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/**
* Twisted Edwards curve. The formula is: ax² + y² = 1 + dx²y².
* For design rationale of types / exports, see weierstrass module documentation.
* @module
*/
/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import {
type AffinePoint,
type BasicCurve,
type Group,
type GroupConstructor,
pippenger,
validateBasic,
wNAF,
} from './curve.js';
import { Field, mod } from './modular.js';
import * as ut from './utils.js';
import { abool, ensureBytes, type FHash, type Hex, memoized } from './utils.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);
/** Edwards curves must declare params a & d. */
export type CurveType = BasicCurve<bigint> & {
a: bigint; // curve param a
d: bigint; // curve param d
hash: FHash; // Hashing
randomBytes: (bytesLength?: number) => Uint8Array; // CSPRNG
adjustScalarBytes?: (bytes: Uint8Array) => Uint8Array; // clears bits to get valid field elemtn
domain?: (data: Uint8Array, ctx: Uint8Array, phflag: boolean) => Uint8Array; // Used for hashing
uvRatio?: (u: bigint, v: bigint) => { isValid: boolean; value: bigint }; // Ratio √(u/v)
prehash?: FHash; // RFC 8032 pre-hashing of messages to sign() / verify()
mapToCurve?: (scalar: bigint[]) => AffinePoint<bigint>; // for hash-to-curve standard
};
export type CurveTypeWithLength = Readonly<CurveType & { nByteLength: number; nBitLength: number }>;
// verification rule is either zip215 or rfc8032 / nist186-5. Consult fromHex:
const VERIFY_DEFAULT = { zip215: true };
function validateOpts(curve: CurveType): CurveTypeWithLength {
const opts = validateBasic(curve);
ut.validateObject(
curve,
{
hash: 'function',
a: 'bigint',
d: 'bigint',
randomBytes: 'function',
},
{
adjustScalarBytes: 'function',
domain: 'function',
uvRatio: 'function',
mapToCurve: 'function',
}
);
// Set defaults
return Object.freeze({ ...opts } as const);
}
/** Instance of Extended Point with coordinates in X, Y, Z, T. */
export interface ExtPointType extends Group<ExtPointType> {
readonly ex: bigint;
readonly ey: bigint;
readonly ez: bigint;
readonly et: bigint;
get x(): bigint;
get y(): bigint;
assertValidity(): void;
multiply(scalar: bigint): ExtPointType;
multiplyUnsafe(scalar: bigint): ExtPointType;
isSmallOrder(): boolean;
isTorsionFree(): boolean;
clearCofactor(): ExtPointType;
toAffine(iz?: bigint): AffinePoint<bigint>;
toRawBytes(isCompressed?: boolean): Uint8Array;
toHex(isCompressed?: boolean): string;
_setWindowSize(windowSize: number): void;
}
/** Static methods of Extended Point with coordinates in X, Y, Z, T. */
export interface ExtPointConstructor extends GroupConstructor<ExtPointType> {
new (x: bigint, y: bigint, z: bigint, t: bigint): ExtPointType;
fromAffine(p: AffinePoint<bigint>): ExtPointType;
fromHex(hex: Hex): ExtPointType;
fromPrivateKey(privateKey: Hex): ExtPointType;
msm(points: ExtPointType[], scalars: bigint[]): ExtPointType;
}
/**
* Edwards Curve interface.
* Main methods: `getPublicKey(priv)`, `sign(msg, priv)`, `verify(sig, msg, pub)`.
*/
export type CurveFn = {
CURVE: ReturnType<typeof validateOpts>;
getPublicKey: (privateKey: Hex) => Uint8Array;
sign: (message: Hex, privateKey: Hex, options?: { context?: Hex }) => Uint8Array;
verify: (
sig: Hex,
message: Hex,
publicKey: Hex,
options?: { context?: Hex; zip215: boolean }
) => boolean;
ExtendedPoint: ExtPointConstructor;
utils: {
randomPrivateKey: () => Uint8Array;
getExtendedPublicKey: (key: Hex) => {
head: Uint8Array;
prefix: Uint8Array;
scalar: bigint;
point: ExtPointType;
pointBytes: Uint8Array;
};
precompute: (windowSize?: number, point?: ExtPointType) => ExtPointType;
};
};
/**
* Creates Twisted Edwards curve with EdDSA signatures.
