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2025-09-29 07:21:46 -04:00
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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { hmac } from '@noble/hashes/hmac';
import { concatBytes, randomBytes } from '@noble/hashes/utils';
import { weierstrass, CurveType } from './abstract/weierstrass.js';
import { CHash } from './abstract/utils.js';
// connects noble-curves to noble-hashes
export function getHash(hash: CHash) {
return {
hash,
hmac: (key: Uint8Array, ...msgs: Uint8Array[]) => hmac(hash, key, concatBytes(...msgs)),
randomBytes,
};
}
// Same API as @noble/hashes, with ability to create curve with custom hash
type CurveDef = Readonly<Omit<CurveType, 'hash' | 'hmac' | 'randomBytes'>>;
export function createCurve(curveDef: CurveDef, defHash: CHash) {
const create = (hash: CHash) => weierstrass({ ...curveDef, ...getHash(hash) });
return Object.freeze({ ...create(defHash), create });
}

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
/**
* BLS (Barreto-Lynn-Scott) family of pairing-friendly curves.
* Implements BLS (Boneh-Lynn-Shacham) signatures.
* Consists of two curves: G1 and G2:
* - G1 is a subgroup of (x, y) E(Fq) over y² = x³ + 4.
* - G2 is a subgroup of ((x₁, x₂+i), (y₁, y₂+i)) E(Fq²) over y² = x³ + 4(1 + i) where i is √-1
* - Gt, created by bilinear (ate) pairing e(G1, G2), consists of p-th roots of unity in
* Fq^k where k is embedding degree. Only degree 12 is currently supported, 24 is not.
* Pairing is used to aggregate and verify signatures.
* We are using Fp for private keys (shorter) and Fp₂ for signatures (longer).
* Some projects may prefer to swap this relation, it is not supported for now.
*/
import { AffinePoint } from './curve.js';
import { IField, getMinHashLength, mapHashToField } from './modular.js';
import { Hex, PrivKey, CHash, bitLen, bitGet, ensureBytes } from './utils.js';
import * as htf from './hash-to-curve.js';
import {
CurvePointsType,
ProjPointType as ProjPointType,
CurvePointsRes,
weierstrassPoints,
} from './weierstrass.js';
type Fp = bigint; // Can be different field?
// prettier-ignore
const _2n = BigInt(2), _3n = BigInt(3);
export type SignatureCoder<Fp2> = {
fromHex(hex: Hex): ProjPointType<Fp2>;
toRawBytes(point: ProjPointType<Fp2>): Uint8Array;
toHex(point: ProjPointType<Fp2>): string;
};
export type CurveType<Fp, Fp2, Fp6, Fp12> = {
G1: Omit<CurvePointsType<Fp>, 'n'> & {
mapToCurve: htf.MapToCurve<Fp>;
htfDefaults: htf.Opts;
};
G2: Omit<CurvePointsType<Fp2>, 'n'> & {
Signature: SignatureCoder<Fp2>;
mapToCurve: htf.MapToCurve<Fp2>;
htfDefaults: htf.Opts;
};
fields: {
Fp: IField<Fp>;
Fr: IField<bigint>;
Fp2: IField<Fp2> & {
reim: (num: Fp2) => { re: bigint; im: bigint };
multiplyByB: (num: Fp2) => Fp2;
frobeniusMap(num: Fp2, power: number): Fp2;
};
Fp6: IField<Fp6>;
Fp12: IField<Fp12> & {
frobeniusMap(num: Fp12, power: number): Fp12;
multiplyBy014(num: Fp12, o0: Fp2, o1: Fp2, o4: Fp2): Fp12;
conjugate(num: Fp12): Fp12;
finalExponentiate(num: Fp12): Fp12;
};
};
params: {
x: bigint;
r: bigint;
};
htfDefaults: htf.Opts;
hash: CHash; // Because we need outputLen for DRBG
randomBytes: (bytesLength?: number) => Uint8Array;
};
export type CurveFn<Fp, Fp2, Fp6, Fp12> = {
getPublicKey: (privateKey: PrivKey) => Uint8Array;
sign: {
(message: Hex, privateKey: PrivKey): Uint8Array;
(message: ProjPointType<Fp2>, privateKey: PrivKey): ProjPointType<Fp2>;
};
verify: (
signature: Hex | ProjPointType<Fp2>,
message: Hex | ProjPointType<Fp2>,
publicKey: Hex | ProjPointType<Fp>
) => boolean;
verifyBatch: (
signature: Hex | ProjPointType<Fp2>,
messages: (Hex | ProjPointType<Fp2>)[],
publicKeys: (Hex | ProjPointType<Fp>)[]
) => boolean;
aggregatePublicKeys: {
(publicKeys: Hex[]): Uint8Array;
(publicKeys: ProjPointType<Fp>[]): ProjPointType<Fp>;
};
aggregateSignatures: {
(signatures: Hex[]): Uint8Array;
(signatures: ProjPointType<Fp2>[]): ProjPointType<Fp2>;
};
millerLoop: (ell: [Fp2, Fp2, Fp2][], g1: [Fp, Fp]) => Fp12;
pairing: (P: ProjPointType<Fp>, Q: ProjPointType<Fp2>, withFinalExponent?: boolean) => Fp12;
G1: CurvePointsRes<Fp> & ReturnType<typeof htf.createHasher<Fp>>;
G2: CurvePointsRes<Fp2> & ReturnType<typeof htf.createHasher<Fp2>>;
Signature: SignatureCoder<Fp2>;
params: {
x: bigint;
r: bigint;
G1b: bigint;
G2b: Fp2;
};
fields: {
Fp: IField<Fp>;
Fp2: IField<Fp2>;
Fp6: IField<Fp6>;
Fp12: IField<Fp12>;
Fr: IField<bigint>;
};
utils: {
randomPrivateKey: () => Uint8Array;
calcPairingPrecomputes: (p: AffinePoint<Fp2>) => [Fp2, Fp2, Fp2][];
};
};
export function bls<Fp2, Fp6, Fp12>(
CURVE: CurveType<Fp, Fp2, Fp6, Fp12>
): CurveFn<Fp, Fp2, Fp6, Fp12> {
// Fields are specific for curve, so for now we'll need to pass them with opts
const { Fp, Fr, Fp2, Fp6, Fp12 } = CURVE.fields;
const BLS_X_LEN = bitLen(CURVE.params.x);
// Pre-compute coefficients for sparse multiplication
// Point addition and point double calculations is reused for coefficients
function calcPairingPrecomputes(p: AffinePoint<Fp2>) {
const { x, y } = p;
// prettier-ignore
const Qx = x, Qy = y, Qz = Fp2.ONE;
// prettier-ignore
let Rx = Qx, Ry = Qy, Rz = Qz;
let ell_coeff: [Fp2, Fp2, Fp2][] = [];
for (let i = BLS_X_LEN - 2; i >= 0; i--) {
// Double
let t0 = Fp2.sqr(Ry); // Ry²
let t1 = Fp2.sqr(Rz); // Rz²
let t2 = Fp2.multiplyByB(Fp2.mul(t1, _3n)); // 3 * T1 * B
let t3 = Fp2.mul(t2, _3n); // 3 * T2
let t4 = Fp2.sub(Fp2.sub(Fp2.sqr(Fp2.add(Ry, Rz)), t1), t0); // (Ry + Rz)² - T1 - T0
ell_coeff.push([
Fp2.sub(t2, t0), // T2 - T0
Fp2.mul(Fp2.sqr(Rx), _3n), // 3 * Rx²
Fp2.neg(t4), // -T4
]);
Rx = Fp2.div(Fp2.mul(Fp2.mul(Fp2.sub(t0, t3), Rx), Ry), _2n); // ((T0 - T3) * Rx * Ry) / 2
Ry = Fp2.sub(Fp2.sqr(Fp2.div(Fp2.add(t0, t3), _2n)), Fp2.mul(Fp2.sqr(t2), _3n)); // ((T0 + T3) / 2)² - 3 * T2²
Rz = Fp2.mul(t0, t4); // T0 * T4
if (bitGet(CURVE.params.x, i)) {
// Addition
let t0 = Fp2.sub(Ry, Fp2.mul(Qy, Rz)); // Ry - Qy * Rz
let t1 = Fp2.sub(Rx, Fp2.mul(Qx, Rz)); // Rx - Qx * Rz
ell_coeff.push([
Fp2.sub(Fp2.mul(t0, Qx), Fp2.mul(t1, Qy)), // T0 * Qx - T1 * Qy
Fp2.neg(t0), // -T0
t1, // T1
]);
let t2 = Fp2.sqr(t1); // T1²
let t3 = Fp2.mul(t2, t1); // T2 * T1
let t4 = Fp2.mul(t2, Rx); // T2 * Rx
let t5 = Fp2.add(Fp2.sub(t3, Fp2.mul(t4, _2n)), Fp2.mul(Fp2.sqr(t0), Rz)); // T3 - 2 * T4 + T0² * Rz
Rx = Fp2.mul(t1, t5); // T1 * T5
Ry = Fp2.sub(Fp2.mul(Fp2.sub(t4, t5), t0), Fp2.mul(t3, Ry)); // (T4 - T5) * T0 - T3 * Ry
Rz = Fp2.mul(Rz, t3); // Rz * T3
}
}
return ell_coeff;
}
function millerLoop(ell: [Fp2, Fp2, Fp2][], g1: [Fp, Fp]): Fp12 {
const { x } = CURVE.params;
const Px = g1[0];
const Py = g1[1];
let f12 = Fp12.ONE;
for (let j = 0, i = BLS_X_LEN - 2; i >= 0; i--, j++) {
const E = ell[j];
f12 = Fp12.multiplyBy014(f12, E[0], Fp2.mul(E[1], Px), Fp2.mul(E[2], Py));
if (bitGet(x, i)) {
j += 1;
const F = ell[j];
f12 = Fp12.multiplyBy014(f12, F[0], Fp2.mul(F[1], Px), Fp2.mul(F[2], Py));
}
if (i !== 0) f12 = Fp12.sqr(f12);
}
return Fp12.conjugate(f12);
}
const utils = {
randomPrivateKey: (): Uint8Array => {
const length = getMinHashLength(Fr.ORDER);
return mapHashToField(CURVE.randomBytes(length), Fr.ORDER);
},
calcPairingPrecomputes,
};
// Point on G1 curve: (x, y)
const G1_ = weierstrassPoints({ n: Fr.ORDER, ...CURVE.G1 });
const G1 = Object.assign(
G1_,
htf.createHasher(G1_.ProjectivePoint, CURVE.G1.mapToCurve, {
...CURVE.htfDefaults,
...CURVE.G1.htfDefaults,
})
);
// Sparse multiplication against precomputed coefficients
// TODO: replace with weakmap?
type withPairingPrecomputes = { _PPRECOMPUTES: [Fp2, Fp2, Fp2][] | undefined };
function pairingPrecomputes(point: G2): [Fp2, Fp2, Fp2][] {
const p = point as G2 & withPairingPrecomputes;
if (p._PPRECOMPUTES) return p._PPRECOMPUTES;
p._PPRECOMPUTES = calcPairingPrecomputes(point.toAffine());
return p._PPRECOMPUTES;
}
// TODO: export
// function clearPairingPrecomputes(point: G2) {
// const p = point as G2 & withPairingPrecomputes;
// p._PPRECOMPUTES = undefined;
// }
// Point on G2 curve (complex numbers): (x₁, x₂+i), (y₁, y₂+i)
const G2_ = weierstrassPoints({ n: Fr.ORDER, ...CURVE.G2 });
const G2 = Object.assign(
G2_,
htf.createHasher(G2_.ProjectivePoint as htf.H2CPointConstructor<Fp2>, CURVE.G2.mapToCurve, {
...CURVE.htfDefaults,
...CURVE.G2.htfDefaults,
})
);
const { Signature } = CURVE.G2;
// Calculates bilinear pairing
function pairing(Q: G1, P: G2, withFinalExponent: boolean = true): Fp12 {
if (Q.equals(G1.ProjectivePoint.ZERO) || P.equals(G2.ProjectivePoint.ZERO))
throw new Error('pairing is not available for ZERO point');
Q.assertValidity();
P.assertValidity();
// Performance: 9ms for millerLoop and ~14ms for exp.
const Qa = Q.toAffine();
const looped = millerLoop(pairingPrecomputes(P), [Qa.x, Qa.y]);
return withFinalExponent ? Fp12.finalExponentiate(looped) : looped;
}
type G1 = typeof G1.ProjectivePoint.BASE;
type G2 = typeof G2.ProjectivePoint.BASE;
type G1Hex = Hex | G1;
type G2Hex = Hex | G2;
function normP1(point: G1Hex): G1 {
return point instanceof G1.ProjectivePoint ? (point as G1) : G1.ProjectivePoint.fromHex(point);
}
function normP2(point: G2Hex): G2 {
return point instanceof G2.ProjectivePoint ? point : Signature.fromHex(point);
}
function normP2Hash(point: G2Hex, htfOpts?: htf.htfBasicOpts): G2 {
return point instanceof G2.ProjectivePoint
? point
: (G2.hashToCurve(ensureBytes('point', point), htfOpts) as G2);
}
// Multiplies generator by private key.
// P = pk x G
function getPublicKey(privateKey: PrivKey): Uint8Array {
return G1.ProjectivePoint.fromPrivateKey(privateKey).toRawBytes(true);
}
// Executes `hashToCurve` on the message and then multiplies the result by private key.
// S = pk x H(m)
function sign(message: Hex, privateKey: PrivKey, htfOpts?: htf.htfBasicOpts): Uint8Array;
function sign(message: G2, privateKey: PrivKey, htfOpts?: htf.htfBasicOpts): G2;
function sign(message: G2Hex, privateKey: PrivKey, htfOpts?: htf.htfBasicOpts): Uint8Array | G2 {
const msgPoint = normP2Hash(message, htfOpts);
msgPoint.assertValidity();
const sigPoint = msgPoint.multiply(G1.normPrivateKeyToScalar(privateKey));
if (message instanceof G2.ProjectivePoint) return sigPoint;
return Signature.toRawBytes(sigPoint);
}
// Checks if pairing of public key & hash is equal to pairing of generator & signature.
// e(P, H(m)) == e(G, S)
function verify(
signature: G2Hex,
message: G2Hex,
publicKey: G1Hex,
htfOpts?: htf.htfBasicOpts
): boolean {
const P = normP1(publicKey);
const Hm = normP2Hash(message, htfOpts);
const G = G1.ProjectivePoint.BASE;
const S = normP2(signature);
// Instead of doing 2 exponentiations, we use property of billinear maps
// and do one exp after multiplying 2 points.
const ePHm = pairing(P.negate(), Hm, false);
const eGS = pairing(G, S, false);
const exp = Fp12.finalExponentiate(Fp12.mul(eGS, ePHm));
return Fp12.eql(exp, Fp12.ONE);
}
// Adds a bunch of public key points together.
// pk1 + pk2 + pk3 = pkA
function aggregatePublicKeys(publicKeys: Hex[]): Uint8Array;
function aggregatePublicKeys(publicKeys: G1[]): G1;
function aggregatePublicKeys(publicKeys: G1Hex[]): Uint8Array | G1 {
if (!publicKeys.length) throw new Error('Expected non-empty array');
const agg = publicKeys.map(normP1).reduce((sum, p) => sum.add(p), G1.ProjectivePoint.ZERO);
const aggAffine = agg; //.toAffine();
if (publicKeys[0] instanceof G1.ProjectivePoint) {
aggAffine.assertValidity();
return aggAffine;
}
// toRawBytes ensures point validity
return aggAffine.toRawBytes(true);
}
// Adds a bunch of signature points together.
function aggregateSignatures(signatures: Hex[]): Uint8Array;
function aggregateSignatures(signatures: G2[]): G2;
function aggregateSignatures(signatures: G2Hex[]): Uint8Array | G2 {
if (!signatures.length) throw new Error('Expected non-empty array');
const agg = signatures.map(normP2).reduce((sum, s) => sum.add(s), G2.ProjectivePoint.ZERO);
const aggAffine = agg; //.toAffine();
if (signatures[0] instanceof G2.ProjectivePoint) {
aggAffine.assertValidity();
return aggAffine;
}
return Signature.toRawBytes(aggAffine);
}
// https://ethresear.ch/t/fast-verification-of-multiple-bls-signatures/5407
// e(G, S) = e(G, SUM(n)(Si)) = MUL(n)(e(G, Si))
function verifyBatch(
signature: G2Hex,
messages: G2Hex[],
publicKeys: G1Hex[],
htfOpts?: htf.htfBasicOpts
): boolean {
// @ts-ignore
// console.log('verifyBatch', bytesToHex(signature as any), messages, publicKeys.map(bytesToHex));
if (!messages.length) throw new Error('Expected non-empty messages array');
if (publicKeys.length !== messages.length)
throw new Error('Pubkey count should equal msg count');
const sig = normP2(signature);
const nMessages = messages.map((i) => normP2Hash(i, htfOpts));
const nPublicKeys = publicKeys.map(normP1);
try {
const paired = [];
for (const message of new Set(nMessages)) {
const groupPublicKey = nMessages.reduce(
(groupPublicKey, subMessage, i) =>
subMessage === message ? groupPublicKey.add(nPublicKeys[i]) : groupPublicKey,
G1.ProjectivePoint.ZERO
);
// const msg = message instanceof PointG2 ? message : await PointG2.hashToCurve(message);
// Possible to batch pairing for same msg with different groupPublicKey here
paired.push(pairing(groupPublicKey, message, false));
}
paired.push(pairing(G1.ProjectivePoint.BASE.negate(), sig, false));
const product = paired.reduce((a, b) => Fp12.mul(a, b), Fp12.ONE);
const exp = Fp12.finalExponentiate(product);
return Fp12.eql(exp, Fp12.ONE);
} catch {
return false;
}
}
G1.ProjectivePoint.BASE._setWindowSize(4);
return {
getPublicKey,
sign,
verify,
verifyBatch,
aggregatePublicKeys,
aggregateSignatures,
millerLoop,
pairing,
G1,
G2,
Signature,
fields: {
Fr,
Fp,
Fp2,
Fp6,
Fp12,
},
params: {
x: CURVE.params.x,
r: CURVE.params.r,
G1b: CURVE.G1.b,
G2b: CURVE.G2.b,
},
utils,
};
}

