y) = (sqrt(156324)*u/v, (1+u)/(1-u))` */ toMontgomery(publicKey) { const { y } = Point.fromBytes(publicKey); const size = lengths.publicKey; const is25519 = size === 32; if (!is25519 && size !== 57) throw new Error('only defined for 25519 and 448'); const u = is25519 ? Fp.div(_1n + y, _1n - y) : Fp.div(y - _1n, y + _1n); return Fp.toBytes(u); }, toMontgomerySecret(secretKey) { const size = lengths.secretKey; (0, utils_ts_1._abytes2)(secretKey, size); const hashed = cHash(secretKey.subarray(0, size)); return adjustScalarBytes(hashed).subarray(0, size); }, /** @deprecated */ randomPrivateKey: randomSecretKey, /** @deprecated */ precompute(windowSize = 8, point = Point.BASE) { return point.precompute(windowSize, false); }, }; return Object.freeze({ keygen, getPublicKey, sign, verify, utils, Point, lengths, }); } function _eddsa_legacy_opts_to_new(c) { const CURVE = { a: c.a, d: c.d, p: c.Fp.ORDER, n: c.n, h: c.h, Gx: c.Gx, Gy: c.Gy, }; const Fp = c.Fp; const Fn = (0, modular_ts_1.Field)(CURVE.n, c.nBitLength, true); const curveOpts = { Fp, Fn, uvRatio: c.uvRatio }; const eddsaOpts = { randomBytes: c.randomBytes, adjustScalarBytes: c.adjustScalarBytes, domain: c.domain, prehash: c.prehash, mapToCurve: c.mapToCurve, }; return { CURVE, curveOpts, hash: c.hash, eddsaOpts }; } function _eddsa_new_output_to_legacy(c, eddsa) { const Point = eddsa.Point; const legacy = Object.assign({}, eddsa, { ExtendedPoint: Point, CURVE: c, nBitLength: Point.Fn.BITS, nByteLength: Point.Fn.BYTES, }); return legacy; } // TODO: remove. Use eddsa function twistedEdwards(c) { const { CURVE, curveOpts, hash, eddsaOpts } = _eddsa_legacy_opts_to_new(c); const Point = edwards(CURVE, curveOpts); const EDDSA = eddsa(Point, hash, eddsaOpts); return _eddsa_new_output_to_legacy(c, EDDSA); } //# sourceMappingURL=edwards.js.map