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index.ts
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index.ts
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/*! noble-ed25519 - MIT License (c) 2019 Paul Miller (paulmillr.com) */
const P = 2n ** 255n - 19n; // ed25519 is twisted edwards curve
const N = 2n ** 252n + 27742317777372353535851937790883648493n; // curve's (group) order
const Gx = 0x216936d3cd6e53fec0a4e231fdd6dc5c692cc7609525a7b2c9562d608f25d51an; // base point x
const Gy = 0x6666666666666666666666666666666666666666666666666666666666666658n; // base point y
const CURVE = { // Curve's formula is −x² + y² = -a + dx²y²
a: -1n, // where a=-1, d = -(121665/121666) == -(121665 * inv(121666)) mod P
d: 37095705934669439343138083508754565189542113879843219016388785533085940283555n,
p: P, n: N, h: 8, Gx, Gy // field prime, curve (group) order, cofactor
};
type Bytes = Uint8Array; type Hex = Bytes | string; // types
const err = (m = ''): never => { throw new Error(m); }; // error helper, messes-up stack trace
const str = (s: unknown): s is string => typeof s === 'string'; // is string
const au8 = (a: unknown, l?: number): Bytes => // is Uint8Array (of specific length)
!(a instanceof Uint8Array) || (typeof l === 'number' && l > 0 && a.length !== l) ?
err('Uint8Array of valid length expected') : a;
const u8n = (data?: any) => new Uint8Array(data); // creates Uint8Array
const toU8 = (a: Hex, len?: number) => au8(str(a) ? h2b(a) : u8n(a), len); // norm(hex/u8a) to u8a
const mod = (a: bigint, b = P) => { let r = a % b; return r >= 0n ? r : b + r; }; // mod division
const isPoint = (p: any) => (p instanceof Point ? p : err('Point expected')); // is xyzt point
let Gpows: Point[] | undefined = undefined; // precomputes for base point G
interface AffinePoint { x: bigint, y: bigint } // Point in 2d xy affine coordinates
class Point { // Point in xyzt extended coordinates
constructor(readonly ex: bigint, readonly ey: bigint, readonly ez: bigint, readonly et: bigint) {}
static readonly BASE = new Point(Gx, Gy, 1n, mod(Gx * Gy)); // Generator / Base point
static readonly ZERO = new Point(0n, 1n, 1n, 0n); // Identity / Zero point
static fromAffine(p: AffinePoint) { return new Point(p.x, p.y, 1n, mod(p.x * p.y)); }
static fromHex(hex: Hex, zip215 = false) { // RFC8032 5.1.3: hex / Uint8Array to Point.
const { d } = CURVE;
hex = toU8(hex, 32);
const normed = hex.slice(); // copy the array to not mess it up
const lastByte = hex[31];
normed[31] = lastByte & ~0x80; // adjust first LE byte = last BE byte
const y = b2n_LE(normed); // decode as little-endian, convert to num
if (zip215 && !(0n <= y && y < 2n ** 256n)) err('bad y coord 1'); // zip215=true [1..2^256-1]
if (!zip215 && !(0n <= y && y < P)) err('bad y coord 2'); // zip215=false [1..P-1]
const y2 = mod(y * y); // y²
const u = mod(y2 - 1n); // u=y²-1
const v = mod(d * y2 + 1n); // v=dy²+1
let { isValid, value: x } = uvRatio(u, v); // (uv³)(uv⁷)^(p-5)/8; square root
if (!isValid) err('bad y coordinate 3'); // not square root: bad point
const isXOdd = (x & 1n) === 1n; // adjust sign of x coordinate
const isLastByteOdd = (lastByte & 0x80) !== 0; // x_0, last bit
if (!zip215 && x === 0n && isLastByteOdd) err('bad y coord 3'); // x=0 and x_0 = 1
if (isLastByteOdd !== isXOdd) x = mod(-x);
return new Point(x, y, 1n, mod(x * y)); // Z=1, T=xy
}
get x() { return this.toAffine().x; } // .x, .y will call expensive toAffine.