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
// Abelian group utilities
import { IField, validateField, nLength } from './modular.js';
import { validateObject } from './utils.js';
const _0n = BigInt(0);
const _1n = BigInt(1);
export type AffinePoint<T> = {
x: T;
y: T;
} & { z?: never; t?: never };
export interface Group<T extends Group<T>> {
double(): T;
negate(): T;
add(other: T): T;
subtract(other: T): T;
equals(other: T): boolean;
multiply(scalar: bigint): T;
}
export type GroupConstructor<T> = {
BASE: T;
ZERO: T;
};
export type Mapper<T> = (i: T[]) => T[];
// Elliptic curve multiplication of Point by scalar. Fragile.
// Scalars should always be less than curve order: this should be checked inside of a curve itself.
// Creates precomputation tables for fast multiplication:
// - private scalar is split by fixed size windows of W bits
// - every window point is collected from window's table & added to accumulator
// - since windows are different, same point inside tables won't be accessed more than once per calc
// - each multiplication is 'Math.ceil(CURVE_ORDER / 𝑊) + 1' point additions (fixed for any scalar)
// - +1 window is neccessary for wNAF
// - wNAF reduces table size: 2x less memory + 2x faster generation, but 10% slower multiplication
// TODO: Research returning 2d JS array of windows, instead of a single window. This would allow
// windows to be in different memory locations
export function wNAF<T extends Group<T>>(c: GroupConstructor<T>, bits: number) {
const constTimeNegate = (condition: boolean, item: T): T => {
const neg = item.negate();
return condition ? neg : item;
};
const opts = (W: number) => {
const windows = Math.ceil(bits / W) + 1; // +1, because
const windowSize = 2 ** (W - 1); // -1 because we skip zero
return { windows, windowSize };
};
return {
constTimeNegate,
// non-const time multiplication ladder
unsafeLadder(elm: T, n: bigint) {
let p = c.ZERO;
let d: T = elm;
while (n > _0n) {
if (n & _1n) p = p.add(d);
d = d.double();
n >>= _1n;
}
return p;
},
/**
* Creates a wNAF precomputation window. Used for caching.
* Default window size is set by `utils.precompute()` and is equal to 8.
* Number of precomputed points depends on the curve size:
* 2^(𝑊1) * (Math.ceil(𝑛 / 𝑊) + 1), where:
* - 𝑊 is the window size
* - 𝑛 is the bitlength of the curve order.
* For a 256-bit curve and window size 8, the number of precomputed points is 128 * 33 = 4224.
* @returns precomputed point tables flattened to a single array
*/
precomputeWindow(elm: T, W: number): Group<T>[] {
const { windows, windowSize } = opts(W);
const points: T[] = [];
let p: T = elm;
let base = p;
for (let window = 0; window < windows; window++) {
base = p;
points.push(base);
// =1, because we skip zero
for (let i = 1; i < windowSize; i++) {
base = base.add(p);
points.push(base);
}
p = base.double();
}
return points;
},
/**
* Implements ec multiplication using precomputed tables and w-ary non-adjacent form.
* @param W window size
* @param precomputes precomputed tables
* @param n scalar (we don't check here, but should be less than curve order)
* @returns real and fake (for const-time) points
*/
wNAF(W: number, precomputes: T[], n: bigint): { p: T; f: T } {
// TODO: maybe check that scalar is less than group order? wNAF behavious is undefined otherwise
// But need to carefully remove other checks before wNAF. ORDER == bits here
const { windows, windowSize } = opts(W);
let p = c.ZERO;
let f = c.BASE;
const mask = BigInt(2 ** W - 1); // Create mask with W ones: 0b1111 for W=4 etc.
const maxNumber = 2 ** W;
const shiftBy = BigInt(W);
for (let window = 0; window < windows; window++) {
const offset = window * windowSize;
// Extract W bits.
let wbits = Number(n & mask);
// Shift number by W bits.
n >>= shiftBy;
// If the bits are bigger than max size, we'll split those.
// +224 => 256 - 32
if (wbits > windowSize) {
wbits -= maxNumber;
n += _1n;
}
// This code was first written with assumption that 'f' and 'p' will never be infinity point:
// since each addition is multiplied by 2 ** W, it cannot cancel each other. However,
// there is negate now: it is possible that negated element from low value
// would be the same as high element, which will create carry into next window.
// It's not obvious how this can fail, but still worth investigating later.
// Check if we're onto Zero point.
// Add random point inside current window to f.
const offset1 = offset;
const offset2 = offset + Math.abs(wbits) - 1; // -1 because we skip zero
const cond1 = window % 2 !== 0;
const cond2 = wbits < 0;
if (wbits === 0) {
// The most important part for const-time getPublicKey
f = f.add(constTimeNegate(cond1, precomputes[offset1]));
} else {
p = p.add(constTimeNegate(cond2, precomputes[offset2]));
}
}
// JIT-compiler should not eliminate f here, since it will later be used in normalizeZ()
// Even if the variable is still unused, there are some checks which will
// throw an exception, so compiler needs to prove they won't happen, which is hard.
// At this point there is a way to F be infinity-point even if p is not,
// which makes it less const-time: around 1 bigint multiply.
return { p, f };
},
wNAFCached(P: T, precomputesMap: Map<T, T[]>, n: bigint, transform: Mapper<T>): { p: T; f: T } {
// @ts-ignore
const W: number = P._WINDOW_SIZE || 1;
// Calculate precomputes on a first run, reuse them after
let comp = precomputesMap.get(P);
if (!comp) {
comp = this.precomputeWindow(P, W) as T[];
if (W !== 1) {
precomputesMap.set(P, transform(comp));
}
}
return this.wNAF(W, comp, n);
},
};
}
// Generic BasicCurve interface: works even for polynomial fields (BLS): P, n, h would be ok.
// Though generator can be different (Fp2 / Fp6 for BLS).
export type BasicCurve<T> = {
Fp: IField<T>; // Field over which we'll do calculations (Fp)
n: bigint; // Curve order, total count of valid points in the field
nBitLength?: number; // bit length of curve order
nByteLength?: number; // byte length of curve order
h: bigint; // cofactor. we can assign default=1, but users will just ignore it w/o validation
hEff?: bigint; // Number to multiply to clear cofactor
Gx: T; // base point X coordinate
Gy: T; // base point Y coordinate
allowInfinityPoint?: boolean; // bls12-381 requires it. ZERO point is valid, but invalid pubkey
};
export function validateBasic<FP, T>(curve: BasicCurve<FP> & T) {
validateField(curve.Fp);
validateObject(
curve,
{
n: 'bigint',
h: 'bigint',
Gx: 'field',
Gy: 'field',
},
{
nBitLength: 'isSafeInteger',
nByteLength: 'isSafeInteger',
}
);
// Set defaults
return Object.freeze({
...nLength(curve.n, curve.nBitLength),
...curve,
...{ p: curve.Fp.ORDER },
} as const);
}

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
// Twisted Edwards curve. The formula is: ax² + y² = 1 + dx²y²
import { mod } from './modular.js';
import * as ut from './utils.js';
import { ensureBytes, FHash, Hex } from './utils.js';
import { Group, GroupConstructor, wNAF, BasicCurve, validateBasic, AffinePoint } from './curve.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
};
// verification rule is either zip215 or rfc8032 / nist186-5. Consult fromHex:
const VERIFY_DEFAULT = { zip215: true };
function validateOpts(curve: CurveType) {
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;
}
// 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;
}
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;
};
};
};
// It is not generic twisted curve for now, but ed25519/ed448 generic implementation
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;
const MASK = _2n << (BigInt(nByteLength * 8) - _1n);
const modP = Fp.create; // Function overrides
// 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) => {
if (ctx.length || phflag) throw new Error('Contexts/pre-hash are not supported');
return data;
}); // NOOP
const inBig = (n: bigint) => typeof n === 'bigint' && _0n < n; // n in [1..]
const inRange = (n: bigint, max: bigint) => inBig(n) && inBig(max) && n < max; // n in [1..max-1]
const in0MaskRange = (n: bigint) => n === _0n || inRange(n, MASK); // n in [0..MASK-1]
function assertInRange(n: bigint, max: bigint) {
// n in [1..max-1]
if (inRange(n, max)) return n;
throw new Error(`Expected valid scalar < ${max}, got ${typeof n} ${n}`);
}
function assertGE0(n: bigint) {
// n in [0..CURVE_ORDER-1]
return n === _0n ? n : assertInRange(n, CURVE_ORDER); // GE = prime subgroup, not full group
}
const pointPrecomputes = new Map<Point, Point[]>();
function isPoint(other: unknown) {
if (!(other instanceof Point)) throw new Error('ExtendedPoint expected');
}
// 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
) {
if (!in0MaskRange(ex)) throw new Error('x required');
if (!in0MaskRange(ey)) throw new Error('y required');
if (!in0MaskRange(ez)) throw new Error('z required');
if (!in0MaskRange(et)) throw new Error('t required');
}
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 || {};
if (!in0MaskRange(x) || !in0MaskRange(y)) throw new Error('invalid affine point');
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);
}
// We calculate precomputes for elliptic curve point multiplication
// using windowed method. This specifies window size and
// stores precomputed values. Usually only base point would be precomputed.
_WINDOW_SIZE?: number;
// "Private method", don't use it directly
_setWindowSize(windowSize: number) {
this._WINDOW_SIZE = windowSize;
pointPrecomputes.delete(this);
}
// Not required for fromHex(), which always creates valid points.
// Could be useful for fromAffine().
assertValidity(): void {
const { a, d } = CURVE;
if (this.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 } = this;
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)');
}
// Compare one point to another.
equals(other: Point): boolean {
isPoint(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;
}
protected 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) {
isPoint(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, pointPrecomputes, n, Point.normalizeZ);
}
// Constant-time multiplication.
multiply(scalar: bigint): Point {
const { p, f } = this.wNAF(assertInRange(scalar, CURVE_ORDER));
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.
multiplyUnsafe(scalar: bigint): Point {
let n = assertGE0(scalar); // 0 <= scalar < CURVE.n
if (n === _0n) return I;
if (this.equals(I) || n === _1n) return this;
if (this.equals(G)) return this.wNAF(n).p;
return wnaf.unsafeLadder(this, n);
}
// 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> {
const { ex: x, ey: y, ez: z } = this;
const is0 = this.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 };
}
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
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);
if (y === _0n) {
// y=0 is allowed
} else {
// RFC8032 prohibits >= p, but ZIP215 doesn't
if (zip215) assertInRange(y, MASK); // zip215=true [1..P-1] (2^255-19-1 for ed25519)
else assertInRange(y, Fp.ORDER); // zip215=false [1..MASK-1] (2^256-1 for ed25519)
}
// 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 = nByteLength;
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
assertGE0(s); // 0 <= s < l
const res = ut.concatBytes(R, ut.numberToBytesLE(s, Fp.BYTES));
return ensureBytes('result', res, nByteLength * 2); // 64-byte signature
}
const verifyOpts: { context?: Hex; zip215?: boolean } = VERIFY_DEFAULT;
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);
if (prehash) msg = prehash(msg); // for ed25519ph, etc
const s = ut.bytesToNumberLE(sig.slice(len, 2 * len));
// zip215: true is good for consensus-critical apps and allows points < 2^256
// zip215: false follows RFC8032 / NIST186-5 and restricts points to CURVE.p
let A, R, SB;
try {
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));
// [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 = Point.BASE): typeof Point.BASE {
point._setWindowSize(windowSize);
point.multiply(BigInt(3));
return point;
},
};
return {
CURVE,
getPublicKey,
sign,
verify,
ExtendedPoint: Point,
utils,
};
}

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import type { Group, GroupConstructor, AffinePoint } from './curve.js';
import { mod, IField } from './modular.js';
import { bytesToNumberBE, CHash, concatBytes, utf8ToBytes, validateObject } from './utils.js';
/**
* * `DST` is a domain separation tag, defined in section 2.2.5
* * `p` characteristic of F, where F is a finite field of characteristic p and order q = p^m
* * `m` is extension degree (1 for prime fields)
* * `k` is the target security target in bits (e.g. 128), from section 5.1
* * `expand` is `xmd` (SHA2, SHA3, BLAKE) or `xof` (SHAKE, BLAKE-XOF)
* * `hash` conforming to `utils.CHash` interface, with `outputLen` / `blockLen` props
*/
type UnicodeOrBytes = string | Uint8Array;
export type Opts = {
DST: UnicodeOrBytes;
p: bigint;
m: number;
k: number;
expand: 'xmd' | 'xof';
hash: CHash;
};
function validateDST(dst: UnicodeOrBytes): Uint8Array {
if (dst instanceof Uint8Array) return dst;
if (typeof dst === 'string') return utf8ToBytes(dst);
throw new Error('DST must be Uint8Array or string');
}
// Octet Stream to Integer. "spec" implementation of os2ip is 2.5x slower vs bytesToNumberBE.
const os2ip = bytesToNumberBE;
// Integer to Octet Stream (numberToBytesBE)
function i2osp(value: number, length: number): Uint8Array {
if (value < 0 || value >= 1 << (8 * length)) {
throw new Error(`bad I2OSP call: value=${value} length=${length}`);
}
const res = Array.from({ length }).fill(0) as number[];
for (let i = length - 1; i >= 0; i--) {
res[i] = value & 0xff;
value >>>= 8;
}
return new Uint8Array(res);
}
function strxor(a: Uint8Array, b: Uint8Array): Uint8Array {
const arr = new Uint8Array(a.length);
for (let i = 0; i < a.length; i++) {
arr[i] = a[i] ^ b[i];
}
return arr;
}
function isBytes(item: unknown): void {
if (!(item instanceof Uint8Array)) throw new Error('Uint8Array expected');
}
function isNum(item: unknown): void {
if (!Number.isSafeInteger(item)) throw new Error('number expected');
}
// Produces a uniformly random byte string using a cryptographic hash function H that outputs b bits
// https://www.rfc-editor.org/rfc/rfc9380#section-5.3.1
export function expand_message_xmd(
msg: Uint8Array,
DST: Uint8Array,
lenInBytes: number,
H: CHash
): Uint8Array {
isBytes(msg);
isBytes(DST);
isNum(lenInBytes);
// https://www.rfc-editor.org/rfc/rfc9380#section-5.3.3
if (DST.length > 255) DST = H(concatBytes(utf8ToBytes('H2C-OVERSIZE-DST-'), DST));
const { outputLen: b_in_bytes, blockLen: r_in_bytes } = H;
const ell = Math.ceil(lenInBytes / b_in_bytes);
if (ell > 255) throw new Error('Invalid xmd length');
const DST_prime = concatBytes(DST, i2osp(DST.length, 1));
const Z_pad = i2osp(0, r_in_bytes);
const l_i_b_str = i2osp(lenInBytes, 2); // len_in_bytes_str
const b = new Array<Uint8Array>(ell);
const b_0 = H(concatBytes(Z_pad, msg, l_i_b_str, i2osp(0, 1), DST_prime));
b[0] = H(concatBytes(b_0, i2osp(1, 1), DST_prime));
for (let i = 1; i <= ell; i++) {
const args = [strxor(b_0, b[i - 1]), i2osp(i + 1, 1), DST_prime];
b[i] = H(concatBytes(...args));
}
const pseudo_random_bytes = concatBytes(...b);
return pseudo_random_bytes.slice(0, lenInBytes);
}
// Produces a uniformly random byte string using an extendable-output function (XOF) H.
// 1. The collision resistance of H MUST be at least k bits.
// 2. H MUST be an XOF that has been proved indifferentiable from
// a random oracle under a reasonable cryptographic assumption.
// https://www.rfc-editor.org/rfc/rfc9380#section-5.3.2
export function expand_message_xof(
msg: Uint8Array,
DST: Uint8Array,
lenInBytes: number,
k: number,
H: CHash
): Uint8Array {
isBytes(msg);
isBytes(DST);
isNum(lenInBytes);
// https://www.rfc-editor.org/rfc/rfc9380#section-5.3.3
// DST = H('H2C-OVERSIZE-DST-' || a_very_long_DST, Math.ceil((lenInBytes * k) / 8));
if (DST.length > 255) {
const dkLen = Math.ceil((2 * k) / 8);
DST = H.create({ dkLen }).update(utf8ToBytes('H2C-OVERSIZE-DST-')).update(DST).digest();
}
if (lenInBytes > 65535 || DST.length > 255)
throw new Error('expand_message_xof: invalid lenInBytes');
return (
H.create({ dkLen: lenInBytes })
.update(msg)
.update(i2osp(lenInBytes, 2))
// 2. DST_prime = DST || I2OSP(len(DST), 1)
.update(DST)
.update(i2osp(DST.length, 1))
.digest()
);
}
/**
* Hashes arbitrary-length byte strings to a list of one or more elements of a finite field F
* https://www.rfc-editor.org/rfc/rfc9380#section-5.2
* @param msg a byte string containing the message to hash
* @param count the number of elements of F to output
* @param options `{DST: string, p: bigint, m: number, k: number, expand: 'xmd' | 'xof', hash: H}`, see above
* @returns [u_0, ..., u_(count - 1)], a list of field elements.
*/
export function hash_to_field(msg: Uint8Array, count: number, options: Opts): bigint[][] {
validateObject(options, {
DST: 'stringOrUint8Array',
p: 'bigint',
m: 'isSafeInteger',
k: 'isSafeInteger',
hash: 'hash',
});
const { p, k, m, hash, expand, DST: _DST } = options;
isBytes(msg);
isNum(count);
const DST = validateDST(_DST);
const log2p = p.toString(2).length;
const L = Math.ceil((log2p + k) / 8); // section 5.1 of ietf draft link above
const len_in_bytes = count * m * L;
let prb; // pseudo_random_bytes
if (expand === 'xmd') {
prb = expand_message_xmd(msg, DST, len_in_bytes, hash);
} else if (expand === 'xof') {
prb = expand_message_xof(msg, DST, len_in_bytes, k, hash);
} else if (expand === '_internal_pass') {
// for internal tests only
prb = msg;
} else {
throw new Error('expand must be "xmd" or "xof"');
}
const u = new Array(count);
for (let i = 0; i < count; i++) {
const e = new Array(m);
for (let j = 0; j < m; j++) {
const elm_offset = L * (j + i * m);
const tv = prb.subarray(elm_offset, elm_offset + L);
e[j] = mod(os2ip(tv), p);
}
u[i] = e;
}
return u;
}
export function isogenyMap<T, F extends IField<T>>(field: F, map: [T[], T[], T[], T[]]) {
// Make same order as in spec
const COEFF = map.map((i) => Array.from(i).reverse());
return (x: T, y: T) => {
const [xNum, xDen, yNum, yDen] = COEFF.map((val) =>
val.reduce((acc, i) => field.add(field.mul(acc, x), i))
);
x = field.div(xNum, xDen); // xNum / xDen
y = field.mul(y, field.div(yNum, yDen)); // y * (yNum / yDev)
return { x, y };
};
}
export interface H2CPoint<T> extends Group<H2CPoint<T>> {
add(rhs: H2CPoint<T>): H2CPoint<T>;
toAffine(iz?: bigint): AffinePoint<T>;
clearCofactor(): H2CPoint<T>;
assertValidity(): void;
}
export interface H2CPointConstructor<T> extends GroupConstructor<H2CPoint<T>> {
fromAffine(ap: AffinePoint<T>): H2CPoint<T>;
}
export type MapToCurve<T> = (scalar: bigint[]) => AffinePoint<T>;
// Separated from initialization opts, so users won't accidentally change per-curve parameters
// (changing DST is ok!)
export type htfBasicOpts = { DST: UnicodeOrBytes };
export function createHasher<T>(
Point: H2CPointConstructor<T>,
mapToCurve: MapToCurve<T>,
def: Opts & { encodeDST?: UnicodeOrBytes }
) {
if (typeof mapToCurve !== 'function') throw new Error('mapToCurve() must be defined');
return {
// Encodes byte string to elliptic curve.
// hash_to_curve from https://www.rfc-editor.org/rfc/rfc9380#section-3
hashToCurve(msg: Uint8Array, options?: htfBasicOpts) {
const u = hash_to_field(msg, 2, { ...def, DST: def.DST, ...options } as Opts);
const u0 = Point.fromAffine(mapToCurve(u[0]));
const u1 = Point.fromAffine(mapToCurve(u[1]));
const P = u0.add(u1).clearCofactor();
P.assertValidity();
return P;
},
// Encodes byte string to elliptic curve.
// encode_to_curve from https://www.rfc-editor.org/rfc/rfc9380#section-3
encodeToCurve(msg: Uint8Array, options?: htfBasicOpts) {
const u = hash_to_field(msg, 1, { ...def, DST: def.encodeDST, ...options } as Opts);
const P = Point.fromAffine(mapToCurve(u[0])).clearCofactor();
P.assertValidity();
return P;
},
};
}