* @example
* import { Field } from '@noble/curves/abstract/modular';
* // Before that, define BigInt-s: a, d, p, n, Gx, Gy, h
* const curve = twistedEdwards({ a, d, Fp: Field(p), n, Gx, Gy, h })
*/
export function twistedEdwards(curveDef: CurveType): CurveFn {
const CURVE = validateOpts(curveDef) as ReturnType<typeof validateOpts>;
const {
Fp,
n: CURVE_ORDER,
prehash: prehash,
hash: cHash,
randomBytes,
nByteLength,
h: cofactor,
} = CURVE;
// 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(nByteLength * 8) - _1n);
const modP = Fp.create; // Function overrides
const Fn = Field(CURVE.n, CURVE.nBitLength);
// sqrt(u/v)
const uvRatio =
CURVE.uvRatio ||
((u: bigint, v: bigint) => {
try {
return { isValid: true, value: Fp.sqrt(u * Fp.inv(v)) };
} catch (e) {
return { isValid: false, value: _0n };
}
});
const adjustScalarBytes = CURVE.adjustScalarBytes || ((bytes: Uint8Array) => bytes); // NOOP
const domain =
CURVE.domain ||
((data: Uint8Array, ctx: Uint8Array, phflag: boolean) => {
abool('phflag', phflag);
if (ctx.length || phflag) throw new Error('Contexts/pre-hash are not supported');
return data;
}); // NOOP
// 0 <= n < MASK
// Coordinates larger than Fp.ORDER are allowed for zip215
function aCoordinate(title: string, n: bigint) {
ut.aInRange('coordinate ' + title, n, _0n, MASK);
}
function assertPoint(other: unknown) {
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 = memoized((p: Point, iz?: bigint): AffinePoint<bigint> => {
const { ex: x, ey: y, ez: z } = p;
const is0 = p.is0();
if (iz == null) iz = is0 ? _8n : (Fp.inv(z) as bigint); // 8 was chosen arbitrarily
const ax = modP(x * iz);
const ay = modP(y * iz);
const zz = modP(z * iz);
if (is0) return { x: _0n, y: _1n };
if (zz !== _1n) throw new Error('invZ was invalid');
return { x: ax, y: ay };
});
const assertValidMemo = memoized((p: Point) => {
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 { ex: X, ey: Y, ez: Z, et: 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 implements ExtPointType {
static readonly BASE = new Point(CURVE.Gx, CURVE.Gy, _1n, modP(CURVE.Gx * CURVE.Gy));
static readonly ZERO = new Point(_0n, _1n, _1n, _0n); // 0, 1, 1, 0
constructor(
readonly ex: bigint,
readonly ey: bigint,
readonly ez: bigint,
readonly et: bigint
) {
aCoordinate('x', ex);
aCoordinate('y', ey);
aCoordinate('z', ez);
aCoordinate('t', et);
Object.freeze(this);
}
get x(): bigint {
return this.toAffine().x;
}
get y(): bigint {
return this.toAffine().y;
}
static fromAffine(p: AffinePoint<bigint>): Point {
if (p instanceof Point) throw new Error('extended point not allowed');
const { x, y } = p || {};
aCoordinate('x', x);
aCoordinate('y', y);
return new Point(x, y, _1n, modP(x * y));
}
static normalizeZ(points: Point[]): Point[] {
const toInv = Fp.invertBatch(points.map((p) => p.ez));
return points.map((p, i) => p.toAffine(toInv[i])).map(Point.fromAffine);
}
// Multiscalar Multiplication
static msm(points: Point[], scalars: bigint[]): Point {
return pippenger(Point, Fn, points, scalars);
}
// "Private method", don't use it directly
_setWindowSize(windowSize: number) {
wnaf.setWindowSize(this, windowSize);
}
// Not required for fromHex(), which always creates valid points.