get y() { return this.toAffine().y; } // Should be used with care.
equals(other: Point): boolean { // equality check: compare points
const { ex: X1, ey: Y1, ez: Z1 } = this;
const { ex: X2, ey: Y2, ez: Z2 } = isPoint(other); // isPoint() checks class equality
const X1Z2 = mod(X1 * Z2), X2Z1 = mod(X2 * Z1);
const Y1Z2 = mod(Y1 * Z2), Y2Z1 = mod(Y2 * Z1);
return X1Z2 === X2Z1 && Y1Z2 === Y2Z1;
}
is0(): boolean { return this.equals(I); }
negate(): Point { // negate: flip over the affine x coordinate
return new Point(mod(-this.ex), this.ey, this.ez, mod(-this.et));
}
double(): Point { // Point doubling. Complete formula.
const { ex: X1, ey: Y1, ez: Z1 } = this; // Cost: 4M + 4S + 1*a + 6add + 1*2
const { a } = CURVE; // https://hyperelliptic.org/EFD/g1p/auto-twisted-extended.html#doubling-dbl-2008-hwcd
const A = mod(X1 * X1); const B = mod(Y1 * Y1); const C = mod(2n * mod(Z1 * Z1));
const D = mod(a * A); const x1y1 = X1 + Y1; const E = mod(mod(x1y1 * x1y1) - A - B);
const G = D + B; const F = G - C; const H = D - B;
const X3 = mod(E * F); const Y3 = mod(G * H); const T3 = mod(E * H); const Z3 = mod(F * G);
return new Point(X3, Y3, Z3, T3);
}
add(other: Point) { // Point addition. Complete formula.
const { ex: X1, ey: Y1, ez: Z1, et: T1 } = this; // Cost: 8M + 1*k + 8add + 1*2.
const { ex: X2, ey: Y2, ez: Z2, et: T2 } = isPoint(other); // doesn't check if other on-curve
const { a, d } = CURVE; // http://hyperelliptic.org/EFD/g1p/auto-twisted-extended-1.html#addition-add-2008-hwcd-3
const A = mod(X1 * X2); const B = mod(Y1 * Y2); const C = mod(T1 * d * T2);
const D = mod(Z1 * Z2); const E = mod((X1 + Y1) * (X2 + Y2) - A - B);
const F = mod(D - C); const G = mod(D + C); const H = mod(B - a * A);
const X3 = mod(E * F); const Y3 = mod(G * H); const T3 = mod(E * H); const Z3 = mod(F * G);
return new Point(X3, Y3, Z3, T3);
}
mul(n: bigint, safe = true): Point { // Multiply point by scalar n
if (n === 0n) return safe === true ? err('cannot multiply by 0') : I;
if (!(typeof n === 'bigint' && 0n < n && n < N)) err('invalid scalar, must be < L');
if (!safe && this.is0() || n === 1n) return this; // safe=true bans 0. safe=false allows 0.
if (this.equals(G)) return wNAF(n).p; // use wNAF precomputes for base points
let p = I, f = G; // init result point & fake point
for (let d: Point = this; n > 0n; d = d.double(), n >>= 1n) { // double-and-add ladder
if (n & 1n) p = p.add(d); // if bit is present, add to point
else if (safe) f = f.add(d); // if not, add to fake for timing safety
}
return p;
}
multiply(scalar: bigint) { return this.mul(scalar); } // Aliases for compatibilty
clearCofactor(): Point { return this.mul(BigInt(CURVE.h), false); } // multiply by cofactor
isSmallOrder(): boolean { return this.clearCofactor().is0(); } // check if P is small order
isTorsionFree(): boolean { // multiply by big number CURVE.n
let p = this.mul(N / 2n, false).double(); // ensures the point is not "bad".