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
// Utilities for modular arithmetics and finite fields
import {
bitMask,
numberToBytesBE,
numberToBytesLE,
bytesToNumberBE,
bytesToNumberLE,
ensureBytes,
validateObject,
} from './utils.js';
// prettier-ignore
const _0n = BigInt(0), _1n = BigInt(1), _2n = BigInt(2), _3n = BigInt(3);
// prettier-ignore
const _4n = BigInt(4), _5n = BigInt(5), _8n = BigInt(8);
// prettier-ignore
const _9n = BigInt(9), _16n = BigInt(16);
// Calculates a modulo b
export function mod(a: bigint, b: bigint): bigint {
const result = a % b;
return result >= _0n ? result : b + result;
}
/**
* Efficiently raise num to power and do modular division.
* Unsafe in some contexts: uses ladder, so can expose bigint bits.
* @example
* pow(2n, 6n, 11n) // 64n % 11n == 9n
*/
// TODO: use field version && remove
export function pow(num: bigint, power: bigint, modulo: bigint): bigint {
if (modulo <= _0n || power < _0n) throw new Error('Expected power/modulo > 0');
if (modulo === _1n) return _0n;
let res = _1n;
while (power > _0n) {
if (power & _1n) res = (res * num) % modulo;
num = (num * num) % modulo;
power >>= _1n;
}
return res;
}
// Does x ^ (2 ^ power) mod p. pow2(30, 4) == 30 ^ (2 ^ 4)
export function pow2(x: bigint, power: bigint, modulo: bigint): bigint {
let res = x;
while (power-- > _0n) {
res *= res;
res %= modulo;
}
return res;
}
// Inverses number over modulo
export function invert(number: bigint, modulo: bigint): bigint {
if (number === _0n || modulo <= _0n) {
throw new Error(`invert: expected positive integers, got n=${number} mod=${modulo}`);
}
// Euclidean GCD https://brilliant.org/wiki/extended-euclidean-algorithm/
// Fermat's little theorem "CT-like" version inv(n) = n^(m-2) mod m is 30x slower.
let a = mod(number, modulo);
let b = modulo;
// prettier-ignore
let x = _0n, y = _1n, u = _1n, v = _0n;
while (a !== _0n) {
// JIT applies optimization if those two lines follow each other
const q = b / a;
const r = b % a;
const m = x - u * q;
const n = y - v * q;
// prettier-ignore
b = a, a = r, x = u, y = v, u = m, v = n;
}
const gcd = b;
if (gcd !== _1n) throw new Error('invert: does not exist');
return mod(x, modulo);
}
/**
* Tonelli-Shanks square root search algorithm.
* 1. https://eprint.iacr.org/2012/685.pdf (page 12)
* 2. Square Roots from 1; 24, 51, 10 to Dan Shanks
* Will start an infinite loop if field order P is not prime.
* @param P field order
* @returns function that takes field Fp (created from P) and number n
*/
export function tonelliShanks(P: bigint) {
// Legendre constant: used to calculate Legendre symbol (a | p),
// which denotes the value of a^((p-1)/2) (mod p).
// (a | p) ≡ 1 if a is a square (mod p)
// (a | p) ≡ -1 if a is not a square (mod p)
// (a | p) ≡ 0 if a ≡ 0 (mod p)
const legendreC = (P - _1n) / _2n;
let Q: bigint, S: number, Z: bigint;
// Step 1: By factoring out powers of 2 from p - 1,
// find q and s such that p - 1 = q*(2^s) with q odd
for (Q = P - _1n, S = 0; Q % _2n === _0n; Q /= _2n, S++);
// Step 2: Select a non-square z such that (z | p) ≡ -1 and set c ≡ zq
for (Z = _2n; Z < P && pow(Z, legendreC, P) !== P - _1n; Z++);
// Fast-path
if (S === 1) {
const p1div4 = (P + _1n) / _4n;
return function tonelliFast<T>(Fp: IField<T>, n: T) {
const root = Fp.pow(n, p1div4);
if (!Fp.eql(Fp.sqr(root), n)) throw new Error('Cannot find square root');
return root;
};
}
// Slow-path
const Q1div2 = (Q + _1n) / _2n;
return function tonelliSlow<T>(Fp: IField<T>, n: T): T {
// Step 0: Check that n is indeed a square: (n | p) should not be ≡ -1
if (Fp.pow(n, legendreC) === Fp.neg(Fp.ONE)) throw new Error('Cannot find square root');
let r = S;
// TODO: will fail at Fp2/etc
let g = Fp.pow(Fp.mul(Fp.ONE, Z), Q); // will update both x and b
let x = Fp.pow(n, Q1div2); // first guess at the square root
let b = Fp.pow(n, Q); // first guess at the fudge factor
while (!Fp.eql(b, Fp.ONE)) {
if (Fp.eql(b, Fp.ZERO)) return Fp.ZERO; // https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm (4. If t = 0, return r = 0)
// Find m such b^(2^m)==1
let m = 1;
for (let t2 = Fp.sqr(b); m < r; m++) {
if (Fp.eql(t2, Fp.ONE)) break;
t2 = Fp.sqr(t2); // t2 *= t2
}
// NOTE: r-m-1 can be bigger than 32, need to convert to bigint before shift, otherwise there will be overflow
const ge = Fp.pow(g, _1n << BigInt(r - m - 1)); // ge = 2^(r-m-1)
g = Fp.sqr(ge); // g = ge * ge
x = Fp.mul(x, ge); // x *= ge
b = Fp.mul(b, g); // b *= g
r = m;
}
return x;
};
}
export function FpSqrt(P: bigint) {
// NOTE: different algorithms can give different roots, it is up to user to decide which one they want.
// For example there is FpSqrtOdd/FpSqrtEven to choice root based on oddness (used for hash-to-curve).
// P ≡ 3 (mod 4)
// √n = n^((P+1)/4)
if (P % _4n === _3n) {
// Not all roots possible!
// const ORDER =
// 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaabn;
// const NUM = 72057594037927816n;
const p1div4 = (P + _1n) / _4n;
return function sqrt3mod4<T>(Fp: IField<T>, n: T) {
const root = Fp.pow(n, p1div4);
// Throw if root**2 != n
if (!Fp.eql(Fp.sqr(root), n)) throw new Error('Cannot find square root');
return root;
};
}
// Atkin algorithm for q ≡ 5 (mod 8), https://eprint.iacr.org/2012/685.pdf (page 10)
if (P % _8n === _5n) {
const c1 = (P - _5n) / _8n;
return function sqrt5mod8<T>(Fp: IField<T>, n: T) {
const n2 = Fp.mul(n, _2n);
const v = Fp.pow(n2, c1);
const nv = Fp.mul(n, v);
const i = Fp.mul(Fp.mul(nv, _2n), v);
const root = Fp.mul(nv, Fp.sub(i, Fp.ONE));
if (!Fp.eql(Fp.sqr(root), n)) throw new Error('Cannot find square root');
return root;
};
}
// P ≡ 9 (mod 16)
if (P % _16n === _9n) {
// NOTE: tonelli is too slow for bls-Fp2 calculations even on start
// Means we cannot use sqrt for constants at all!
//
// const c1 = Fp.sqrt(Fp.negate(Fp.ONE)); // 1. c1 = sqrt(-1) in F, i.e., (c1^2) == -1 in F
// const c2 = Fp.sqrt(c1); // 2. c2 = sqrt(c1) in F, i.e., (c2^2) == c1 in F
// const c3 = Fp.sqrt(Fp.negate(c1)); // 3. c3 = sqrt(-c1) in F, i.e., (c3^2) == -c1 in F
// const c4 = (P + _7n) / _16n; // 4. c4 = (q + 7) / 16 # Integer arithmetic
// sqrt = (x) => {
// let tv1 = Fp.pow(x, c4); // 1. tv1 = x^c4
// let tv2 = Fp.mul(c1, tv1); // 2. tv2 = c1 * tv1
// const tv3 = Fp.mul(c2, tv1); // 3. tv3 = c2 * tv1
// let tv4 = Fp.mul(c3, tv1); // 4. tv4 = c3 * tv1
// const e1 = Fp.equals(Fp.square(tv2), x); // 5. e1 = (tv2^2) == x
// const e2 = Fp.equals(Fp.square(tv3), x); // 6. e2 = (tv3^2) == x
// tv1 = Fp.cmov(tv1, tv2, e1); // 7. tv1 = CMOV(tv1, tv2, e1) # Select tv2 if (tv2^2) == x
// tv2 = Fp.cmov(tv4, tv3, e2); // 8. tv2 = CMOV(tv4, tv3, e2) # Select tv3 if (tv3^2) == x
// const e3 = Fp.equals(Fp.square(tv2), x); // 9. e3 = (tv2^2) == x
// return Fp.cmov(tv1, tv2, e3); // 10. z = CMOV(tv1, tv2, e3) # Select the sqrt from tv1 and tv2
// }
}
// Other cases: Tonelli-Shanks algorithm
return tonelliShanks(P);
}
// Little-endian check for first LE bit (last BE bit);
export const isNegativeLE = (num: bigint, modulo: bigint) => (mod(num, modulo) & _1n) === _1n;
// Field is not always over prime: for example, Fp2 has ORDER(q)=p^m
export interface IField<T> {
ORDER: bigint;
BYTES: number;
BITS: number;
MASK: bigint;
ZERO: T;
ONE: T;
// 1-arg
create: (num: T) => T;
isValid: (num: T) => boolean;
is0: (num: T) => boolean;
neg(num: T): T;
inv(num: T): T;
sqrt(num: T): T;
sqr(num: T): T;
// 2-args
eql(lhs: T, rhs: T): boolean;
add(lhs: T, rhs: T): T;
sub(lhs: T, rhs: T): T;
mul(lhs: T, rhs: T | bigint): T;
pow(lhs: T, power: bigint): T;
div(lhs: T, rhs: T | bigint): T;
// N for NonNormalized (for now)
addN(lhs: T, rhs: T): T;
subN(lhs: T, rhs: T): T;
mulN(lhs: T, rhs: T | bigint): T;
sqrN(num: T): T;
// Optional
// Should be same as sgn0 function in
// [RFC9380](https://www.rfc-editor.org/rfc/rfc9380#section-4.1).
// NOTE: sgn0 is 'negative in LE', which is same as odd. And negative in LE is kinda strange definition anyway.
isOdd?(num: T): boolean; // Odd instead of even since we have it for Fp2
// legendre?(num: T): T;
pow(lhs: T, power: bigint): T;
invertBatch: (lst: T[]) => T[];
toBytes(num: T): Uint8Array;
fromBytes(bytes: Uint8Array): T;
// If c is False, CMOV returns a, otherwise it returns b.
cmov(a: T, b: T, c: boolean): T;
}
// prettier-ignore
const FIELD_FIELDS = [
'create', 'isValid', 'is0', 'neg', 'inv', 'sqrt', 'sqr',
'eql', 'add', 'sub', 'mul', 'pow', 'div',
'addN', 'subN', 'mulN', 'sqrN'
] as const;
export function validateField<T>(field: IField<T>) {
const initial = {
ORDER: 'bigint',
MASK: 'bigint',
BYTES: 'isSafeInteger',
BITS: 'isSafeInteger',
} as Record<string, string>;
const opts = FIELD_FIELDS.reduce((map, val: string) => {
map[val] = 'function';
return map;
}, initial);
return validateObject(field, opts);
}
// Generic field functions
/**
* Same as `pow` but for Fp: non-constant-time.
* Unsafe in some contexts: uses ladder, so can expose bigint bits.
*/
export function FpPow<T>(f: IField<T>, num: T, power: bigint): T {
// Should have same speed as pow for bigints
// TODO: benchmark!
if (power < _0n) throw new Error('Expected power > 0');
if (power === _0n) return f.ONE;
if (power === _1n) return num;
let p = f.ONE;
let d = num;
while (power > _0n) {
if (power & _1n) p = f.mul(p, d);
d = f.sqr(d);
power >>= _1n;
}
return p;
}
/**
* Efficiently invert an array of Field elements.
* `inv(0)` will return `undefined` here: make sure to throw an error.
*/
export function FpInvertBatch<T>(f: IField<T>, nums: T[]): T[] {
const tmp = new Array(nums.length);
// Walk from first to last, multiply them by each other MOD p
const lastMultiplied = nums.reduce((acc, num, i) => {
if (f.is0(num)) return acc;
tmp[i] = acc;
return f.mul(acc, num);
}, f.ONE);
// Invert last element
const inverted = f.inv(lastMultiplied);
// Walk from last to first, multiply them by inverted each other MOD p
nums.reduceRight((acc, num, i) => {
if (f.is0(num)) return acc;
tmp[i] = f.mul(acc, tmp[i]);
return f.mul(acc, num);
}, inverted);
return tmp;
}
export function FpDiv<T>(f: IField<T>, lhs: T, rhs: T | bigint): T {
return f.mul(lhs, typeof rhs === 'bigint' ? invert(rhs, f.ORDER) : f.inv(rhs));
}
// This function returns True whenever the value x is a square in the field F.
export function FpIsSquare<T>(f: IField<T>) {
const legendreConst = (f.ORDER - _1n) / _2n; // Integer arithmetic
return (x: T): boolean => {
const p = f.pow(x, legendreConst);
return f.eql(p, f.ZERO) || f.eql(p, f.ONE);
};
}
// CURVE.n lengths
export function nLength(n: bigint, nBitLength?: number) {
// Bit size, byte size of CURVE.n
const _nBitLength = nBitLength !== undefined ? nBitLength : n.toString(2).length;
const nByteLength = Math.ceil(_nBitLength / 8);
return { nBitLength: _nBitLength, nByteLength };
}
type FpField = IField<bigint> & Required<Pick<IField<bigint>, 'isOdd'>>;
/**
* Initializes a finite field over prime. **Non-primes are not supported.**
* Do not init in loop: slow. Very fragile: always run a benchmark on a change.
* Major performance optimizations:
* * a) denormalized operations like mulN instead of mul
* * b) same object shape: never add or remove keys
* * c) Object.freeze
* @param ORDER prime positive bigint
* @param bitLen how many bits the field consumes
* @param isLE (def: false) if encoding / decoding should be in little-endian
* @param redef optional faster redefinitions of sqrt and other methods
*/
export function Field(
ORDER: bigint,
bitLen?: number,
isLE = false,
redef: Partial<IField<bigint>> = {}
): Readonly<FpField> {
if (ORDER <= _0n) throw new Error(`Expected Field ORDER > 0, got ${ORDER}`);
const { nBitLength: BITS, nByteLength: BYTES } = nLength(ORDER, bitLen);
if (BYTES > 2048) throw new Error('Field lengths over 2048 bytes are not supported');
const sqrtP = FpSqrt(ORDER);
const f: Readonly<FpField> = Object.freeze({
ORDER,
BITS,
BYTES,
MASK: bitMask(BITS),
ZERO: _0n,
ONE: _1n,
create: (num) => mod(num, ORDER),
isValid: (num) => {
if (typeof num !== 'bigint')
throw new Error(`Invalid field element: expected bigint, got ${typeof num}`);
return _0n <= num && num < ORDER; // 0 is valid element, but it's not invertible
},
is0: (num) => num === _0n,
isOdd: (num) => (num & _1n) === _1n,
neg: (num) => mod(-num, ORDER),
eql: (lhs, rhs) => lhs === rhs,
sqr: (num) => mod(num * num, ORDER),
add: (lhs, rhs) => mod(lhs + rhs, ORDER),
sub: (lhs, rhs) => mod(lhs - rhs, ORDER),
mul: (lhs, rhs) => mod(lhs * rhs, ORDER),
pow: (num, power) => FpPow(f, num, power),
div: (lhs, rhs) => mod(lhs * invert(rhs, ORDER), ORDER),
// Same as above, but doesn't normalize
sqrN: (num) => num * num,
addN: (lhs, rhs) => lhs + rhs,
subN: (lhs, rhs) => lhs - rhs,
mulN: (lhs, rhs) => lhs * rhs,
inv: (num) => invert(num, ORDER),
sqrt: redef.sqrt || ((n) => sqrtP(f, n)),
invertBatch: (lst) => FpInvertBatch(f, lst),
// TODO: do we really need constant cmov?
// We don't have const-time bigints anyway, so probably will be not very useful
cmov: (a, b, c) => (c ? b : a),
toBytes: (num) => (isLE ? numberToBytesLE(num, BYTES) : numberToBytesBE(num, BYTES)),
fromBytes: (bytes) => {
if (bytes.length !== BYTES)
throw new Error(`Fp.fromBytes: expected ${BYTES}, got ${bytes.length}`);
return isLE ? bytesToNumberLE(bytes) : bytesToNumberBE(bytes);
},
} as FpField);
return Object.freeze(f);
}
export function FpSqrtOdd<T>(Fp: IField<T>, elm: T) {
if (!Fp.isOdd) throw new Error(`Field doesn't have isOdd`);
const root = Fp.sqrt(elm);
return Fp.isOdd(root) ? root : Fp.neg(root);
}
export function FpSqrtEven<T>(Fp: IField<T>, elm: T) {
if (!Fp.isOdd) throw new Error(`Field doesn't have isOdd`);
const root = Fp.sqrt(elm);
return Fp.isOdd(root) ? Fp.neg(root) : root;
}
/**
* "Constant-time" private key generation utility.
* Same as mapKeyToField, but accepts less bytes (40 instead of 48 for 32-byte field).
* Which makes it slightly more biased, less secure.
* @deprecated use mapKeyToField instead
*/
export function hashToPrivateScalar(
hash: string | Uint8Array,
groupOrder: bigint,
isLE = false
): bigint {
hash = ensureBytes('privateHash', hash);
const hashLen = hash.length;
const minLen = nLength(groupOrder).nByteLength + 8;
if (minLen < 24 || hashLen < minLen || hashLen > 1024)
throw new Error(`hashToPrivateScalar: expected ${minLen}-1024 bytes of input, got ${hashLen}`);
const num = isLE ? bytesToNumberLE(hash) : bytesToNumberBE(hash);
return mod(num, groupOrder - _1n) + _1n;
}
/**
* Returns total number of bytes consumed by the field element.
* For example, 32 bytes for usual 256-bit weierstrass curve.
* @param fieldOrder number of field elements, usually CURVE.n
* @returns byte length of field
*/
export function getFieldBytesLength(fieldOrder: bigint): number {
if (typeof fieldOrder !== 'bigint') throw new Error('field order must be bigint');
const bitLength = fieldOrder.toString(2).length;
return Math.ceil(bitLength / 8);
}
/**
* Returns minimal amount of bytes that can be safely reduced
* by field order.
* Should be 2^-128 for 128-bit curve such as P256.
* @param fieldOrder number of field elements, usually CURVE.n
* @returns byte length of target hash
*/
export function getMinHashLength(fieldOrder: bigint): number {
const length = getFieldBytesLength(fieldOrder);
return length + Math.ceil(length / 2);
}
/**
* "Constant-time" private key generation utility.
* Can take (n + n/2) or more bytes of uniform input e.g. from CSPRNG or KDF
* and convert them into private scalar, with the modulo bias being negligible.
* Needs at least 48 bytes of input for 32-byte private key.
* https://research.kudelskisecurity.com/2020/07/28/the-definitive-guide-to-modulo-bias-and-how-to-avoid-it/
* FIPS 186-5, A.2 https://csrc.nist.gov/publications/detail/fips/186/5/final
* RFC 9380, https://www.rfc-editor.org/rfc/rfc9380#section-5
* @param hash hash output from SHA3 or a similar function
* @param groupOrder size of subgroup - (e.g. secp256k1.CURVE.n)
* @param isLE interpret hash bytes as LE num
* @returns valid private scalar
*/
export function mapHashToField(key: Uint8Array, fieldOrder: bigint, isLE = false): Uint8Array {
const len = key.length;
const fieldLen = getFieldBytesLength(fieldOrder);
const minLen = getMinHashLength(fieldOrder);
// No small numbers: need to understand bias story. No huge numbers: easier to detect JS timings.
if (len < 16 || len < minLen || len > 1024)
throw new Error(`expected ${minLen}-1024 bytes of input, got ${len}`);
const num = isLE ? bytesToNumberBE(key) : bytesToNumberLE(key);
// `mod(x, 11)` can sometimes produce 0. `mod(x, 10) + 1` is the same, but no 0
const reduced = mod(num, fieldOrder - _1n) + _1n;
return isLE ? numberToBytesLE(reduced, fieldLen) : numberToBytesBE(reduced, fieldLen);
}