// Could be useful for fromAffine().
assertValidity(): void {
assertValidMemo(this);
}
// Compare one point to another.
equals(other: Point): boolean {
assertPoint(other);
const { ex: X1, ey: Y1, ez: Z1 } = this;
const { ex: X2, ey: Y2, ez: 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(): boolean {
return this.equals(Point.ZERO);
}
negate(): Point {
// Flips point sign to a negative one (-x, y in affine coords)
return new Point(modP(-this.ex), this.ey, this.ez, modP(-this.et));
}
// 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(): Point {
const { a } = CURVE;
const { ex: X1, ey: Y1, ez: 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: Point) {
assertPoint(other);
const { a, d } = CURVE;
const { ex: X1, ey: Y1, ez: Z1, et: T1 } = this;
const { ex: X2, ey: Y2, ez: Z2, et: T2 } = other;
// Faster algo for adding 2 Extended Points when curve's a=-1.
// http://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-4
// Cost: 8M + 8add + 2*2.
// Note: It does not check whether the `other` point is valid.
if (a === BigInt(-1)) {
const A = modP((Y1 - X1) * (Y2 + X2));
const B = modP((Y1 + X1) * (Y2 - X2));
const F = modP(B - A);
if (F === _0n) return this.double(); // Same point. Tests say it doesn't affect timing
const C = modP(Z1 * _2n * T2);
const D = modP(T1 * _2n * Z2);
const E = D + C;
const G = B + A;
const H = D - C;
const X3 = modP(E * F);
const Y3 = modP(G * H);
const T3 = modP(E * H);
const Z3 = modP(F * G);
return new Point(X3, Y3, Z3, T3);
}
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: Point): Point {
return this.add(other.negate());
}
private wNAF(n: bigint): { p: Point; f: Point } {
return wnaf.wNAFCached(this, n, Point.normalizeZ);
}
// Constant-time multiplication.
multiply(scalar: bigint): Point {
const n = scalar;
ut.aInRange('scalar', n, _1n, CURVE_ORDER); // 1 <= scalar < L
const { p, f } = this.wNAF(n);
return Point.normalizeZ([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: bigint, acc = Point.ZERO): Point {
const n = scalar;
ut.aInRange('scalar', n, _0n, CURVE_ORDER); // 0 <= scalar < L
if (n === _0n) return I;
if (this.is0() || n === _1n) return this;
return wnaf.wNAFCachedUnsafe(this, n, Point.normalizeZ, 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(): boolean {
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(): boolean {
return wnaf.unsafeLadder(this, CURVE_ORDER).is0();
}
// Converts Extended point to default (x, y) coordinates.
// Can accept precomputed Z^-1 - for example, from invertBatch.
toAffine(iz?: bigint): AffinePoint<bigint> {
return toAffineMemo(this, iz);
}
clearCofactor(): Point {
const { h: cofactor } = CURVE;
if (cofactor === _1n) return this;
return this.multiplyUnsafe(cofactor);
}
// Converts hash string or Uint8Array to Point.
// Uses algo from RFC8032 5.1.3.
static fromHex(hex: Hex, zip215 = false): Point {
const { d, a } = CURVE;
const len = Fp.BYTES;
hex = ensureBytes('pointHex', hex, len); // copy hex to a new array
abool('zip215', zip215);
const normed = hex.slice(); // copy again, we'll manipulate it
const lastByte = hex[len - 1]; // select last byte
normed[len - 1] = lastByte & ~0x80; // clear last bit
const y = ut.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;
ut.aInRange('pointHex.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('Point.fromHex: 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('Point.fromHex: 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 fromPrivateKey(privKey: Hex) {
return getExtendedPublicKey(privKey).point;
}
toRawBytes(): Uint8Array {
const { x, y } = this.toAffine();
const bytes = ut.numberToBytesLE(y, Fp.BYTES); // each y has 2 x values (x, -y)
bytes[bytes.length - 1] |= x & _1n ? 0x80 : 0; // when compressing, it's enough to store y
return bytes; // and use the last byte to encode sign of x
}
toHex(): string {
return ut.bytesToHex(this.toRawBytes()); // Same as toRawBytes, but returns string.