if (N % 2n) p = p.add(this); // P^(N+1) // P*N == (P*(N/2))*2+P
return p.is0();
}
toAffine(): AffinePoint { // converts point to 2d xy affine point
const { ex: x, ey: y, ez: z } = this; // (x, y, z, t) ∋ (x=x/z, y=y/z, t=xy)
if (this.equals(I)) return { x: 0n, y: 1n }; // fast-path for zero point
const iz = invert(z); // z^-1: invert z
if (mod(z * iz) !== 1n) err('invalid inverse'); // (z * z^-1) must be 1, otherwise bad math
return { x: mod(x * iz), y: mod(y * iz) } // x = x*z^-1; y = y*z^-1
}
toRawBytes(): Bytes { // Encode to Uint8Array
const { x, y } = this.toAffine(); // convert to affine 2d point
const b = n2b_32LE(y); // encode number to 32 bytes
b[31] |= x & 1n ? 0x80 : 0; // store sign in first LE byte
return b;
}
toHex(): string { return b2h(this.toRawBytes()); } // encode to hex string
}
const { BASE: G, ZERO: I } = Point; // Generator, identity points
const padh = (num: number | bigint, pad: number) => num.toString(16).padStart(pad, '0')
const b2h = (b: Bytes): string => Array.from(b).map(e => padh(e, 2)).join(''); // bytes to hex
const h2b = (hex: string): Bytes => { // hex to bytes
const l = hex.length; // error if not string,
if (!str(hex) || l % 2) err('hex invalid 1'); // or has odd length like 3, 5.
const arr = u8n(l / 2); // create result array
for (let i = 0; i < arr.length; i++) {
const j = i * 2;
const h = hex.slice(j, j + 2); // hexByte. slice is faster than substr
const b = Number.parseInt(h, 16); // byte, created from string part
if (Number.isNaN(b) || b < 0) err('hex invalid 2'); // byte must be valid 0 <= byte < 256
arr[i] = b;
}
return arr;
};
const n2b_32LE = (num: bigint) => h2b(padh(num, 32 * 2)).reverse(); // number to bytes LE
const b2n_LE = (b: Bytes): bigint => BigInt('0x' + b2h(u8n(au8(b)).reverse())); // bytes LE to num
const concatB = (...arrs: Bytes[]) => { // concatenate Uint8Array-s
const r = u8n(arrs.reduce((sum, a) => sum + au8(a).length, 0)); // create u8a of summed length
let pad = 0; // walk through each array,
arrs.forEach(a => {r.set(a, pad); pad += a.length}); // ensure they have proper type
return r;
};
const invert = (num: bigint, md = P): bigint => { // modular inversion
if (num === 0n || md <= 0n) err('no inverse n=' + num + ' mod=' + md); // no neg exponent for now
let a = mod(num, md), b = md, x = 0n, y = 1n, u = 1n, v = 0n;
while (a !== 0n) { // uses euclidean gcd algorithm
const q = b / a, r = b % a; // not constant-time
const m = x - u * q, n = y - v * q;
b = a, a = r, x = u, y = v, u = m, v = n;
}
return b === 1n ? mod(x, md) : err('no inverse'); // b is gcd at this point
};
const pow2 = (x: bigint, power: bigint): bigint => { // pow2(x, 4) == x^(2^4)
let r = x;
while (power-- > 0n) { r *= r; r %= P; }
return r;
}
const pow_2_252_3 = (x: bigint) => { // x^(2^252-3) unrolled util for square root
const x2 = (x * x) % P; // x^2, bits 1
const b2 = (x2 * x) % P; // x^3, bits 11
const b4 = (pow2(b2, 2n) * b2) % P; // x^(2^4-1), bits 1111
const b5 = (pow2(b4, 1n) * x) % P; // x^(2^5-1), bits 11111
const b10 = (pow2(b5, 5n) * b5) % P; // x^(2^10)
const b20 = (pow2(b10, 10n) * b10) % P; // x^(2^20)
const b40 = (pow2(b20, 20n) * b20) % P; // x^(2^40)
const b80 = (pow2(b40, 40n) * b40) % P; // x^(2^80)
const b160 = (pow2(b80, 80n) * b80) % P; // x^(2^160)
const b240 = (pow2(b160, 80n) * b80) % P; // x^(2^240)
const b250 = (pow2(b240, 10n) * b10) % P; // x^(2^250)
const pow_p_5_8 = (pow2(b250, 2n) * x) % P; // < To pow to (p+3)/8, multiply it by x.