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { mod, pow } from './modular.js';
import { bytesToNumberLE, ensureBytes, numberToBytesLE, validateObject } from './utils.js';
const _0n = BigInt(0);
const _1n = BigInt(1);
type Hex = string | Uint8Array;
export type CurveType = {
P: bigint; // finite field prime
nByteLength: number;
adjustScalarBytes?: (bytes: Uint8Array) => Uint8Array;
domain?: (data: Uint8Array, ctx: Uint8Array, phflag: boolean) => Uint8Array;
a: bigint;
montgomeryBits: number;
powPminus2?: (x: bigint) => bigint;
xyToU?: (x: bigint, y: bigint) => bigint;
Gu: bigint;
randomBytes?: (bytesLength?: number) => Uint8Array;
};
export type CurveFn = {
scalarMult: (scalar: Hex, u: Hex) => Uint8Array;
scalarMultBase: (scalar: Hex) => Uint8Array;
getSharedSecret: (privateKeyA: Hex, publicKeyB: Hex) => Uint8Array;
getPublicKey: (privateKey: Hex) => Uint8Array;
utils: { randomPrivateKey: () => Uint8Array };
GuBytes: Uint8Array;
};
function validateOpts(curve: CurveType) {
validateObject(
curve,
{
a: 'bigint',
},
{
montgomeryBits: 'isSafeInteger',
nByteLength: 'isSafeInteger',
adjustScalarBytes: 'function',
domain: 'function',
powPminus2: 'function',
Gu: 'bigint',
}
);
// Set defaults
return Object.freeze({ ...curve } as const);
}
// NOTE: not really montgomery curve, just bunch of very specific methods for X25519/X448 (RFC 7748, https://www.rfc-editor.org/rfc/rfc7748)
// Uses only one coordinate instead of two
export function montgomery(curveDef: CurveType): CurveFn {
const CURVE = validateOpts(curveDef);
const { P } = CURVE;
const modP = (n: bigint) => mod(n, P);
const montgomeryBits = CURVE.montgomeryBits;
const montgomeryBytes = Math.ceil(montgomeryBits / 8);
const fieldLen = CURVE.nByteLength;
const adjustScalarBytes = CURVE.adjustScalarBytes || ((bytes: Uint8Array) => bytes);
const powPminus2 = CURVE.powPminus2 || ((x: bigint) => pow(x, P - BigInt(2), P));
// cswap from RFC7748. But it is not from RFC7748!
/*
cswap(swap, x_2, x_3):
dummy = mask(swap) AND (x_2 XOR x_3)
x_2 = x_2 XOR dummy
x_3 = x_3 XOR dummy
Return (x_2, x_3)
Where mask(swap) is the all-1 or all-0 word of the same length as x_2
and x_3, computed, e.g., as mask(swap) = 0 - swap.
*/
function cswap(swap: bigint, x_2: bigint, x_3: bigint): [bigint, bigint] {
const dummy = modP(swap * (x_2 - x_3));
x_2 = modP(x_2 - dummy);
x_3 = modP(x_3 + dummy);
return [x_2, x_3];
}
// Accepts 0 as well
function assertFieldElement(n: bigint): bigint {
if (typeof n === 'bigint' && _0n <= n && n < P) return n;
throw new Error('Expected valid scalar 0 < scalar < CURVE.P');
}
// x25519 from 4
// The constant a24 is (486662 - 2) / 4 = 121665 for curve25519/X25519
const a24 = (CURVE.a - BigInt(2)) / BigInt(4);
/**
*
* @param pointU u coordinate (x) on Montgomery Curve 25519
* @param scalar by which the point would be multiplied
* @returns new Point on Montgomery curve
*/
function montgomeryLadder(pointU: bigint, scalar: bigint): bigint {
const u = assertFieldElement(pointU);
// Section 5: Implementations MUST accept non-canonical values and process them as
// if they had been reduced modulo the field prime.
const k = assertFieldElement(scalar);
const x_1 = u;
let x_2 = _1n;
let z_2 = _0n;
let x_3 = u;
let z_3 = _1n;
let swap = _0n;
let sw: [bigint, bigint];
for (let t = BigInt(montgomeryBits - 1); t >= _0n; t--) {
const k_t = (k >> t) & _1n;
swap ^= k_t;
sw = cswap(swap, x_2, x_3);
x_2 = sw[0];
x_3 = sw[1];
sw = cswap(swap, z_2, z_3);
z_2 = sw[0];
z_3 = sw[1];
swap = k_t;
const A = x_2 + z_2;
const AA = modP(A * A);
const B = x_2 - z_2;
const BB = modP(B * B);
const E = AA - BB;
const C = x_3 + z_3;
const D = x_3 - z_3;
const DA = modP(D * A);
const CB = modP(C * B);
const dacb = DA + CB;
const da_cb = DA - CB;
x_3 = modP(dacb * dacb);
z_3 = modP(x_1 * modP(da_cb * da_cb));
x_2 = modP(AA * BB);
z_2 = modP(E * (AA + modP(a24 * E)));
}
// (x_2, x_3) = cswap(swap, x_2, x_3)
sw = cswap(swap, x_2, x_3);
x_2 = sw[0];
x_3 = sw[1];
// (z_2, z_3) = cswap(swap, z_2, z_3)
sw = cswap(swap, z_2, z_3);
z_2 = sw[0];
z_3 = sw[1];
// z_2^(p - 2)
const z2 = powPminus2(z_2);
// Return x_2 * (z_2^(p - 2))
return modP(x_2 * z2);
}
function encodeUCoordinate(u: bigint): Uint8Array {
return numberToBytesLE(modP(u), montgomeryBytes);
}
function decodeUCoordinate(uEnc: Hex): bigint {
// Section 5: When receiving such an array, implementations of X25519
// MUST mask the most significant bit in the final byte.
// This is very ugly way, but it works because fieldLen-1 is outside of bounds for X448, so this becomes NOOP
// fieldLen - scalaryBytes = 1 for X448 and = 0 for X25519
const u = ensureBytes('u coordinate', uEnc, montgomeryBytes);
// u[fieldLen-1] crashes QuickJS (TypeError: out-of-bound numeric index)
if (fieldLen === montgomeryBytes) u[fieldLen - 1] &= 127; // 0b0111_1111
return bytesToNumberLE(u);
}
function decodeScalar(n: Hex): bigint {
const bytes = ensureBytes('scalar', n);
if (bytes.length !== montgomeryBytes && bytes.length !== fieldLen)
throw new Error(`Expected ${montgomeryBytes} or ${fieldLen} bytes, got ${bytes.length}`);
return bytesToNumberLE(adjustScalarBytes(bytes));
}
function scalarMult(scalar: Hex, u: Hex): Uint8Array {
const pointU = decodeUCoordinate(u);
const _scalar = decodeScalar(scalar);
const pu = montgomeryLadder(pointU, _scalar);
// The result was not contributory
// https://cr.yp.to/ecdh.html#validate
if (pu === _0n) throw new Error('Invalid private or public key received');
return encodeUCoordinate(pu);
}
// Computes public key from private. By doing scalar multiplication of base point.
const GuBytes = encodeUCoordinate(CURVE.Gu);
function scalarMultBase(scalar: Hex): Uint8Array {
return scalarMult(scalar, GuBytes);
}
return {
scalarMult,
scalarMultBase,
getSharedSecret: (privateKey: Hex, publicKey: Hex) => scalarMult(privateKey, publicKey),
getPublicKey: (privateKey: Hex): Uint8Array => scalarMultBase(privateKey),
utils: { randomPrivateKey: () => CURVE.randomBytes!(CURVE.nByteLength) },
GuBytes: GuBytes,
};
}

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
// Poseidon Hash: https://eprint.iacr.org/2019/458.pdf, https://www.poseidon-hash.info
import { IField, FpPow, validateField } from './modular.js';
// We don't provide any constants, since different implementations use different constants.
// For reference constants see './test/poseidon.test.js'.
export type PoseidonOpts = {
Fp: IField<bigint>;
t: number;
roundsFull: number;
roundsPartial: number;
sboxPower?: number;
reversePartialPowIdx?: boolean; // Hack for stark
mds: bigint[][];
roundConstants: bigint[][];
};
export function validateOpts(opts: PoseidonOpts) {
const { Fp, mds, reversePartialPowIdx: rev, roundConstants: rc } = opts;
const { roundsFull, roundsPartial, sboxPower, t } = opts;
validateField(Fp);
for (const i of ['t', 'roundsFull', 'roundsPartial'] as const) {
if (typeof opts[i] !== 'number' || !Number.isSafeInteger(opts[i]))
throw new Error(`Poseidon: invalid param ${i}=${opts[i]} (${typeof opts[i]})`);
}
// MDS is TxT matrix
if (!Array.isArray(mds) || mds.length !== t) throw new Error('Poseidon: wrong MDS matrix');
const _mds = mds.map((mdsRow) => {
if (!Array.isArray(mdsRow) || mdsRow.length !== t)
throw new Error(`Poseidon MDS matrix row: ${mdsRow}`);
return mdsRow.map((i) => {
if (typeof i !== 'bigint') throw new Error(`Poseidon MDS matrix value=${i}`);
return Fp.create(i);
});
});
if (rev !== undefined && typeof rev !== 'boolean')
throw new Error(`Poseidon: invalid param reversePartialPowIdx=${rev}`);
if (roundsFull % 2 !== 0) throw new Error(`Poseidon roundsFull is not even: ${roundsFull}`);
const rounds = roundsFull + roundsPartial;
if (!Array.isArray(rc) || rc.length !== rounds)
throw new Error('Poseidon: wrong round constants');
const roundConstants = rc.map((rc) => {
if (!Array.isArray(rc) || rc.length !== t)
throw new Error(`Poseidon wrong round constants: ${rc}`);
return rc.map((i) => {
if (typeof i !== 'bigint' || !Fp.isValid(i))
throw new Error(`Poseidon wrong round constant=${i}`);
return Fp.create(i);
});
});
if (!sboxPower || ![3, 5, 7].includes(sboxPower))
throw new Error(`Poseidon wrong sboxPower=${sboxPower}`);
const _sboxPower = BigInt(sboxPower);
let sboxFn = (n: bigint) => FpPow(Fp, n, _sboxPower);
// Unwrapped sbox power for common cases (195->142μs)
if (sboxPower === 3) sboxFn = (n: bigint) => Fp.mul(Fp.sqrN(n), n);
else if (sboxPower === 5) sboxFn = (n: bigint) => Fp.mul(Fp.sqrN(Fp.sqrN(n)), n);
return Object.freeze({ ...opts, rounds, sboxFn, roundConstants, mds: _mds });
}
export function splitConstants(rc: bigint[], t: number) {
if (typeof t !== 'number') throw new Error('poseidonSplitConstants: wrong t');
if (!Array.isArray(rc) || rc.length % t) throw new Error('poseidonSplitConstants: wrong rc');
const res = [];
let tmp = [];
for (let i = 0; i < rc.length; i++) {
tmp.push(rc[i]);
if (tmp.length === t) {
res.push(tmp);
tmp = [];
}
}
return res;
}
export function poseidon(opts: PoseidonOpts) {
const _opts = validateOpts(opts);
const { Fp, mds, roundConstants, rounds, roundsPartial, sboxFn, t } = _opts;
const halfRoundsFull = _opts.roundsFull / 2;
const partialIdx = _opts.reversePartialPowIdx ? t - 1 : 0;
const poseidonRound = (values: bigint[], isFull: boolean, idx: number) => {
values = values.map((i, j) => Fp.add(i, roundConstants[idx][j]));
if (isFull) values = values.map((i) => sboxFn(i));
else values[partialIdx] = sboxFn(values[partialIdx]);
// Matrix multiplication
values = mds.map((i) => i.reduce((acc, i, j) => Fp.add(acc, Fp.mulN(i, values[j])), Fp.ZERO));
return values;
};
const poseidonHash = function poseidonHash(values: bigint[]) {
if (!Array.isArray(values) || values.length !== t)
throw new Error(`Poseidon: wrong values (expected array of bigints with length ${t})`);
values = values.map((i) => {
if (typeof i !== 'bigint') throw new Error(`Poseidon: wrong value=${i} (${typeof i})`);
return Fp.create(i);
});
let round = 0;
// Apply r_f/2 full rounds.
for (let i = 0; i < halfRoundsFull; i++) values = poseidonRound(values, true, round++);
// Apply r_p partial rounds.
for (let i = 0; i < roundsPartial; i++) values = poseidonRound(values, false, round++);
// Apply r_f/2 full rounds.
for (let i = 0; i < halfRoundsFull; i++) values = poseidonRound(values, true, round++);
if (round !== rounds)
throw new Error(`Poseidon: wrong number of rounds: last round=${round}, total=${rounds}`);
return values;
};
// For verification in tests
poseidonHash.roundConstants = roundConstants;
return poseidonHash;
}

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
// 100 lines of code in the file are duplicated from noble-hashes (utils).
// This is OK: `abstract` directory does not use noble-hashes.
// User may opt-in into using different hashing library. This way, noble-hashes
// won't be included into their bundle.
const _0n = BigInt(0);
const _1n = BigInt(1);
const _2n = BigInt(2);
const u8a = (a: any): a is Uint8Array => a instanceof Uint8Array;
export type Hex = Uint8Array | string; // hex strings are accepted for simplicity
export type PrivKey = Hex | bigint; // bigints are accepted to ease learning curve
export type CHash = {
(message: Uint8Array | string): Uint8Array;
blockLen: number;
outputLen: number;
create(opts?: { dkLen?: number }): any; // For shake
};
export type FHash = (message: Uint8Array | string) => Uint8Array;
const hexes = /* @__PURE__ */ Array.from({ length: 256 }, (_, i) =>
i.toString(16).padStart(2, '0')
);
/**
* @example bytesToHex(Uint8Array.from([0xca, 0xfe, 0x01, 0x23])) // 'cafe0123'
*/
export function bytesToHex(bytes: Uint8Array): string {
if (!u8a(bytes)) throw new Error('Uint8Array expected');
// pre-caching improves the speed 6x
let hex = '';
for (let i = 0; i < bytes.length; i++) {
hex += hexes[bytes[i]];
}
return hex;
}
export function numberToHexUnpadded(num: number | bigint): string {
const hex = num.toString(16);
return hex.length & 1 ? `0${hex}` : hex;
}
export function hexToNumber(hex: string): bigint {
if (typeof hex !== 'string') throw new Error('hex string expected, got ' + typeof hex);
// Big Endian
return BigInt(hex === '' ? '0' : `0x${hex}`);
}
/**
* @example hexToBytes('cafe0123') // Uint8Array.from([0xca, 0xfe, 0x01, 0x23])
*/
export function hexToBytes(hex: string): Uint8Array {
if (typeof hex !== 'string') throw new Error('hex string expected, got ' + typeof hex);
const len = hex.length;
if (len % 2) throw new Error('padded hex string expected, got unpadded hex of length ' + len);
const array = new Uint8Array(len / 2);
for (let i = 0; i < array.length; i++) {
const j = i * 2;
const hexByte = hex.slice(j, j + 2);
const byte = Number.parseInt(hexByte, 16);
if (Number.isNaN(byte) || byte < 0) throw new Error('Invalid byte sequence');
array[i] = byte;
}
return array;
}
// BE: Big Endian, LE: Little Endian
export function bytesToNumberBE(bytes: Uint8Array): bigint {
return hexToNumber(bytesToHex(bytes));
}
export function bytesToNumberLE(bytes: Uint8Array): bigint {
if (!u8a(bytes)) throw new Error('Uint8Array expected');
return hexToNumber(bytesToHex(Uint8Array.from(bytes).reverse()));
}
export function numberToBytesBE(n: number | bigint, len: number): Uint8Array {
return hexToBytes(n.toString(16).padStart(len * 2, '0'));
}
export function numberToBytesLE(n: number | bigint, len: number): Uint8Array {
return numberToBytesBE(n, len).reverse();
}
// Unpadded, rarely used
export function numberToVarBytesBE(n: number | bigint): Uint8Array {
return hexToBytes(numberToHexUnpadded(n));
}
/**
* Takes hex string or Uint8Array, converts to Uint8Array.
* Validates output length.
* Will throw error for other types.
* @param title descriptive title for an error e.g. 'private key'
* @param hex hex string or Uint8Array
* @param expectedLength optional, will compare to result array's length
* @returns
*/
export function ensureBytes(title: string, hex: Hex, expectedLength?: number): Uint8Array {
let res: Uint8Array;
if (typeof hex === 'string') {
try {
res = hexToBytes(hex);
} catch (e) {
throw new Error(`${title} must be valid hex string, got "${hex}". Cause: ${e}`);
}
} else if (u8a(hex)) {
// Uint8Array.from() instead of hash.slice() because node.js Buffer
// is instance of Uint8Array, and its slice() creates **mutable** copy
res = Uint8Array.from(hex);
} else {
throw new Error(`${title} must be hex string or Uint8Array`);
}
const len = res.length;
if (typeof expectedLength === 'number' && len !== expectedLength)
throw new Error(`${title} expected ${expectedLength} bytes, got ${len}`);
return res;
}
/**
* Copies several Uint8Arrays into one.
*/
export function concatBytes(...arrays: Uint8Array[]): Uint8Array {
const r = new Uint8Array(arrays.reduce((sum, a) => sum + a.length, 0));
let pad = 0; // walk through each item, ensure they have proper type
arrays.forEach((a) => {
if (!u8a(a)) throw new Error('Uint8Array expected');
r.set(a, pad);
pad += a.length;
});
return r;
}
export function equalBytes(b1: Uint8Array, b2: Uint8Array) {
// We don't care about timing attacks here
if (b1.length !== b2.length) return false;
for (let i = 0; i < b1.length; i++) if (b1[i] !== b2[i]) return false;
return true;
}
// Global symbols in both browsers and Node.js since v11
// See https://github.com/microsoft/TypeScript/issues/31535
declare const TextEncoder: any;
/**
* @example utf8ToBytes('abc') // new Uint8Array([97, 98, 99])
*/
export function utf8ToBytes(str: string): Uint8Array {
if (typeof str !== 'string') throw new Error(`utf8ToBytes expected string, got ${typeof str}`);
return new Uint8Array(new TextEncoder().encode(str)); // https://bugzil.la/1681809
}
// Bit operations
/**
* Calculates amount of bits in a bigint.
* Same as `n.toString(2).length`
*/
export function bitLen(n: bigint) {
let len;
for (len = 0; n > _0n; n >>= _1n, len += 1);
return len;
}
/**
* Gets single bit at position.
* NOTE: first bit position is 0 (same as arrays)
* Same as `!!+Array.from(n.toString(2)).reverse()[pos]`
*/
export function bitGet(n: bigint, pos: number) {
return (n >> BigInt(pos)) & _1n;
}
/**
* Sets single bit at position.
*/
export const bitSet = (n: bigint, pos: number, value: boolean) => {
return n | ((value ? _1n : _0n) << BigInt(pos));
};
/**
* Calculate mask for N bits. Not using ** operator with bigints because of old engines.
* Same as BigInt(`0b${Array(i).fill('1').join('')}`)
*/
export const bitMask = (n: number) => (_2n << BigInt(n - 1)) - _1n;
// DRBG
const u8n = (data?: any) => new Uint8Array(data); // creates Uint8Array
const u8fr = (arr: any) => Uint8Array.from(arr); // another shortcut
type Pred<T> = (v: Uint8Array) => T | undefined;
/**
* Minimal HMAC-DRBG from NIST 800-90 for RFC6979 sigs.
* @returns function that will call DRBG until 2nd arg returns something meaningful
* @example
* const drbg = createHmacDRBG<Key>(32, 32, hmac);
* drbg(seed, bytesToKey); // bytesToKey must return Key or undefined
*/
export function createHmacDrbg<T>(
hashLen: number,
qByteLen: number,
hmacFn: (key: Uint8Array, ...messages: Uint8Array[]) => Uint8Array
): (seed: Uint8Array, predicate: Pred<T>) => T {
if (typeof hashLen !== 'number' || hashLen < 2) throw new Error('hashLen must be a number');
if (typeof qByteLen !== 'number' || qByteLen < 2) throw new Error('qByteLen must be a number');
if (typeof hmacFn !== 'function') throw new Error('hmacFn must be a function');
// Step B, Step C: set hashLen to 8*ceil(hlen/8)
let v = u8n(hashLen); // Minimal non-full-spec HMAC-DRBG from NIST 800-90 for RFC6979 sigs.
let k = u8n(hashLen); // Steps B and C of RFC6979 3.2: set hashLen, in our case always same
let i = 0; // Iterations counter, will throw when over 1000
const reset = () => {
v.fill(1);
k.fill(0);
i = 0;
};
const h = (...b: Uint8Array[]) => hmacFn(k, v, ...b); // hmac(k)(v, ...values)
const reseed = (seed = u8n()) => {
// HMAC-DRBG reseed() function. Steps D-G
k = h(u8fr([0x00]), seed); // k = hmac(k || v || 0x00 || seed)
v = h(); // v = hmac(k || v)
if (seed.length === 0) return;
k = h(u8fr([0x01]), seed); // k = hmac(k || v || 0x01 || seed)
v = h(); // v = hmac(k || v)
};
const gen = () => {
// HMAC-DRBG generate() function
if (i++ >= 1000) throw new Error('drbg: tried 1000 values');
let len = 0;
const out: Uint8Array[] = [];
while (len < qByteLen) {
v = h();
const sl = v.slice();
out.push(sl);
len += v.length;
}
return concatBytes(...out);
};
const genUntil = (seed: Uint8Array, pred: Pred<T>): T => {
reset();
reseed(seed); // Steps D-G
let res: T | undefined = undefined; // Step H: grind until k is in [1..n-1]
while (!(res = pred(gen()))) reseed();
reset();
return res;
};
return genUntil;
}
// Validating curves and fields
const validatorFns = {
bigint: (val: any) => typeof val === 'bigint',
function: (val: any) => typeof val === 'function',
boolean: (val: any) => typeof val === 'boolean',
string: (val: any) => typeof val === 'string',
stringOrUint8Array: (val: any) => typeof val === 'string' || val instanceof Uint8Array,
isSafeInteger: (val: any) => Number.isSafeInteger(val),
array: (val: any) => Array.isArray(val),
field: (val: any, object: any) => (object as any).Fp.isValid(val),
hash: (val: any) => typeof val === 'function' && Number.isSafeInteger(val.outputLen),
} as const;
type Validator = keyof typeof validatorFns;
type ValMap<T extends Record<string, any>> = { [K in keyof T]?: Validator };
// type Record<K extends string | number | symbol, T> = { [P in K]: T; }
export function validateObject<T extends Record<string, any>>(
object: T,
validators: ValMap<T>,
optValidators: ValMap<T> = {}
) {
const checkField = (fieldName: keyof T, type: Validator, isOptional: boolean) => {
const checkVal = validatorFns[type];
if (typeof checkVal !== 'function')
throw new Error(`Invalid validator "${type}", expected function`);
const val = object[fieldName as keyof typeof object];
if (isOptional && val === undefined) return;
if (!checkVal(val, object)) {
throw new Error(
`Invalid param ${String(fieldName)}=${val} (${typeof val}), expected ${type}`
);
}
};
for (const [fieldName, type] of Object.entries(validators)) checkField(fieldName, type!, false);
for (const [fieldName, type] of Object.entries(optValidators)) checkField(fieldName, type!, true);
return object;
}
// validate type tests
// const o: { a: number; b: number; c: number } = { a: 1, b: 5, c: 6 };
// const z0 = validateObject(o, { a: 'isSafeInteger' }, { c: 'bigint' }); // Ok!
// // Should fail type-check
// const z1 = validateObject(o, { a: 'tmp' }, { c: 'zz' });
// const z2 = validateObject(o, { a: 'isSafeInteger' }, { c: 'zz' });
// const z3 = validateObject(o, { test: 'boolean', z: 'bug' });
// const z4 = validateObject(o, { a: 'boolean', z: 'bug' });