}
}
const { BASE: G, ZERO: I } = Point;
const wnaf = wNAF(Point, nByteLength * 8);
function modN(a: bigint) {
return mod(a, CURVE_ORDER);
}
// Little-endian SHA512 with modulo n
function modN_LE(hash: Uint8Array): bigint {
return modN(ut.bytesToNumberLE(hash));
}
/** Convenience method that creates public key and other stuff. RFC8032 5.1.5 */
function getExtendedPublicKey(key: Hex) {
const len = Fp.BYTES;
key = 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 = 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
const point = G.multiply(scalar); // Point on Edwards curve aka public key
const pointBytes = point.toRawBytes(); // Uint8Array representation
return { head, prefix, scalar, point, pointBytes };
}
// Calculates EdDSA pub key. RFC8032 5.1.5. Privkey is hashed. Use first half with 3 bits cleared
function getPublicKey(privKey: Hex): Uint8Array {
return getExtendedPublicKey(privKey).pointBytes;
}
// int('LE', SHA512(dom2(F, C) || msgs)) mod N
function hashDomainToScalar(context: Hex = new Uint8Array(), ...msgs: Uint8Array[]) {
const msg = ut.concatBytes(...msgs);
return modN_LE(cHash(domain(msg, ensureBytes('context', context), !!prehash)));
}
/** Signs message with privateKey. RFC8032 5.1.6 */
function sign(msg: Hex, privKey: Hex, options: { context?: Hex } = {}): Uint8Array {
msg = ensureBytes('message', msg);
if (prehash) msg = prehash(msg); // for ed25519ph etc.
const { prefix, scalar, pointBytes } = getExtendedPublicKey(privKey);
const r = hashDomainToScalar(options.context, prefix, msg); // r = dom2(F, C) || prefix || PH(M)
const R = G.multiply(r).toRawBytes(); // R = rG
const k = hashDomainToScalar(options.context, R, pointBytes, msg); // R || A || PH(M)
const s = modN(r + k * scalar); // S = (r + k * s) mod L
ut.aInRange('signature.s', s, _0n, CURVE_ORDER); // 0 <= s < l
const res = ut.concatBytes(R, ut.numberToBytesLE(s, Fp.BYTES));
return ensureBytes('result', res, Fp.BYTES * 2); // 64-byte signature
}
const verifyOpts: { context?: Hex; zip215?: boolean } = VERIFY_DEFAULT;
/**
* Verifies EdDSA signature against message and public key. RFC8032 5.1.7.
* An extended group equation is checked.
*/
function verify(sig: Hex, msg: Hex, publicKey: Hex, options = verifyOpts): boolean {
const { context, zip215 } = options;
const len = Fp.BYTES; // Verifies EdDSA signature against message and public key. RFC8032 5.1.7.
sig = ensureBytes('signature', sig, 2 * len); // An extended group equation is checked.
msg = ensureBytes('message', msg);
publicKey = ensureBytes('publicKey', publicKey, len);
if (zip215 !== undefined) abool('zip215', zip215);
if (prehash) msg = prehash(msg); // for ed25519ph, etc
const s = ut.bytesToNumberLE(sig.slice(len, 2 * 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.fromHex(publicKey, zip215);
R = Point.fromHex(sig.slice(0, len), zip215);
SB = G.multiplyUnsafe(s); // 0 <= s < l is done inside
} catch (error) {
return false;
}
if (!zip215 && A.isSmallOrder()) return false;
const k = hashDomainToScalar(context, R.toRawBytes(), A.toRawBytes(), msg);
const RkA = R.add(A.multiplyUnsafe(k));
// Extended group equation
// [8][S]B = [8]R + [8][k]A'
return RkA.subtract(SB).clearCofactor().equals(Point.ZERO);
}
G._setWindowSize(8); // Enable precomputes. Slows down first publicKey computation by 20ms.
const utils = {
getExtendedPublicKey,
// ed25519 private keys are uniform 32b. No need to check for modulo bias, like in secp256k1.
randomPrivateKey: (): Uint8Array => randomBytes(Fp.BYTES),
/**
* We're doing scalar multiplication (used in getPublicKey etc) with precomputed BASE_POINT
* values. This slows down first getPublicKey() by milliseconds (see Speed section),
* but allows to speed-up subsequent getPublicKey() calls up to 20x.
* @param windowSize 2, 4, 8, 16
*/
precompute(windowSize = 8, point: ExtPointType = Point.BASE): ExtPointType {
point._setWindowSize(windowSize);
point.multiply(BigInt(3));
return point;
},
};
return {
CURVE,
getPublicKey,
sign,
verify,
ExtendedPoint: Point,
utils,
};
}
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