return { pow_p_5_8, b2 };
}
const RM1 = 19681161376707505956807079304988542015446066515923890162744021073123829784752n; // √-1
const uvRatio = (u: bigint, v: bigint): { isValid: boolean, value: bigint } => { // for sqrt comp
const v3 = mod(v * v * v); // v³
const v7 = mod(v3 * v3 * v); // v⁷
const pow = pow_2_252_3(u * v7).pow_p_5_8; // (uv⁷)^(p-5)/8
let x = mod(u * v3 * pow); // (uv³)(uv⁷)^(p-5)/8
const vx2 = mod(v * x * x); // vx²
const root1 = x; // First root candidate
const root2 = mod(x * RM1); // Second root candidate; RM1 is √-1
const useRoot1 = vx2 === u; // If vx² = u (mod p), x is a square root
const useRoot2 = vx2 === mod(-u); // If vx² = -u, set x <-- x * 2^((p-1)/4)
const noRoot = vx2 === mod(-u * RM1); // 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 ((mod(x) & 1n) === 1n) x = mod(-x); // edIsNegative
return { isValid: useRoot1 || useRoot2, value: x };
}
const modL_LE = (hash: Bytes): bigint => mod(b2n_LE(hash), N); // modulo L; but little-endian
type Sha512FnSync = undefined | ((...messages: Bytes[]) => Bytes);
let _shaS: Sha512FnSync;
const sha512a = (...m: Bytes[]) => etc.sha512Async(...m); // Async SHA512
const sha512s = (...m: Bytes[]) => // Sync SHA512, not set by default
typeof _shaS === 'function' ? _shaS(...m) : err('etc.sha512Sync not set');
type ExtK = { head: Bytes, prefix: Bytes, scalar: bigint, point: Point, pointBytes: Bytes };
const hash2extK = (hashed: Bytes): ExtK => { // RFC8032 5.1.5
const head = hashed.slice(0, 32); // slice creates a copy, unlike subarray
head[0] &= 248; // Clamp bits: 0b1111_1000,
head[31] &= 127; // 0b0111_1111,
head[31] |= 64; // 0b0100_0000
const prefix = hashed.slice(32, 64); // private key "prefix"
const scalar = modL_LE(head); // modular division over curve order
const point = G.mul(scalar); // public key point
const pointBytes = point.toRawBytes(); // point serialized to Uint8Array
return { head, prefix, scalar, point, pointBytes };
}
// RFC8032 5.1.5; getPublicKey async, sync. Hash priv key and extract point.