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { sha256 } from '@noble/hashes/sha256';
import { weierstrass } from './abstract/weierstrass.js';
import { getHash } from './_shortw_utils.js';
import { Field } from './abstract/modular.js';
/**
* bn254 pairing-friendly curve.
* Previously known as alt_bn_128, when it had 128-bit security.
* Recent research shown it's weaker, the naming has been adjusted to its prime bit count.
* https://github.com/zcash/zcash/issues/2502
*/
export const bn254 = weierstrass({
a: BigInt(0),
b: BigInt(3),
Fp: Field(BigInt('0x30644e72e131a029b85045b68181585d97816a916871ca8d3c208c16d87cfd47')),
n: BigInt('0x30644e72e131a029b85045b68181585d2833e84879b9709143e1f593f0000001'),
Gx: BigInt(1),
Gy: BigInt(2),
h: BigInt(1),
...getHash(sha256),
});

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { sha512 } from '@noble/hashes/sha512';
import { concatBytes, randomBytes, utf8ToBytes } from '@noble/hashes/utils';
import { ExtPointType, twistedEdwards } from './abstract/edwards.js';
import { montgomery } from './abstract/montgomery.js';
import { Field, FpSqrtEven, isNegativeLE, mod, pow2 } from './abstract/modular.js';
import {
bytesToHex,
bytesToNumberLE,
ensureBytes,
equalBytes,
Hex,
numberToBytesLE,
} from './abstract/utils.js';
import { createHasher, htfBasicOpts, expand_message_xmd } from './abstract/hash-to-curve.js';
import { AffinePoint } from './abstract/curve.js';
/**
* ed25519 Twisted Edwards curve with following addons:
* - X25519 ECDH
* - Ristretto cofactor elimination
* - Elligator hash-to-group / point indistinguishability
*/
const ED25519_P = BigInt(
'57896044618658097711785492504343953926634992332820282019728792003956564819949'
);
// √(-1) aka √(a) aka 2^((p-1)/4)
const ED25519_SQRT_M1 = BigInt(
'19681161376707505956807079304988542015446066515923890162744021073123829784752'
);
// prettier-ignore
const _0n = BigInt(0), _1n = BigInt(1), _2n = BigInt(2), _5n = BigInt(5);
// prettier-ignore
const _10n = BigInt(10), _20n = BigInt(20), _40n = BigInt(40), _80n = BigInt(80);
function ed25519_pow_2_252_3(x: bigint) {
const P = ED25519_P;
const x2 = (x * x) % P;
const b2 = (x2 * x) % P; // x^3, 11
const b4 = (pow2(b2, _2n, P) * b2) % P; // x^15, 1111
const b5 = (pow2(b4, _1n, P) * x) % P; // x^31
const b10 = (pow2(b5, _5n, P) * b5) % P;
const b20 = (pow2(b10, _10n, P) * b10) % P;
const b40 = (pow2(b20, _20n, P) * b20) % P;
const b80 = (pow2(b40, _40n, P) * b40) % P;
const b160 = (pow2(b80, _80n, P) * b80) % P;
const b240 = (pow2(b160, _80n, P) * b80) % P;
const b250 = (pow2(b240, _10n, P) * b10) % P;
const pow_p_5_8 = (pow2(b250, _2n, P) * x) % P;
// ^ To pow to (p+3)/8, multiply it by x.
return { pow_p_5_8, b2 };
}
function adjustScalarBytes(bytes: Uint8Array): Uint8Array {
// Section 5: For X25519, in order to decode 32 random bytes as an integer scalar,
// set the three least significant bits of the first byte
bytes[0] &= 248; // 0b1111_1000
// and the most significant bit of the last to zero,
bytes[31] &= 127; // 0b0111_1111
// set the second most significant bit of the last byte to 1
bytes[31] |= 64; // 0b0100_0000
return bytes;
}
// sqrt(u/v)
function uvRatio(u: bigint, v: bigint): { isValid: boolean; value: bigint } {
const P = ED25519_P;
const v3 = mod(v * v * v, P); // v³
const v7 = mod(v3 * v3 * v, P); // v⁷
// (p+3)/8 and (p-5)/8
const pow = ed25519_pow_2_252_3(u * v7).pow_p_5_8;
let x = mod(u * v3 * pow, P); // (uv³)(uv⁷)^(p-5)/8
const vx2 = mod(v * x * x, P); // vx²
const root1 = x; // First root candidate
const root2 = mod(x * ED25519_SQRT_M1, P); // Second root candidate
const useRoot1 = vx2 === u; // If vx² = u (mod p), x is a square root
const useRoot2 = vx2 === mod(-u, P); // If vx² = -u, set x <-- x * 2^((p-1)/4)
const noRoot = vx2 === mod(-u * ED25519_SQRT_M1, P); // There is no valid root, vx² = -u√(-1)
if (useRoot1) x = root1;
if (useRoot2 || noRoot) x = root2; // We return root2 anyway, for const-time
if (isNegativeLE(x, P)) x = mod(-x, P);
return { isValid: useRoot1 || useRoot2, value: x };
}
// Just in case
export const ED25519_TORSION_SUBGROUP = [
'0100000000000000000000000000000000000000000000000000000000000000',
'c7176a703d4dd84fba3c0b760d10670f2a2053fa2c39ccc64ec7fd7792ac037a',
'0000000000000000000000000000000000000000000000000000000000000080',
'26e8958fc2b227b045c3f489f2ef98f0d5dfac05d3c63339b13802886d53fc05',
'ecffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff7f',
'26e8958fc2b227b045c3f489f2ef98f0d5dfac05d3c63339b13802886d53fc85',
'0000000000000000000000000000000000000000000000000000000000000000',
'c7176a703d4dd84fba3c0b760d10670f2a2053fa2c39ccc64ec7fd7792ac03fa',
];
const Fp = Field(ED25519_P, undefined, true);
const ed25519Defaults = {
// Param: a
a: BigInt(-1), // Fp.create(-1) is proper; our way still works and is faster
// d is equal to -121665/121666 over finite field.
// Negative number is P - number, and division is invert(number, P)
d: BigInt('37095705934669439343138083508754565189542113879843219016388785533085940283555'),
// Finite field 𝔽p over which we'll do calculations; 2n**255n - 19n
Fp,
// Subgroup order: how many points curve has
// 2n**252n + 27742317777372353535851937790883648493n;
n: BigInt('7237005577332262213973186563042994240857116359379907606001950938285454250989'),
// Cofactor
h: BigInt(8),
// Base point (x, y) aka generator point
Gx: BigInt('15112221349535400772501151409588531511454012693041857206046113283949847762202'),
Gy: BigInt('46316835694926478169428394003475163141307993866256225615783033603165251855960'),
hash: sha512,
randomBytes,
adjustScalarBytes,
// dom2
// Ratio of u to v. Allows us to combine inversion and square root. Uses algo from RFC8032 5.1.3.
// Constant-time, u/√v
uvRatio,
} as const;
export const ed25519 = /* @__PURE__ */ twistedEdwards(ed25519Defaults);
function ed25519_domain(data: Uint8Array, ctx: Uint8Array, phflag: boolean) {
if (ctx.length > 255) throw new Error('Context is too big');
return concatBytes(
utf8ToBytes('SigEd25519 no Ed25519 collisions'),
new Uint8Array([phflag ? 1 : 0, ctx.length]),
ctx,
data
);
}
export const ed25519ctx = /* @__PURE__ */ twistedEdwards({
...ed25519Defaults,
domain: ed25519_domain,
});
export const ed25519ph = /* @__PURE__ */ twistedEdwards({
...ed25519Defaults,
domain: ed25519_domain,
prehash: sha512,
});
export const x25519 = /* @__PURE__ */ (() =>
montgomery({
P: ED25519_P,
a: BigInt(486662),
montgomeryBits: 255, // n is 253 bits
nByteLength: 32,
Gu: BigInt(9),
powPminus2: (x: bigint): bigint => {
const P = ED25519_P;
// x^(p-2) aka x^(2^255-21)
const { pow_p_5_8, b2 } = ed25519_pow_2_252_3(x);
return mod(pow2(pow_p_5_8, BigInt(3), P) * b2, P);
},
adjustScalarBytes,
randomBytes,
}))();
/**
* Converts ed25519 public key to x25519 public key. Uses formula:
* * `(u, v) = ((1+y)/(1-y), sqrt(-486664)*u/x)`
* * `(x, y) = (sqrt(-486664)*u/v, (u-1)/(u+1))`
* @example
* const someonesPub = ed25519.getPublicKey(ed25519.utils.randomPrivateKey());
* const aPriv = x25519.utils.randomPrivateKey();
* x25519.getSharedSecret(aPriv, edwardsToMontgomeryPub(someonesPub))
*/
export function edwardsToMontgomeryPub(edwardsPub: Hex): Uint8Array {
const { y } = ed25519.ExtendedPoint.fromHex(edwardsPub);
const _1n = BigInt(1);
return Fp.toBytes(Fp.create((_1n + y) * Fp.inv(_1n - y)));
}
export const edwardsToMontgomery = edwardsToMontgomeryPub; // deprecated
/**
* Converts ed25519 secret key to x25519 secret key.
* @example
* const someonesPub = x25519.getPublicKey(x25519.utils.randomPrivateKey());
* const aPriv = ed25519.utils.randomPrivateKey();
* x25519.getSharedSecret(edwardsToMontgomeryPriv(aPriv), someonesPub)
*/
export function edwardsToMontgomeryPriv(edwardsPriv: Uint8Array): Uint8Array {
const hashed = ed25519Defaults.hash(edwardsPriv.subarray(0, 32));
return ed25519Defaults.adjustScalarBytes(hashed).subarray(0, 32);
}
// Hash To Curve Elligator2 Map (NOTE: different from ristretto255 elligator)
// NOTE: very important part is usage of FpSqrtEven for ELL2_C1_EDWARDS, since
// SageMath returns different root first and everything falls apart
const ELL2_C1 = (Fp.ORDER + BigInt(3)) / BigInt(8); // 1. c1 = (q + 3) / 8 # Integer arithmetic
const ELL2_C2 = Fp.pow(_2n, ELL2_C1); // 2. c2 = 2^c1
const ELL2_C3 = Fp.sqrt(Fp.neg(Fp.ONE)); // 3. c3 = sqrt(-1)
const ELL2_C4 = (Fp.ORDER - BigInt(5)) / BigInt(8); // 4. c4 = (q - 5) / 8 # Integer arithmetic
const ELL2_J = BigInt(486662);
// prettier-ignore
function map_to_curve_elligator2_curve25519(u: bigint) {
let tv1 = Fp.sqr(u); // 1. tv1 = u^2
tv1 = Fp.mul(tv1, _2n); // 2. tv1 = 2 * tv1
let xd = Fp.add(tv1, Fp.ONE); // 3. xd = tv1 + 1 # Nonzero: -1 is square (mod p), tv1 is not
let x1n = Fp.neg(ELL2_J); // 4. x1n = -J # x1 = x1n / xd = -J / (1 + 2 * u^2)
let tv2 = Fp.sqr(xd); // 5. tv2 = xd^2
let gxd = Fp.mul(tv2, xd); // 6. gxd = tv2 * xd # gxd = xd^3
let gx1 = Fp.mul(tv1, ELL2_J); // 7. gx1 = J * tv1 # x1n + J * xd
gx1 = Fp.mul(gx1, x1n); // 8. gx1 = gx1 * x1n # x1n^2 + J * x1n * xd
gx1 = Fp.add(gx1, tv2); // 9. gx1 = gx1 + tv2 # x1n^2 + J * x1n * xd + xd^2
gx1 = Fp.mul(gx1, x1n); // 10. gx1 = gx1 * x1n # x1n^3 + J * x1n^2 * xd + x1n * xd^2
let tv3 = Fp.sqr(gxd); // 11. tv3 = gxd^2
tv2 = Fp.sqr(tv3); // 12. tv2 = tv3^2 # gxd^4
tv3 = Fp.mul(tv3, gxd); // 13. tv3 = tv3 * gxd # gxd^3
tv3 = Fp.mul(tv3, gx1); // 14. tv3 = tv3 * gx1 # gx1 * gxd^3
tv2 = Fp.mul(tv2, tv3); // 15. tv2 = tv2 * tv3 # gx1 * gxd^7
let y11 = Fp.pow(tv2, ELL2_C4); // 16. y11 = tv2^c4 # (gx1 * gxd^7)^((p - 5) / 8)
y11 = Fp.mul(y11, tv3); // 17. y11 = y11 * tv3 # gx1*gxd^3*(gx1*gxd^7)^((p-5)/8)
let y12 = Fp.mul(y11, ELL2_C3); // 18. y12 = y11 * c3
tv2 = Fp.sqr(y11); // 19. tv2 = y11^2
tv2 = Fp.mul(tv2, gxd); // 20. tv2 = tv2 * gxd
let e1 = Fp.eql(tv2, gx1); // 21. e1 = tv2 == gx1
let y1 = Fp.cmov(y12, y11, e1); // 22. y1 = CMOV(y12, y11, e1) # If g(x1) is square, this is its sqrt
let x2n = Fp.mul(x1n, tv1); // 23. x2n = x1n * tv1 # x2 = x2n / xd = 2 * u^2 * x1n / xd
let y21 = Fp.mul(y11, u); // 24. y21 = y11 * u
y21 = Fp.mul(y21, ELL2_C2); // 25. y21 = y21 * c2
let y22 = Fp.mul(y21, ELL2_C3); // 26. y22 = y21 * c3
let gx2 = Fp.mul(gx1, tv1); // 27. gx2 = gx1 * tv1 # g(x2) = gx2 / gxd = 2 * u^2 * g(x1)
tv2 = Fp.sqr(y21); // 28. tv2 = y21^2
tv2 = Fp.mul(tv2, gxd); // 29. tv2 = tv2 * gxd
let e2 = Fp.eql(tv2, gx2); // 30. e2 = tv2 == gx2
let y2 = Fp.cmov(y22, y21, e2); // 31. y2 = CMOV(y22, y21, e2) # If g(x2) is square, this is its sqrt
tv2 = Fp.sqr(y1); // 32. tv2 = y1^2
tv2 = Fp.mul(tv2, gxd); // 33. tv2 = tv2 * gxd
let e3 = Fp.eql(tv2, gx1); // 34. e3 = tv2 == gx1
let xn = Fp.cmov(x2n, x1n, e3); // 35. xn = CMOV(x2n, x1n, e3) # If e3, x = x1, else x = x2
let y = Fp.cmov(y2, y1, e3); // 36. y = CMOV(y2, y1, e3) # If e3, y = y1, else y = y2
let e4 = Fp.isOdd(y); // 37. e4 = sgn0(y) == 1 # Fix sign of y
y = Fp.cmov(y, Fp.neg(y), e3 !== e4); // 38. y = CMOV(y, -y, e3 XOR e4)
return { xMn: xn, xMd: xd, yMn: y, yMd: _1n }; // 39. return (xn, xd, y, 1)
}
const ELL2_C1_EDWARDS = FpSqrtEven(Fp, Fp.neg(BigInt(486664))); // sgn0(c1) MUST equal 0
function map_to_curve_elligator2_edwards25519(u: bigint) {
const { xMn, xMd, yMn, yMd } = map_to_curve_elligator2_curve25519(u); // 1. (xMn, xMd, yMn, yMd) =
// map_to_curve_elligator2_curve25519(u)
let xn = Fp.mul(xMn, yMd); // 2. xn = xMn * yMd
xn = Fp.mul(xn, ELL2_C1_EDWARDS); // 3. xn = xn * c1
let xd = Fp.mul(xMd, yMn); // 4. xd = xMd * yMn # xn / xd = c1 * xM / yM
let yn = Fp.sub(xMn, xMd); // 5. yn = xMn - xMd
let yd = Fp.add(xMn, xMd); // 6. yd = xMn + xMd # (n / d - 1) / (n / d + 1) = (n - d) / (n + d)
let tv1 = Fp.mul(xd, yd); // 7. tv1 = xd * yd
let e = Fp.eql(tv1, Fp.ZERO); // 8. e = tv1 == 0
xn = Fp.cmov(xn, Fp.ZERO, e); // 9. xn = CMOV(xn, 0, e)
xd = Fp.cmov(xd, Fp.ONE, e); // 10. xd = CMOV(xd, 1, e)
yn = Fp.cmov(yn, Fp.ONE, e); // 11. yn = CMOV(yn, 1, e)
yd = Fp.cmov(yd, Fp.ONE, e); // 12. yd = CMOV(yd, 1, e)
const inv = Fp.invertBatch([xd, yd]); // batch division
return { x: Fp.mul(xn, inv[0]), y: Fp.mul(yn, inv[1]) }; // 13. return (xn, xd, yn, yd)
}
const htf = /* @__PURE__ */ (() =>
createHasher(
ed25519.ExtendedPoint,
(scalars: bigint[]) => map_to_curve_elligator2_edwards25519(scalars[0]),
{
DST: 'edwards25519_XMD:SHA-512_ELL2_RO_',
encodeDST: 'edwards25519_XMD:SHA-512_ELL2_NU_',
p: Fp.ORDER,
m: 1,
k: 128,
expand: 'xmd',
hash: sha512,
}
))();
export const hashToCurve = /* @__PURE__ */ (() => htf.hashToCurve)();
export const encodeToCurve = /* @__PURE__ */ (() => htf.encodeToCurve)();
function assertRstPoint(other: unknown) {
if (!(other instanceof RistPoint)) throw new Error('RistrettoPoint expected');
}
// √(-1) aka √(a) aka 2^((p-1)/4)
const SQRT_M1 = ED25519_SQRT_M1;
// √(ad - 1)
const SQRT_AD_MINUS_ONE = BigInt(
'25063068953384623474111414158702152701244531502492656460079210482610430750235'
);
// 1 / √(a-d)
const INVSQRT_A_MINUS_D = BigInt(
'54469307008909316920995813868745141605393597292927456921205312896311721017578'
);
// 1-d²
const ONE_MINUS_D_SQ = BigInt(
'1159843021668779879193775521855586647937357759715417654439879720876111806838'
);
// (d-1)²
const D_MINUS_ONE_SQ = BigInt(
'40440834346308536858101042469323190826248399146238708352240133220865137265952'
);
// Calculates 1/√(number)
const invertSqrt = (number: bigint) => uvRatio(_1n, number);
const MAX_255B = BigInt('0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff');
const bytes255ToNumberLE = (bytes: Uint8Array) =>
ed25519.CURVE.Fp.create(bytesToNumberLE(bytes) & MAX_255B);
type ExtendedPoint = ExtPointType;
// Computes Elligator map for Ristretto
// https://ristretto.group/formulas/elligator.html
function calcElligatorRistrettoMap(r0: bigint): ExtendedPoint {
const { d } = ed25519.CURVE;
const P = ed25519.CURVE.Fp.ORDER;
const mod = ed25519.CURVE.Fp.create;
const r = mod(SQRT_M1 * r0 * r0); // 1
const Ns = mod((r + _1n) * ONE_MINUS_D_SQ); // 2
let c = BigInt(-1); // 3
const D = mod((c - d * r) * mod(r + d)); // 4
let { isValid: Ns_D_is_sq, value: s } = uvRatio(Ns, D); // 5
let s_ = mod(s * r0); // 6
if (!isNegativeLE(s_, P)) s_ = mod(-s_);
if (!Ns_D_is_sq) s = s_; // 7
if (!Ns_D_is_sq) c = r; // 8
const Nt = mod(c * (r - _1n) * D_MINUS_ONE_SQ - D); // 9
const s2 = s * s;
const W0 = mod((s + s) * D); // 10
const W1 = mod(Nt * SQRT_AD_MINUS_ONE); // 11
const W2 = mod(_1n - s2); // 12
const W3 = mod(_1n + s2); // 13
return new ed25519.ExtendedPoint(mod(W0 * W3), mod(W2 * W1), mod(W1 * W3), mod(W0 * W2));
}
/**
* Each ed25519/ExtendedPoint has 8 different equivalent points. This can be
* a source of bugs for protocols like ring signatures. Ristretto was created to solve this.
* Ristretto point operates in X:Y:Z:T extended coordinates like ExtendedPoint,
* but it should work in its own namespace: do not combine those two.
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448
*/
class RistPoint {
static BASE: RistPoint;
static ZERO: RistPoint;
// Private property to discourage combining ExtendedPoint + RistrettoPoint
// Always use Ristretto encoding/decoding instead.
constructor(private readonly ep: ExtendedPoint) {}
static fromAffine(ap: AffinePoint<bigint>) {
return new RistPoint(ed25519.ExtendedPoint.fromAffine(ap));
}
/**
* Takes uniform output of 64-byte hash function like sha512 and converts it to `RistrettoPoint`.
* The hash-to-group operation applies Elligator twice and adds the results.
* **Note:** this is one-way map, there is no conversion from point to hash.
* https://ristretto.group/formulas/elligator.html
* @param hex 64-byte output of a hash function
*/
static hashToCurve(hex: Hex): RistPoint {
hex = ensureBytes('ristrettoHash', hex, 64);
const r1 = bytes255ToNumberLE(hex.slice(0, 32));
const R1 = calcElligatorRistrettoMap(r1);
const r2 = bytes255ToNumberLE(hex.slice(32, 64));
const R2 = calcElligatorRistrettoMap(r2);
return new RistPoint(R1.add(R2));
}
/**
* Converts ristretto-encoded string to ristretto point.
* https://ristretto.group/formulas/decoding.html
* @param hex Ristretto-encoded 32 bytes. Not every 32-byte string is valid ristretto encoding
*/
static fromHex(hex: Hex): RistPoint {
hex = ensureBytes('ristrettoHex', hex, 32);
const { a, d } = ed25519.CURVE;
const P = ed25519.CURVE.Fp.ORDER;
const mod = ed25519.CURVE.Fp.create;
const emsg = 'RistrettoPoint.fromHex: the hex is not valid encoding of RistrettoPoint';
const s = bytes255ToNumberLE(hex);
// 1. Check that s_bytes is the canonical encoding of a field element, or else abort.
// 3. Check that s is non-negative, or else abort
if (!equalBytes(numberToBytesLE(s, 32), hex) || isNegativeLE(s, P)) throw new Error(emsg);
const s2 = mod(s * s);
const u1 = mod(_1n + a * s2); // 4 (a is -1)
const u2 = mod(_1n - a * s2); // 5
const u1_2 = mod(u1 * u1);
const u2_2 = mod(u2 * u2);
const v = mod(a * d * u1_2 - u2_2); // 6
const { isValid, value: I } = invertSqrt(mod(v * u2_2)); // 7
const Dx = mod(I * u2); // 8
const Dy = mod(I * Dx * v); // 9
let x = mod((s + s) * Dx); // 10
if (isNegativeLE(x, P)) x = mod(-x); // 10
const y = mod(u1 * Dy); // 11
const t = mod(x * y); // 12
if (!isValid || isNegativeLE(t, P) || y === _0n) throw new Error(emsg);
return new RistPoint(new ed25519.ExtendedPoint(x, y, _1n, t));
}
/**
* Encodes ristretto point to Uint8Array.
* https://ristretto.group/formulas/encoding.html
*/
toRawBytes(): Uint8Array {
let { ex: x, ey: y, ez: z, et: t } = this.ep;
const P = ed25519.CURVE.Fp.ORDER;
const mod = ed25519.CURVE.Fp.create;
const u1 = mod(mod(z + y) * mod(z - y)); // 1
const u2 = mod(x * y); // 2
// Square root always exists
const u2sq = mod(u2 * u2);
const { value: invsqrt } = invertSqrt(mod(u1 * u2sq)); // 3
const D1 = mod(invsqrt * u1); // 4
const D2 = mod(invsqrt * u2); // 5
const zInv = mod(D1 * D2 * t); // 6
let D: bigint; // 7
if (isNegativeLE(t * zInv, P)) {
let _x = mod(y * SQRT_M1);
let _y = mod(x * SQRT_M1);
x = _x;
y = _y;
D = mod(D1 * INVSQRT_A_MINUS_D);
} else {
D = D2; // 8
}
if (isNegativeLE(x * zInv, P)) y = mod(-y); // 9
let s = mod((z - y) * D); // 10 (check footer's note, no sqrt(-a))
if (isNegativeLE(s, P)) s = mod(-s);
return numberToBytesLE(s, 32); // 11
}
toHex(): string {
return bytesToHex(this.toRawBytes());
}
toString(): string {
return this.toHex();
}
// Compare one point to another.
equals(other: RistPoint): boolean {
assertRstPoint(other);
const { ex: X1, ey: Y1 } = this.ep;
const { ex: X2, ey: Y2 } = other.ep;
const mod = ed25519.CURVE.Fp.create;
// (x1 * y2 == y1 * x2) | (y1 * y2 == x1 * x2)
const one = mod(X1 * Y2) === mod(Y1 * X2);
const two = mod(Y1 * Y2) === mod(X1 * X2);
return one || two;
}
add(other: RistPoint): RistPoint {
assertRstPoint(other);
return new RistPoint(this.ep.add(other.ep));
}
subtract(other: RistPoint): RistPoint {
assertRstPoint(other);
return new RistPoint(this.ep.subtract(other.ep));
}
multiply(scalar: bigint): RistPoint {
return new RistPoint(this.ep.multiply(scalar));
}
multiplyUnsafe(scalar: bigint): RistPoint {
return new RistPoint(this.ep.multiplyUnsafe(scalar));
}
}
export const RistrettoPoint = /* @__PURE__ */ (() => {
if (!RistPoint.BASE) RistPoint.BASE = new RistPoint(ed25519.ExtendedPoint.BASE);
if (!RistPoint.ZERO) RistPoint.ZERO = new RistPoint(ed25519.ExtendedPoint.ZERO);
return RistPoint;
})();
// Hashing to ristretto255. https://www.rfc-editor.org/rfc/rfc9380#appendix-B
export const hashToRistretto255 = (msg: Uint8Array, options: htfBasicOpts) => {
const d = options.DST;
const DST = typeof d === 'string' ? utf8ToBytes(d) : d;
const uniform_bytes = expand_message_xmd(msg, DST, 64, sha512);
const P = RistPoint.hashToCurve(uniform_bytes);
return P;
};
export const hash_to_ristretto255 = hashToRistretto255; // legacy