const getExtendedPublicKeyAsync = (priv: Hex) => sha512a(toU8(priv, 32)).then(hash2extK);
const getExtendedPublicKey = (priv: Hex) => hash2extK(sha512s(toU8(priv, 32)))
const getPublicKeyAsync = (priv: Hex): Promise<Bytes> =>
getExtendedPublicKeyAsync(priv).then(p => p.pointBytes)
const getPublicKey = (priv: Hex): Bytes => getExtendedPublicKey(priv).pointBytes;
type Finishable<T> = { // Reduces logic duplication between
hashable: Bytes, finish: (hashed: Bytes) => T // sync & async versions of sign(), verify()
} // hashable=start(); finish(hash(hashable));
function hashFinish<T>(asynchronous: true, res: Finishable<T>): Promise<T>;
function hashFinish<T>(asynchronous: false, res: Finishable<T>): T;
function hashFinish<T>(asynchronous: boolean, res: Finishable<T>) {
if (asynchronous) return sha512a(res.hashable).then(res.finish);
return res.finish(sha512s(res.hashable));
}
const _sign = (e: ExtK, rBytes: Bytes, msg: Bytes): Finishable<Bytes> => { // sign() shared code
const { pointBytes: P, scalar: s } = e;
const r = modL_LE(rBytes); // r was created outside, reduce it modulo L
const R = G.mul(r).toRawBytes(); // R = [r]B
const hashable = concatB(R, P, msg); // dom2(F, C) || R || A || PH(M)
const finish = (hashed: Bytes): Bytes => { // k = SHA512(dom2(F, C) || R || A || PH(M))
const S = mod(r + modL_LE(hashed) * s, N); // S = (r + k * s) mod L; 0 <= s < l
return au8(concatB(R, n2b_32LE(S)), 64); // 64-byte sig: 32b R.x + 32b LE(S)
}
return { hashable, finish };
};
const signAsync = async (msg: Hex, privKey: Hex): Promise<Bytes> => {
const m = toU8(msg); // RFC8032 5.1.6: sign msg with key async
const e = await getExtendedPublicKeyAsync(privKey); // pub,prfx
const rBytes = await sha512a(e.prefix, m); // r = SHA512(dom2(F, C) || prefix || PH(M))
return hashFinish(true, _sign(e, rBytes, m)); // gen R, k, S, then 64-byte signature
};
const sign = (msg: Hex, privKey: Hex): Bytes => {
const m = toU8(msg); // RFC8032 5.1.6: sign msg with key sync
const e = getExtendedPublicKey(privKey); // pub,prfx
const rBytes = sha512s(e.prefix, m); // r = SHA512(dom2(F, C) || prefix || PH(M))
return hashFinish(false, _sign(e, rBytes, m)); // gen R, k, S, then 64-byte signature
};
const dvo = { zip215: true };
const _verify = (sig: Hex, msg: Hex, pub: Hex, opts = dvo): Finishable<boolean> => {
msg = toU8(msg); // Message hex str/Bytes
sig = toU8(sig, 64); // Signature hex str/Bytes, must be 64 bytes
const { zip215 } = opts; // switch between zip215 and rfc8032 verif
let A: Point, R: Point, s: bigint, SB: Point, hashable = new Uint8Array();
try {
A = Point.fromHex(pub, zip215); // public key A decoded
R = Point.fromHex(sig.slice(0, 32), zip215); // 0 <= R < 2^256: ZIP215 R can be >= P
s = b2n_LE(sig.slice(32, 64)); // Decode second half as an integer S
SB = G.mul(s, false); // in the range 0 <= s < L
hashable = concatB(R.toRawBytes(), A.toRawBytes(), msg); // dom2(F, C) || R || A || PH(M)
} catch (error) {}
const finish = (hashed: Bytes): boolean => { // k = SHA512(dom2(F, C) || R || A || PH(M))
if (SB == null) return false; // false if try-catch catched an error
if (!zip215 && A.isSmallOrder()) return false; // false for SBS: Strongly Binding Signature
const k = modL_LE(hashed); // decode in little-endian, modulo L
const RkA = R.add(A.mul(k, false)); // [8]R + [8][k]A'
return RkA.add(SB.negate()).clearCofactor().is0(); // [8][S]B = [8]R + [8][k]A'
}
return { hashable, finish };
};
// RFC8032 5.1.7: verification async, sync
const verifyAsync = async (s: Hex, m: Hex, p: Hex, opts = dvo) =>
hashFinish(true, _verify(s, m, p, opts));
const verify = (s: Hex, m: Hex, p: Hex, opts = dvo) =>
hashFinish(false, _verify(s, m, p, opts));