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { shake256 } from '@noble/hashes/sha3';
import { concatBytes, randomBytes, utf8ToBytes, wrapConstructor } from '@noble/hashes/utils';
import { ExtPointType, twistedEdwards } from './abstract/edwards.js';
import { mod, pow2, Field, isNegativeLE } from './abstract/modular.js';
import { montgomery } from './abstract/montgomery.js';
import { createHasher, htfBasicOpts, expand_message_xof } from './abstract/hash-to-curve.js';
import {
bytesToHex,
bytesToNumberLE,
ensureBytes,
equalBytes,
Hex,
numberToBytesLE,
} from './abstract/utils.js';
import { AffinePoint } from './abstract/curve.js';
/**
* Edwards448 (not Ed448-Goldilocks) curve with following addons:
* - X448 ECDH
* - Decaf cofactor elimination
* - Elligator hash-to-group / point indistinguishability
* Conforms to RFC 8032 https://www.rfc-editor.org/rfc/rfc8032.html#section-5.2
*/
const shake256_114 = wrapConstructor(() => shake256.create({ dkLen: 114 }));
const shake256_64 = wrapConstructor(() => shake256.create({ dkLen: 64 }));
const ed448P = BigInt(
'726838724295606890549323807888004534353641360687318060281490199180612328166730772686396383698676545930088884461843637361053498018365439'
);
// prettier-ignore
const _1n = BigInt(1), _2n = BigInt(2), _3n = BigInt(3), _4n = BigInt(4), _11n = BigInt(11);
// prettier-ignore
const _22n = BigInt(22), _44n = BigInt(44), _88n = BigInt(88), _223n = BigInt(223);
// powPminus3div4 calculates z = x^k mod p, where k = (p-3)/4.
// Used for efficient square root calculation.
// ((P-3)/4).toString(2) would produce bits [223x 1, 0, 222x 1]
function ed448_pow_Pminus3div4(x: bigint): bigint {
const P = ed448P;
const b2 = (x * x * x) % P;
const b3 = (b2 * b2 * x) % P;
const b6 = (pow2(b3, _3n, P) * b3) % P;
const b9 = (pow2(b6, _3n, P) * b3) % P;
const b11 = (pow2(b9, _2n, P) * b2) % P;
const b22 = (pow2(b11, _11n, P) * b11) % P;
const b44 = (pow2(b22, _22n, P) * b22) % P;
const b88 = (pow2(b44, _44n, P) * b44) % P;
const b176 = (pow2(b88, _88n, P) * b88) % P;
const b220 = (pow2(b176, _44n, P) * b44) % P;
const b222 = (pow2(b220, _2n, P) * b2) % P;
const b223 = (pow2(b222, _1n, P) * x) % P;
return (pow2(b223, _223n, P) * b222) % P;
}
function adjustScalarBytes(bytes: Uint8Array): Uint8Array {
// Section 5: Likewise, for X448, set the two least significant bits of the first byte to 0, and the most
// significant bit of the last byte to 1.
bytes[0] &= 252; // 0b11111100
// and the most significant bit of the last byte to 1.
bytes[55] |= 128; // 0b10000000
// NOTE: is is NOOP for 56 bytes scalars (X25519/X448)
bytes[56] = 0; // Byte outside of group (456 buts vs 448 bits)
return bytes;
}
// Constant-time ratio of u to v. Allows to combine inversion and square root u/√v.
// Uses algo from RFC8032 5.1.3.
function uvRatio(u: bigint, v: bigint): { isValid: boolean; value: bigint } {
const P = ed448P;
// https://www.rfc-editor.org/rfc/rfc8032#section-5.2.3
// To compute the square root of (u/v), the first step is to compute the
// candidate root x = (u/v)^((p+1)/4). This can be done using the
// following trick, to use a single modular powering for both the
// inversion of v and the square root:
// x = (u/v)^((p+1)/4) = u³v(u⁵v³)^((p-3)/4) (mod p)
const u2v = mod(u * u * v, P); // u²v
const u3v = mod(u2v * u, P); // u³v
const u5v3 = mod(u3v * u2v * v, P); // u⁵v³
const root = ed448_pow_Pminus3div4(u5v3);
const x = mod(u3v * root, P);
// Verify that root is exists
const x2 = mod(x * x, P); // x²
// If vx² = u, the recovered x-coordinate is x. Otherwise, no
// square root exists, and the decoding fails.
return { isValid: mod(x2 * v, P) === u, value: x };
}
const Fp = Field(ed448P, 456, true);
const ED448_DEF = {
// Param: a
a: BigInt(1),
// -39081. Negative number is P - number
d: BigInt(
'726838724295606890549323807888004534353641360687318060281490199180612328166730772686396383698676545930088884461843637361053498018326358'
),
// Finite field 𝔽p over which we'll do calculations; 2n**448n - 2n**224n - 1n
Fp,
// Subgroup order: how many points curve has;
// 2n**446n - 13818066809895115352007386748515426880336692474882178609894547503885n
n: BigInt(
'181709681073901722637330951972001133588410340171829515070372549795146003961539585716195755291692375963310293709091662304773755859649779'
),
nBitLength: 456,
// Cofactor
h: BigInt(4),
// Base point (x, y) aka generator point
Gx: BigInt(
'224580040295924300187604334099896036246789641632564134246125461686950415467406032909029192869357953282578032075146446173674602635247710'
),
Gy: BigInt(
'298819210078481492676017930443930673437544040154080242095928241372331506189835876003536878655418784733982303233503462500531545062832660'
),
// SHAKE256(dom4(phflag,context)||x, 114)
hash: shake256_114,
randomBytes,
adjustScalarBytes,
// dom4
domain: (data: Uint8Array, ctx: Uint8Array, phflag: boolean) => {
if (ctx.length > 255) throw new Error(`Context is too big: ${ctx.length}`);
return concatBytes(
utf8ToBytes('SigEd448'),
new Uint8Array([phflag ? 1 : 0, ctx.length]),
ctx,
data
);
},
uvRatio,
} as const;
export const ed448 = /* @__PURE__ */ twistedEdwards(ED448_DEF);
// NOTE: there is no ed448ctx, since ed448 supports ctx by default
export const ed448ph = /* @__PURE__ */ twistedEdwards({ ...ED448_DEF, prehash: shake256_64 });
export const x448 = /* @__PURE__ */ (() =>
montgomery({
a: BigInt(156326),
montgomeryBits: 448,
nByteLength: 57,
P: ed448P,
Gu: BigInt(5),
powPminus2: (x: bigint): bigint => {
const P = ed448P;
const Pminus3div4 = ed448_pow_Pminus3div4(x);
const Pminus3 = pow2(Pminus3div4, BigInt(2), P);
return mod(Pminus3 * x, P); // Pminus3 * x = Pminus2
},
adjustScalarBytes,
randomBytes,
}))();
/**
* Converts edwards448 public key to x448 public key. Uses formula:
* * `(u, v) = ((y-1)/(y+1), sqrt(156324)*u/x)`
* * `(x, y) = (sqrt(156324)*u/v, (1+u)/(1-u))`
* @example
* const aPub = ed448.getPublicKey(utils.randomPrivateKey());
* x448.getSharedSecret(edwardsToMontgomery(aPub), edwardsToMontgomery(someonesPub))
*/
export function edwardsToMontgomeryPub(edwardsPub: string | Uint8Array): Uint8Array {
const { y } = ed448.ExtendedPoint.fromHex(edwardsPub);
const _1n = BigInt(1);
return Fp.toBytes(Fp.create((y - _1n) * Fp.inv(y + _1n)));
}
export const edwardsToMontgomery = edwardsToMontgomeryPub; // deprecated
// Hash To Curve Elligator2 Map
const ELL2_C1 = (Fp.ORDER - BigInt(3)) / BigInt(4); // 1. c1 = (q - 3) / 4 # Integer arithmetic
const ELL2_J = BigInt(156326);
function map_to_curve_elligator2_curve448(u: bigint) {
let tv1 = Fp.sqr(u); // 1. tv1 = u^2
let e1 = Fp.eql(tv1, Fp.ONE); // 2. e1 = tv1 == 1
tv1 = Fp.cmov(tv1, Fp.ZERO, e1); // 3. tv1 = CMOV(tv1, 0, e1) # If Z * u^2 == -1, set tv1 = 0
let xd = Fp.sub(Fp.ONE, tv1); // 4. xd = 1 - tv1
let x1n = Fp.neg(ELL2_J); // 5. x1n = -J
let tv2 = Fp.sqr(xd); // 6. tv2 = xd^2
let gxd = Fp.mul(tv2, xd); // 7. gxd = tv2 * xd # gxd = xd^3
let gx1 = Fp.mul(tv1, Fp.neg(ELL2_J)); // 8. gx1 = -J * tv1 # x1n + J * xd
gx1 = Fp.mul(gx1, x1n); // 9. gx1 = gx1 * x1n # x1n^2 + J * x1n * xd
gx1 = Fp.add(gx1, tv2); // 10. gx1 = gx1 + tv2 # x1n^2 + J * x1n * xd + xd^2
gx1 = Fp.mul(gx1, x1n); // 11. gx1 = gx1 * x1n # x1n^3 + J * x1n^2 * xd + x1n * xd^2
let tv3 = Fp.sqr(gxd); // 12. tv3 = gxd^2
tv2 = Fp.mul(gx1, gxd); // 13. tv2 = gx1 * gxd # gx1 * gxd
tv3 = Fp.mul(tv3, tv2); // 14. tv3 = tv3 * tv2 # gx1 * gxd^3
let y1 = Fp.pow(tv3, ELL2_C1); // 15. y1 = tv3^c1 # (gx1 * gxd^3)^((p - 3) / 4)
y1 = Fp.mul(y1, tv2); // 16. y1 = y1 * tv2 # gx1 * gxd * (gx1 * gxd^3)^((p - 3) / 4)
let x2n = Fp.mul(x1n, Fp.neg(tv1)); // 17. x2n = -tv1 * x1n # x2 = x2n / xd = -1 * u^2 * x1n / xd
let y2 = Fp.mul(y1, u); // 18. y2 = y1 * u
y2 = Fp.cmov(y2, Fp.ZERO, e1); // 19. y2 = CMOV(y2, 0, e1)
tv2 = Fp.sqr(y1); // 20. tv2 = y1^2
tv2 = Fp.mul(tv2, gxd); // 21. tv2 = tv2 * gxd
let e2 = Fp.eql(tv2, gx1); // 22. e2 = tv2 == gx1
let xn = Fp.cmov(x2n, x1n, e2); // 23. xn = CMOV(x2n, x1n, e2) # If e2, x = x1, else x = x2
let y = Fp.cmov(y2, y1, e2); // 24. y = CMOV(y2, y1, e2) # If e2, y = y1, else y = y2
let e3 = Fp.isOdd(y); // 25. e3 = sgn0(y) == 1 # Fix sign of y
y = Fp.cmov(y, Fp.neg(y), e2 !== e3); // 26. y = CMOV(y, -y, e2 XOR e3)
return { xn, xd, yn: y, yd: Fp.ONE }; // 27. return (xn, xd, y, 1)
}
function map_to_curve_elligator2_edwards448(u: bigint) {
let { xn, xd, yn, yd } = map_to_curve_elligator2_curve448(u); // 1. (xn, xd, yn, yd) = map_to_curve_elligator2_curve448(u)
let xn2 = Fp.sqr(xn); // 2. xn2 = xn^2
let xd2 = Fp.sqr(xd); // 3. xd2 = xd^2
let xd4 = Fp.sqr(xd2); // 4. xd4 = xd2^2
let yn2 = Fp.sqr(yn); // 5. yn2 = yn^2
let yd2 = Fp.sqr(yd); // 6. yd2 = yd^2
let xEn = Fp.sub(xn2, xd2); // 7. xEn = xn2 - xd2
let tv2 = Fp.sub(xEn, xd2); // 8. tv2 = xEn - xd2
xEn = Fp.mul(xEn, xd2); // 9. xEn = xEn * xd2
xEn = Fp.mul(xEn, yd); // 10. xEn = xEn * yd
xEn = Fp.mul(xEn, yn); // 11. xEn = xEn * yn
xEn = Fp.mul(xEn, _4n); // 12. xEn = xEn * 4
tv2 = Fp.mul(tv2, xn2); // 13. tv2 = tv2 * xn2
tv2 = Fp.mul(tv2, yd2); // 14. tv2 = tv2 * yd2
let tv3 = Fp.mul(yn2, _4n); // 15. tv3 = 4 * yn2
let tv1 = Fp.add(tv3, yd2); // 16. tv1 = tv3 + yd2
tv1 = Fp.mul(tv1, xd4); // 17. tv1 = tv1 * xd4
let xEd = Fp.add(tv1, tv2); // 18. xEd = tv1 + tv2
tv2 = Fp.mul(tv2, xn); // 19. tv2 = tv2 * xn
let tv4 = Fp.mul(xn, xd4); // 20. tv4 = xn * xd4
let yEn = Fp.sub(tv3, yd2); // 21. yEn = tv3 - yd2
yEn = Fp.mul(yEn, tv4); // 22. yEn = yEn * tv4
yEn = Fp.sub(yEn, tv2); // 23. yEn = yEn - tv2
tv1 = Fp.add(xn2, xd2); // 24. tv1 = xn2 + xd2
tv1 = Fp.mul(tv1, xd2); // 25. tv1 = tv1 * xd2
tv1 = Fp.mul(tv1, xd); // 26. tv1 = tv1 * xd
tv1 = Fp.mul(tv1, yn2); // 27. tv1 = tv1 * yn2
tv1 = Fp.mul(tv1, BigInt(-2)); // 28. tv1 = -2 * tv1
let yEd = Fp.add(tv2, tv1); // 29. yEd = tv2 + tv1
tv4 = Fp.mul(tv4, yd2); // 30. tv4 = tv4 * yd2
yEd = Fp.add(yEd, tv4); // 31. yEd = yEd + tv4
tv1 = Fp.mul(xEd, yEd); // 32. tv1 = xEd * yEd
let e = Fp.eql(tv1, Fp.ZERO); // 33. e = tv1 == 0
xEn = Fp.cmov(xEn, Fp.ZERO, e); // 34. xEn = CMOV(xEn, 0, e)
xEd = Fp.cmov(xEd, Fp.ONE, e); // 35. xEd = CMOV(xEd, 1, e)
yEn = Fp.cmov(yEn, Fp.ONE, e); // 36. yEn = CMOV(yEn, 1, e)
yEd = Fp.cmov(yEd, Fp.ONE, e); // 37. yEd = CMOV(yEd, 1, e)
const inv = Fp.invertBatch([xEd, yEd]); // batch division
return { x: Fp.mul(xEn, inv[0]), y: Fp.mul(yEn, inv[1]) }; // 38. return (xEn, xEd, yEn, yEd)
}
const htf = /* @__PURE__ */ (() =>
createHasher(
ed448.ExtendedPoint,
(scalars: bigint[]) => map_to_curve_elligator2_edwards448(scalars[0]),
{
DST: 'edwards448_XOF:SHAKE256_ELL2_RO_',
encodeDST: 'edwards448_XOF:SHAKE256_ELL2_NU_',
p: Fp.ORDER,
m: 1,
k: 224,
expand: 'xof',
hash: shake256,
}
))();
export const hashToCurve = /* @__PURE__ */ (() => htf.hashToCurve)();
export const encodeToCurve = /* @__PURE__ */ (() => htf.encodeToCurve)();
function assertDcfPoint(other: unknown) {
if (!(other instanceof DcfPoint)) throw new Error('DecafPoint expected');
}
// 1-d
const ONE_MINUS_D = BigInt('39082');
// 1-2d
const ONE_MINUS_TWO_D = BigInt('78163');
// √(-d)
const SQRT_MINUS_D = BigInt(
'98944233647732219769177004876929019128417576295529901074099889598043702116001257856802131563896515373927712232092845883226922417596214'
);
// 1 / √(-d)
const INVSQRT_MINUS_D = BigInt(
'315019913931389607337177038330951043522456072897266928557328499619017160722351061360252776265186336876723201881398623946864393857820716'
);
// Calculates 1/√(number)
const invertSqrt = (number: bigint) => uvRatio(_1n, number);
const MAX_448B = BigInt(
'0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff'
);
const bytes448ToNumberLE = (bytes: Uint8Array) =>
ed448.CURVE.Fp.create(bytesToNumberLE(bytes) & MAX_448B);
type ExtendedPoint = ExtPointType;
// Computes Elligator map for Decaf
// https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448-07#name-element-derivation-2
function calcElligatorDecafMap(r0: bigint): ExtendedPoint {
const { d } = ed448.CURVE;
const P = ed448.CURVE.Fp.ORDER;
const mod = ed448.CURVE.Fp.create;
const r = mod(-(r0 * r0)); // 1
const u0 = mod(d * (r - _1n)); // 2
const u1 = mod((u0 + _1n) * (u0 - r)); // 3
const { isValid: was_square, value: v } = uvRatio(ONE_MINUS_TWO_D, mod((r + _1n) * u1)); // 4
let v_prime = v; // 5
if (!was_square) v_prime = mod(r0 * v);
let sgn = _1n; // 6
if (!was_square) sgn = mod(-_1n);
const s = mod(v_prime * (r + _1n)); // 7
let s_abs = s;
if (isNegativeLE(s, P)) s_abs = mod(-s);
const s2 = s * s;
const W0 = mod(s_abs * _2n); // 8
const W1 = mod(s2 + _1n); // 9
const W2 = mod(s2 - _1n); // 10
const W3 = mod(v_prime * s * (r - _1n) * ONE_MINUS_TWO_D + sgn); // 11
return new ed448.ExtendedPoint(mod(W0 * W3), mod(W2 * W1), mod(W1 * W3), mod(W0 * W2));
}
/**
* Each ed448/ExtendedPoint has 4 different equivalent points. This can be
* a source of bugs for protocols like ring signatures. Decaf was created to solve this.
* Decaf point operates in X:Y:Z:T extended coordinates like ExtendedPoint,
* but it should work in its own namespace: do not combine those two.
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448
*/
class DcfPoint {
static BASE: DcfPoint;
static ZERO: DcfPoint;
// Private property to discourage combining ExtendedPoint + DecafPoint
// Always use Decaf encoding/decoding instead.
constructor(private readonly ep: ExtendedPoint) {}
static fromAffine(ap: AffinePoint<bigint>) {
return new DcfPoint(ed448.ExtendedPoint.fromAffine(ap));
}
/**
* Takes uniform output of 112-byte hash function like shake256 and converts it to `DecafPoint`.
* The hash-to-group operation applies Elligator twice and adds the results.
* **Note:** this is one-way map, there is no conversion from point to hash.
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448-07#name-element-derivation-2
* @param hex 112-byte output of a hash function
*/
static hashToCurve(hex: Hex): DcfPoint {
hex = ensureBytes('decafHash', hex, 112);
const r1 = bytes448ToNumberLE(hex.slice(0, 56));
const R1 = calcElligatorDecafMap(r1);
const r2 = bytes448ToNumberLE(hex.slice(56, 112));
const R2 = calcElligatorDecafMap(r2);
return new DcfPoint(R1.add(R2));
}
/**
* Converts decaf-encoded string to decaf point.
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448-07#name-decode-2
* @param hex Decaf-encoded 56 bytes. Not every 56-byte string is valid decaf encoding
*/
static fromHex(hex: Hex): DcfPoint {
hex = ensureBytes('decafHex', hex, 56);
const { d } = ed448.CURVE;
const P = ed448.CURVE.Fp.ORDER;
const mod = ed448.CURVE.Fp.create;
const emsg = 'DecafPoint.fromHex: the hex is not valid encoding of DecafPoint';
const s = bytes448ToNumberLE(hex);
// 1. Check that s_bytes is the canonical encoding of a field element, or else abort.
// 2. Check that s is non-negative, or else abort
if (!equalBytes(numberToBytesLE(s, 56), hex) || isNegativeLE(s, P)) throw new Error(emsg);
const s2 = mod(s * s); // 1
const u1 = mod(_1n + s2); // 2
const u1sq = mod(u1 * u1);
const u2 = mod(u1sq - _4n * d * s2); // 3
const { isValid, value: invsqrt } = invertSqrt(mod(u2 * u1sq)); // 4
let u3 = mod((s + s) * invsqrt * u1 * SQRT_MINUS_D); // 5
if (isNegativeLE(u3, P)) u3 = mod(-u3);
const x = mod(u3 * invsqrt * u2 * INVSQRT_MINUS_D); // 6
const y = mod((_1n - s2) * invsqrt * u1); // 7
const t = mod(x * y); // 8
if (!isValid) throw new Error(emsg);
return new DcfPoint(new ed448.ExtendedPoint(x, y, _1n, t));
}
/**
* Encodes decaf point to Uint8Array.
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448-07#name-encode-2
*/
toRawBytes(): Uint8Array {
let { ex: x, ey: _y, ez: z, et: t } = this.ep;
const P = ed448.CURVE.Fp.ORDER;
const mod = ed448.CURVE.Fp.create;
const u1 = mod(mod(x + t) * mod(x - t)); // 1
const x2 = mod(x * x);
const { value: invsqrt } = invertSqrt(mod(u1 * ONE_MINUS_D * x2)); // 2
let ratio = mod(invsqrt * u1 * SQRT_MINUS_D); // 3
if (isNegativeLE(ratio, P)) ratio = mod(-ratio);
const u2 = mod(INVSQRT_MINUS_D * ratio * z - t); // 4
let s = mod(ONE_MINUS_D * invsqrt * x * u2); // 5
if (isNegativeLE(s, P)) s = mod(-s);
return numberToBytesLE(s, 56);
}
toHex(): string {
return bytesToHex(this.toRawBytes());
}
toString(): string {
return this.toHex();
}
// Compare one point to another.
// https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448-07#name-equals-2
equals(other: DcfPoint): boolean {
assertDcfPoint(other);
const { ex: X1, ey: Y1 } = this.ep;
const { ex: X2, ey: Y2 } = other.ep;
const mod = ed448.CURVE.Fp.create;
// (x1 * y2 == y1 * x2)
return mod(X1 * Y2) === mod(Y1 * X2);
}
add(other: DcfPoint): DcfPoint {
assertDcfPoint(other);
return new DcfPoint(this.ep.add(other.ep));
}
subtract(other: DcfPoint): DcfPoint {
assertDcfPoint(other);
return new DcfPoint(this.ep.subtract(other.ep));
}
multiply(scalar: bigint): DcfPoint {
return new DcfPoint(this.ep.multiply(scalar));
}
multiplyUnsafe(scalar: bigint): DcfPoint {
return new DcfPoint(this.ep.multiplyUnsafe(scalar));
}
}
export const DecafPoint = /* @__PURE__ */ (() => {
// decaf448 base point is ed448 base x 2
// https://github.com/dalek-cryptography/curve25519-dalek/blob/59837c6ecff02b77b9d5ff84dbc239d0cf33ef90/vendor/ristretto.sage#L699
if (!DcfPoint.BASE) DcfPoint.BASE = new DcfPoint(ed448.ExtendedPoint.BASE).multiply(_2n);
if (!DcfPoint.ZERO) DcfPoint.ZERO = new DcfPoint(ed448.ExtendedPoint.ZERO);
return DcfPoint;
})();
// Hashing to decaf448. https://www.rfc-editor.org/rfc/rfc9380#appendix-C
export const hashToDecaf448 = (msg: Uint8Array, options: htfBasicOpts) => {
const d = options.DST;
const DST = typeof d === 'string' ? utf8ToBytes(d) : d;
const uniform_bytes = expand_message_xof(msg, DST, 112, 224, shake256);
const P = DcfPoint.hashToCurve(uniform_bytes);
return P;
};
export const hash_to_decaf448 = hashToDecaf448; // legacy