declare const globalThis: Record<string, any> | undefined; // Typescript symbol present in browsers
const cr = () => // We support: 1) browsers 2) node.js 19+
typeof globalThis === 'object' && 'crypto' in globalThis ? globalThis.crypto : undefined;
const etc = {
bytesToHex: b2h, hexToBytes: h2b, concatBytes: concatB,
mod, invert,
randomBytes: (len = 32): Bytes => { // CSPRNG (random number generator)
const crypto = cr(); // Can be shimmed in node.js <= 18 to prevent error:
// import { webcrypto } from 'node:crypto';
// if (!globalThis.crypto) globalThis.crypto = webcrypto;
if (!crypto || !crypto.getRandomValues) err('crypto.getRandomValues must be defined');
return crypto.getRandomValues(u8n(len));
},
sha512Async: async (...messages: Bytes[]): Promise<Bytes> => {
const crypto = cr();
if (!crypto || !crypto.subtle) err('crypto.subtle or etc.sha512Async must be defined');
const m = concatB(...messages);
return u8n(await crypto.subtle.digest('SHA-512', m.buffer));
},
sha512Sync: undefined as Sha512FnSync, // Actual logic below
};
Object.defineProperties(etc, { sha512Sync: { // Allow setting it once. Next sets will be ignored
configurable: false, get() { return _shaS; }, set(f) { if (!_shaS) _shaS = f; },
} });
const utils = {
getExtendedPublicKeyAsync, getExtendedPublicKey,
randomPrivateKey: (): Bytes => etc.randomBytes(32),
precompute(w=8, p: Point = G) { p.multiply(3n); w; return p; }, // no-op
}
const W = 8; // Precomputes-related code. W = window size
const precompute = () => { // They give 12x faster getPublicKey(),
const points: Point[] = []; // 10x sign(), 2x verify(). To achieve this,
const windows = 256 / W + 1; // app needs to spend 40ms+ to calculate
let p = G, b = p; // a lot of points related to base point G.
for (let w = 0; w < windows; w++) { // Points are stored in array and used
b = p; // any time Gx multiplication is done.
points.push(b); // They consume 16-32 MiB of RAM.
for (let i = 1; i < 2 ** (W - 1); i++) { b = b.add(p); points.push(b); }
p = b.double(); // Precomputes don't speed-up getSharedKey,
} // which multiplies user point by scalar,
return points; // when precomputes are using base point
}
const wNAF = (n: bigint): { p: Point; f: Point } => { // w-ary non-adjacent form (wNAF) method.
// Compared to other point mult methods,
const comp = Gpows || (Gpows = precompute()); // stores 2x less points using subtraction
const neg = (cnd: boolean, p: Point) => { let n = p.negate(); return cnd ? n : p; } // negate
let p = I, f = G; // f must be G, or could become I in the end
const windows = 1 + 256 / W; // W=8 17 windows
const wsize = 2 ** (W - 1); // W=8 128 window size
const mask = BigInt(2 ** W - 1); // W=8 will create mask 0b11111111
const maxNum = 2 ** W; // W=8 256
const shiftBy = BigInt(W); // W=8 8
for (let w = 0; w < windows; w++) {
const off = w * wsize;
let wbits = Number(n & mask); // extract W bits.
n >>= shiftBy; // shift number by W bits.
if (wbits > wsize) { wbits -= maxNum; n += 1n; } // split if bits > max: +224 => 256-32
const off1 = off, off2 = off + Math.abs(wbits) - 1; // offsets, evaluate both
const cnd1 = w % 2 !== 0, cnd2 = wbits < 0; // conditions, evaluate both
if (wbits === 0) {
f = f.add(neg(cnd1, comp[off1])); // bits are 0: add garbage to fake point
} else { // ^ can't add off2, off2 = I
p = p.add(neg(cnd2, comp[off2])); // bits are 1: add to result point
}
}
return { p, f } // return both real and fake points for JIT
}; // !! you can disable precomputes by commenting-out call of the wNAF() inside Point#mul()
export { getPublicKey, getPublicKeyAsync, sign, verify, // Remove the export to easily use in REPL
signAsync, verifyAsync, CURVE, etc, utils, Point as ExtendedPoint } // envs like browser console