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throw new Error('Incorrect usage. Import submodules instead');

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { sha512 } from '@noble/hashes/sha512';
import { concatBytes, randomBytes, utf8ToBytes } from '@noble/hashes/utils';
import { twistedEdwards } from './abstract/edwards.js';
import { blake2s } from '@noble/hashes/blake2s';
import { Field } from './abstract/modular.js';
/**
* jubjub Twisted Edwards curve.
* https://neuromancer.sk/std/other/JubJub
* jubjub does not use EdDSA, so `hash`/sha512 params are passed because interface expects them.
*/
export const jubjub = /* @__PURE__ */ twistedEdwards({
// Params: a, d
a: BigInt('0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000000'),
d: BigInt('0x2a9318e74bfa2b48f5fd9207e6bd7fd4292d7f6d37579d2601065fd6d6343eb1'),
// Finite field 𝔽p over which we'll do calculations
// Same value as bls12-381 Fr (not Fp)
Fp: Field(BigInt('0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001')),
// Subgroup order: how many points curve has
n: BigInt('0xe7db4ea6533afa906673b0101343b00a6682093ccc81082d0970e5ed6f72cb7'),
// Cofactor
h: BigInt(8),
// Base point (x, y) aka generator point
Gx: BigInt('0x11dafe5d23e1218086a365b99fbf3d3be72f6afd7d1f72623e6b071492d1122b'),
Gy: BigInt('0x1d523cf1ddab1a1793132e78c866c0c33e26ba5cc220fed7cc3f870e59d292aa'),
hash: sha512,
randomBytes,
} as const);
const GH_FIRST_BLOCK = utf8ToBytes(
'096b36a5804bfacef1691e173c366a47ff5ba84a44f26ddd7e8d9f79d5b42df0'
);
// Returns point at JubJub curve which is prime order and not zero
export function groupHash(tag: Uint8Array, personalization: Uint8Array) {
const h = blake2s.create({ personalization, dkLen: 32 });
h.update(GH_FIRST_BLOCK);
h.update(tag);
// NOTE: returns ExtendedPoint, in case it will be multiplied later
let p = jubjub.ExtendedPoint.fromHex(h.digest());
// NOTE: cannot replace with isSmallOrder, returns Point*8
p = p.multiply(jubjub.CURVE.h);
if (p.equals(jubjub.ExtendedPoint.ZERO)) throw new Error('Point has small order');
return p;
}
export function findGroupHash(m: Uint8Array, personalization: Uint8Array) {
const tag = concatBytes(m, new Uint8Array([0]));
for (let i = 0; i < 256; i++) {
tag[tag.length - 1] = i;
try {
return groupHash(tag, personalization);
} catch (e) {}
}
throw new Error('findGroupHash tag overflow');
}

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { createCurve } from './_shortw_utils.js';
import { sha256 } from '@noble/hashes/sha256';
import { Field } from './abstract/modular.js';
import { mapToCurveSimpleSWU } from './abstract/weierstrass.js';
import { createHasher } from './abstract/hash-to-curve.js';
// NIST secp256r1 aka p256
// https://www.secg.org/sec2-v2.pdf, https://neuromancer.sk/std/nist/P-256
const Fp = Field(BigInt('0xffffffff00000001000000000000000000000000ffffffffffffffffffffffff'));
const CURVE_A = Fp.create(BigInt('-3'));
const CURVE_B = BigInt('0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e27d2604b');
// prettier-ignore
export const p256 = createCurve({
a: CURVE_A, // Equation params: a, b
b: CURVE_B,
Fp, // Field: 2n**224n * (2n**32n-1n) + 2n**192n + 2n**96n-1n
// Curve order, total count of valid points in the field
n: BigInt('0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9cac2fc632551'),
// Base (generator) point (x, y)
Gx: BigInt('0x6b17d1f2e12c4247f8bce6e563a440f277037d812deb33a0f4a13945d898c296'),
Gy: BigInt('0x4fe342e2fe1a7f9b8ee7eb4a7c0f9e162bce33576b315ececbb6406837bf51f5'),
h: BigInt(1),
lowS: false,
} as const, sha256);
export const secp256r1 = p256;
const mapSWU = /* @__PURE__ */ (() =>
mapToCurveSimpleSWU(Fp, {
A: CURVE_A,
B: CURVE_B,
Z: Fp.create(BigInt('-10')),
}))();
const htf = /* @__PURE__ */ (() =>
createHasher(secp256r1.ProjectivePoint, (scalars: bigint[]) => mapSWU(scalars[0]), {
DST: 'P256_XMD:SHA-256_SSWU_RO_',
encodeDST: 'P256_XMD:SHA-256_SSWU_NU_',
p: Fp.ORDER,
m: 1,
k: 128,
expand: 'xmd',
hash: sha256,
}))();
export const hashToCurve = /* @__PURE__ */ (() => htf.hashToCurve)();
export const encodeToCurve = /* @__PURE__ */ (() => htf.encodeToCurve)();

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thrower_daemon/node_modules/@noble/curves/src/p384.ts generated vendored Normal file
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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { createCurve } from './_shortw_utils.js';
import { sha384 } from '@noble/hashes/sha512';
import { Field } from './abstract/modular.js';
import { mapToCurveSimpleSWU } from './abstract/weierstrass.js';
import { createHasher } from './abstract/hash-to-curve.js';
// NIST secp384r1 aka p384
// https://www.secg.org/sec2-v2.pdf, https://neuromancer.sk/std/nist/P-384
// Field over which we'll do calculations.
// prettier-ignore
const P = BigInt('0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffff0000000000000000ffffffff');
const Fp = Field(P);
const CURVE_A = Fp.create(BigInt('-3'));
// prettier-ignore
const CURVE_B = BigInt('0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5013875ac656398d8a2ed19d2a85c8edd3ec2aef');
// prettier-ignore
export const p384 = createCurve({
a: CURVE_A, // Equation params: a, b
b: CURVE_B,
Fp, // Field: 2n**384n - 2n**128n - 2n**96n + 2n**32n - 1n
// Curve order, total count of valid points in the field.
n: BigInt('0xffffffffffffffffffffffffffffffffffffffffffffffffc7634d81f4372ddf581a0db248b0a77aecec196accc52973'),
// Base (generator) point (x, y)
Gx: BigInt('0xaa87ca22be8b05378eb1c71ef320ad746e1d3b628ba79b9859f741e082542a385502f25dbf55296c3a545e3872760ab7'),
Gy: BigInt('0x3617de4a96262c6f5d9e98bf9292dc29f8f41dbd289a147ce9da3113b5f0b8c00a60b1ce1d7e819d7a431d7c90ea0e5f'),
h: BigInt(1),
lowS: false,
} as const, sha384);
export const secp384r1 = p384;
const mapSWU = /* @__PURE__ */ (() =>
mapToCurveSimpleSWU(Fp, {
A: CURVE_A,
B: CURVE_B,
Z: Fp.create(BigInt('-12')),
}))();
const htf = /* @__PURE__ */ (() =>
createHasher(secp384r1.ProjectivePoint, (scalars: bigint[]) => mapSWU(scalars[0]), {
DST: 'P384_XMD:SHA-384_SSWU_RO_',
encodeDST: 'P384_XMD:SHA-384_SSWU_NU_',
p: Fp.ORDER,
m: 1,
k: 192,
expand: 'xmd',
hash: sha384,
}))();
export const hashToCurve = /* @__PURE__ */ (() => htf.hashToCurve)();
export const encodeToCurve = /* @__PURE__ */ (() => htf.encodeToCurve)();

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { createCurve } from './_shortw_utils.js';
import { sha512 } from '@noble/hashes/sha512';
import { Field } from './abstract/modular.js';
import { mapToCurveSimpleSWU } from './abstract/weierstrass.js';
import { createHasher } from './abstract/hash-to-curve.js';
// NIST secp521r1 aka p521
// Note that it's 521, which differs from 512 of its hash function.
// https://www.secg.org/sec2-v2.pdf, https://neuromancer.sk/std/nist/P-521
// Field over which we'll do calculations.
// prettier-ignore
const P = BigInt('0x1ffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff');
const Fp = Field(P);
const CURVE = {
a: Fp.create(BigInt('-3')),
b: BigInt(
'0x0051953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b489918ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451fd46b503f00'
),
Fp,
n: BigInt(
'0x01fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb71e91386409'
),
Gx: BigInt(
'0x00c6858e06b70404e9cd9e3ecb662395b4429c648139053fb521f828af606b4d3dbaa14b5e77efe75928fe1dc127a2ffa8de3348b3c1856a429bf97e7e31c2e5bd66'
),
Gy: BigInt(
'0x011839296a789a3bc0045c8a5fb42c7d1bd998f54449579b446817afbd17273e662c97ee72995ef42640c550b9013fad0761353c7086a272c24088be94769fd16650'
),
h: BigInt(1),
};
// prettier-ignore
export const p521 = createCurve({
a: CURVE.a, // Equation params: a, b
b: CURVE.b,
Fp, // Field: 2n**521n - 1n
// Curve order, total count of valid points in the field
n: CURVE.n,
Gx: CURVE.Gx, // Base point (x, y) aka generator point
Gy: CURVE.Gy,
h: CURVE.h,
lowS: false,
allowedPrivateKeyLengths: [130, 131, 132] // P521 keys are variable-length. Normalize to 132b
} as const, sha512);
export const secp521r1 = p521;
const mapSWU = /* @__PURE__ */ (() =>
mapToCurveSimpleSWU(Fp, {
A: CURVE.a,
B: CURVE.b,
Z: Fp.create(BigInt('-4')),
}))();
const htf = /* @__PURE__ */ (() =>
createHasher(secp521r1.ProjectivePoint, (scalars: bigint[]) => mapSWU(scalars[0]), {
DST: 'P521_XMD:SHA-512_SSWU_RO_',
encodeDST: 'P521_XMD:SHA-512_SSWU_NU_',
p: Fp.ORDER,
m: 1,
k: 256,
expand: 'xmd',
hash: sha512,
}))();
export const hashToCurve = /* @__PURE__ */ (() => htf.hashToCurve)();
export const encodeToCurve = /* @__PURE__ */ (() => htf.encodeToCurve)();

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { sha256 } from '@noble/hashes/sha256';
import { weierstrass } from './abstract/weierstrass.js';
import { getHash } from './_shortw_utils.js';
import * as mod from './abstract/modular.js';
export const p = BigInt('0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001');
export const q = BigInt('0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001');
// https://neuromancer.sk/std/other/Pallas
export const pallas = weierstrass({
a: BigInt(0),
b: BigInt(5),
Fp: mod.Field(p),
n: q,
Gx: mod.mod(BigInt(-1), p),
Gy: BigInt(2),
h: BigInt(1),
...getHash(sha256),
});
// https://neuromancer.sk/std/other/Vesta
export const vesta = weierstrass({
a: BigInt(0),
b: BigInt(5),
Fp: mod.Field(q),
n: p,
Gx: mod.mod(BigInt(-1), q),
Gy: BigInt(2),
h: BigInt(1),
...getHash(sha256),
});

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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { sha256 } from '@noble/hashes/sha256';
import { randomBytes } from '@noble/hashes/utils';
import { Field, mod, pow2 } from './abstract/modular.js';
import { ProjPointType as PointType, mapToCurveSimpleSWU } from './abstract/weierstrass.js';
import type { Hex, PrivKey } from './abstract/utils.js';
import { bytesToNumberBE, concatBytes, ensureBytes, numberToBytesBE } from './abstract/utils.js';
import { createHasher, isogenyMap } from './abstract/hash-to-curve.js';
import { createCurve } from './_shortw_utils.js';
const secp256k1P = BigInt('0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f');
const secp256k1N = BigInt('0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141');
const _1n = BigInt(1);
const _2n = BigInt(2);
const divNearest = (a: bigint, b: bigint) => (a + b / _2n) / b;
/**
* √n = n^((p+1)/4) for fields p = 3 mod 4. We unwrap the loop and multiply bit-by-bit.
* (P+1n/4n).toString(2) would produce bits [223x 1, 0, 22x 1, 4x 0, 11, 00]
*/
function sqrtMod(y: bigint): bigint {
const P = secp256k1P;
// prettier-ignore
const _3n = BigInt(3), _6n = BigInt(6), _11n = BigInt(11), _22n = BigInt(22);
// prettier-ignore
const _23n = BigInt(23), _44n = BigInt(44), _88n = BigInt(88);
const b2 = (y * y * y) % P; // x^3, 11
const b3 = (b2 * b2 * y) % P; // x^7
const b6 = (pow2(b3, _3n, P) * b3) % P;
const b9 = (pow2(b6, _3n, P) * b3) % P;
const b11 = (pow2(b9, _2n, P) * b2) % P;
const b22 = (pow2(b11, _11n, P) * b11) % P;
const b44 = (pow2(b22, _22n, P) * b22) % P;
const b88 = (pow2(b44, _44n, P) * b44) % P;
const b176 = (pow2(b88, _88n, P) * b88) % P;
const b220 = (pow2(b176, _44n, P) * b44) % P;
const b223 = (pow2(b220, _3n, P) * b3) % P;
const t1 = (pow2(b223, _23n, P) * b22) % P;
const t2 = (pow2(t1, _6n, P) * b2) % P;
const root = pow2(t2, _2n, P);
if (!Fp.eql(Fp.sqr(root), y)) throw new Error('Cannot find square root');
return root;
}
const Fp = Field(secp256k1P, undefined, undefined, { sqrt: sqrtMod });
export const secp256k1 = createCurve(
{
a: BigInt(0), // equation params: a, b
b: BigInt(7), // Seem to be rigid: bitcointalk.org/index.php?topic=289795.msg3183975#msg3183975
Fp, // Field's prime: 2n**256n - 2n**32n - 2n**9n - 2n**8n - 2n**7n - 2n**6n - 2n**4n - 1n
n: secp256k1N, // Curve order, total count of valid points in the field
// Base point (x, y) aka generator point
Gx: BigInt('55066263022277343669578718895168534326250603453777594175500187360389116729240'),
Gy: BigInt('32670510020758816978083085130507043184471273380659243275938904335757337482424'),
h: BigInt(1), // Cofactor
lowS: true, // Allow only low-S signatures by default in sign() and verify()
/**
* secp256k1 belongs to Koblitz curves: it has efficiently computable endomorphism.
* Endomorphism uses 2x less RAM, speeds up precomputation by 2x and ECDH / key recovery by 20%.
* For precomputed wNAF it trades off 1/2 init time & 1/3 ram for 20% perf hit.
* Explanation: https://gist.github.com/paulmillr/eb670806793e84df628a7c434a873066
*/
endo: {
beta: BigInt('0x7ae96a2b657c07106e64479eac3434e99cf0497512f58995c1396c28719501ee'),
splitScalar: (k: bigint) => {
const n = secp256k1N;
const a1 = BigInt('0x3086d221a7d46bcde86c90e49284eb15');
const b1 = -_1n * BigInt('0xe4437ed6010e88286f547fa90abfe4c3');
const a2 = BigInt('0x114ca50f7a8e2f3f657c1108d9d44cfd8');
const b2 = a1;
const POW_2_128 = BigInt('0x100000000000000000000000000000000'); // (2n**128n).toString(16)
const c1 = divNearest(b2 * k, n);
const c2 = divNearest(-b1 * k, n);
let k1 = mod(k - c1 * a1 - c2 * a2, n);
let k2 = mod(-c1 * b1 - c2 * b2, n);
const k1neg = k1 > POW_2_128;
const k2neg = k2 > POW_2_128;
if (k1neg) k1 = n - k1;
if (k2neg) k2 = n - k2;
if (k1 > POW_2_128 || k2 > POW_2_128) {
throw new Error('splitScalar: Endomorphism failed, k=' + k);
}
return { k1neg, k1, k2neg, k2 };
},
},
},
sha256
);
// Schnorr signatures are superior to ECDSA from above. Below is Schnorr-specific BIP0340 code.
// https://github.com/bitcoin/bips/blob/master/bip-0340.mediawiki
const _0n = BigInt(0);
const fe = (x: bigint) => typeof x === 'bigint' && _0n < x && x < secp256k1P;
const ge = (x: bigint) => typeof x === 'bigint' && _0n < x && x < secp256k1N;
/** An object mapping tags to their tagged hash prefix of [SHA256(tag) | SHA256(tag)] */
const TAGGED_HASH_PREFIXES: { [tag: string]: Uint8Array } = {};
function taggedHash(tag: string, ...messages: Uint8Array[]): Uint8Array {
let tagP = TAGGED_HASH_PREFIXES[tag];
if (tagP === undefined) {
const tagH = sha256(Uint8Array.from(tag, (c) => c.charCodeAt(0)));
tagP = concatBytes(tagH, tagH);
TAGGED_HASH_PREFIXES[tag] = tagP;
}
return sha256(concatBytes(tagP, ...messages));
}
// ECDSA compact points are 33-byte. Schnorr is 32: we strip first byte 0x02 or 0x03
const pointToBytes = (point: PointType<bigint>) => point.toRawBytes(true).slice(1);
const numTo32b = (n: bigint) => numberToBytesBE(n, 32);
const modP = (x: bigint) => mod(x, secp256k1P);
const modN = (x: bigint) => mod(x, secp256k1N);
const Point = secp256k1.ProjectivePoint;
const GmulAdd = (Q: PointType<bigint>, a: bigint, b: bigint) =>
Point.BASE.multiplyAndAddUnsafe(Q, a, b);
// Calculate point, scalar and bytes
function schnorrGetExtPubKey(priv: PrivKey) {
let d_ = secp256k1.utils.normPrivateKeyToScalar(priv); // same method executed in fromPrivateKey
let p = Point.fromPrivateKey(d_); // P = d'⋅G; 0 < d' < n check is done inside
const scalar = p.hasEvenY() ? d_ : modN(-d_);
return { scalar: scalar, bytes: pointToBytes(p) };
}
/**
* lift_x from BIP340. Convert 32-byte x coordinate to elliptic curve point.
* @returns valid point checked for being on-curve
*/
function lift_x(x: bigint): PointType<bigint> {
if (!fe(x)) throw new Error('bad x: need 0 < x < p'); // Fail if x ≥ p.
const xx = modP(x * x);
const c = modP(xx * x + BigInt(7)); // Let c = x³ + 7 mod p.
let y = sqrtMod(c); // Let y = c^(p+1)/4 mod p.
if (y % _2n !== _0n) y = modP(-y); // Return the unique point P such that x(P) = x and
const p = new Point(x, y, _1n); // y(P) = y if y mod 2 = 0 or y(P) = p-y otherwise.
p.assertValidity();
return p;
}
/**
* Create tagged hash, convert it to bigint, reduce modulo-n.
*/
function challenge(...args: Uint8Array[]): bigint {
return modN(bytesToNumberBE(taggedHash('BIP0340/challenge', ...args)));
}
/**
* Schnorr public key is just `x` coordinate of Point as per BIP340.
*/
function schnorrGetPublicKey(privateKey: Hex): Uint8Array {
return schnorrGetExtPubKey(privateKey).bytes; // d'=int(sk). Fail if d'=0 or d'≥n. Ret bytes(d'⋅G)
}
/**
* Creates Schnorr signature as per BIP340. Verifies itself before returning anything.
* auxRand is optional and is not the sole source of k generation: bad CSPRNG won't be dangerous.
*/
function schnorrSign(
message: Hex,
privateKey: PrivKey,
auxRand: Hex = randomBytes(32)
): Uint8Array {
const m = ensureBytes('message', message);
const { bytes: px, scalar: d } = schnorrGetExtPubKey(privateKey); // checks for isWithinCurveOrder
const a = ensureBytes('auxRand', auxRand, 32); // Auxiliary random data a: a 32-byte array
const t = numTo32b(d ^ bytesToNumberBE(taggedHash('BIP0340/aux', a))); // Let t be the byte-wise xor of bytes(d) and hash/aux(a)
const rand = taggedHash('BIP0340/nonce', t, px, m); // Let rand = hash/nonce(t || bytes(P) || m)
const k_ = modN(bytesToNumberBE(rand)); // Let k' = int(rand) mod n
if (k_ === _0n) throw new Error('sign failed: k is zero'); // Fail if k' = 0.
const { bytes: rx, scalar: k } = schnorrGetExtPubKey(k_); // Let R = k'⋅G.
const e = challenge(rx, px, m); // Let e = int(hash/challenge(bytes(R) || bytes(P) || m)) mod n.
const sig = new Uint8Array(64); // Let sig = bytes(R) || bytes((k + ed) mod n).
sig.set(rx, 0);
sig.set(numTo32b(modN(k + e * d)), 32);
// If Verify(bytes(P), m, sig) (see below) returns failure, abort
if (!schnorrVerify(sig, m, px)) throw new Error('sign: Invalid signature produced');
return sig;
}
/**
* Verifies Schnorr signature.
* Will swallow errors & return false except for initial type validation of arguments.
*/
function schnorrVerify(signature: Hex, message: Hex, publicKey: Hex): boolean {
const sig = ensureBytes('signature', signature, 64);
const m = ensureBytes('message', message);
const pub = ensureBytes('publicKey', publicKey, 32);
try {
const P = lift_x(bytesToNumberBE(pub)); // P = lift_x(int(pk)); fail if that fails
const r = bytesToNumberBE(sig.subarray(0, 32)); // Let r = int(sig[0:32]); fail if r ≥ p.
if (!fe(r)) return false;
const s = bytesToNumberBE(sig.subarray(32, 64)); // Let s = int(sig[32:64]); fail if s ≥ n.
if (!ge(s)) return false;
const e = challenge(numTo32b(r), pointToBytes(P), m); // int(challenge(bytes(r)||bytes(P)||m))%n
const R = GmulAdd(P, s, modN(-e)); // R = s⋅G - e⋅P
if (!R || !R.hasEvenY() || R.toAffine().x !== r) return false; // -eP == (n-e)P
return true; // Fail if is_infinite(R) / not has_even_y(R) / x(R) ≠ r.
} catch (error) {
return false;
}
}
export const schnorr = /* @__PURE__ */ (() => ({
getPublicKey: schnorrGetPublicKey,
sign: schnorrSign,
verify: schnorrVerify,
utils: {
randomPrivateKey: secp256k1.utils.randomPrivateKey,
lift_x,
pointToBytes,
numberToBytesBE,
bytesToNumberBE,
taggedHash,
mod,
},
}))();
const isoMap = /* @__PURE__ */ (() =>
isogenyMap(
Fp,
[
// xNum
[
'0x8e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38daaaaa8c7',
'0x7d3d4c80bc321d5b9f315cea7fd44c5d595d2fc0bf63b92dfff1044f17c6581',
'0x534c328d23f234e6e2a413deca25caece4506144037c40314ecbd0b53d9dd262',
'0x8e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38daaaaa88c',
],
// xDen
[
'0xd35771193d94918a9ca34ccbb7b640dd86cd409542f8487d9fe6b745781eb49b',
'0xedadc6f64383dc1df7c4b2d51b54225406d36b641f5e41bbc52a56612a8c6d14',
'0x0000000000000000000000000000000000000000000000000000000000000001', // LAST 1
],
// yNum
[
'0x4bda12f684bda12f684bda12f684bda12f684bda12f684bda12f684b8e38e23c',
'0xc75e0c32d5cb7c0fa9d0a54b12a0a6d5647ab046d686da6fdffc90fc201d71a3',
'0x29a6194691f91a73715209ef6512e576722830a201be2018a765e85a9ecee931',
'0x2f684bda12f684bda12f684bda12f684bda12f684bda12f684bda12f38e38d84',
],
// yDen
[
'0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffff93b',
'0x7a06534bb8bdb49fd5e9e6632722c2989467c1bfc8e8d978dfb425d2685c2573',
'0x6484aa716545ca2cf3a70c3fa8fe337e0a3d21162f0d6299a7bf8192bfd2a76f',
'0x0000000000000000000000000000000000000000000000000000000000000001', // LAST 1
],
].map((i) => i.map((j) => BigInt(j))) as [bigint[], bigint[], bigint[], bigint[]]
))();
const mapSWU = /* @__PURE__ */ (() =>
mapToCurveSimpleSWU(Fp, {
A: BigInt('0x3f8731abdd661adca08a5558f0f5d272e953d363cb6f0e5d405447c01a444533'),
B: BigInt('1771'),
Z: Fp.create(BigInt('-11')),
}))();
const htf = /* @__PURE__ */ (() =>
createHasher(
secp256k1.ProjectivePoint,
(scalars: bigint[]) => {
const { x, y } = mapSWU(Fp.create(scalars[0]));
return isoMap(x, y);
},
{
DST: 'secp256k1_XMD:SHA-256_SSWU_RO_',
encodeDST: 'secp256k1_XMD:SHA-256_SSWU_NU_',
p: Fp.ORDER,
m: 1,
k: 128,
expand: 'xmd',
hash: sha256,
}
))();
export const hashToCurve = /* @__PURE__ */ (() => htf.hashToCurve)();
export const encodeToCurve = /* @__PURE__ */ (() => htf.encodeToCurve)();