@noble/curves
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Audited & minimal JS implementation of elliptic curve cryptography
1,189 lines • 54.5 kB
JavaScript
/**
* Short Weierstrass curve methods. The formula is: y² = x³ + ax + b.
*
* ### Design rationale for types
*
* * Interaction between classes from different curves should fail:
* `k256.Point.BASE.add(p256.Point.BASE)`
* * For this purpose we want to use `instanceof` operator, which is fast and works during runtime
* * Different calls of `curve()` would return different classes -
* `curve(params) !== curve(params)`: if somebody decided to monkey-patch their curve,
* it won't affect others
*
* TypeScript can't infer types for classes created inside a function. Classes is one instance
* of nominative types in TypeScript and interfaces only check for shape, so it's hard to create
* unique type for every function call.
*
* We can use generic types via some param, like curve opts, but that would:
* 1. Enable interaction between `curve(params)` and `curve(params)` (curves of same params)
* which is hard to debug.
* 2. Params can be generic and we can't enforce them to be constant value:
* if somebody creates curve from non-constant params,
* it would be allowed to interact with other curves with non-constant params
*
* @todo https://www.typescriptlang.org/docs/handbook/release-notes/typescript-2-7.html#unique-symbol
* @module
*/
/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { hmac } from '@noble/hashes/hmac.js';
import { _validateObject, abool, abytes, aInRange, bitMask, bytesToHex, bytesToNumberBE, concatBytes, createHmacDrbg, ensureBytes, hexToBytes, inRange, isBytes, memoized, numberToHexUnpadded, randomBytes, } from "../utils.js";
import { _createCurveFields, mulEndoUnsafe, negateCt, normalizeZ, pippenger, wNAF, } from "./curve.js";
import { Field, FpInvertBatch, getMinHashLength, mapHashToField, validateField, } from "./modular.js";
function validateSigVerOpts(opts) {
if (opts.lowS !== undefined)
abool('lowS', opts.lowS);
if (opts.prehash !== undefined)
abool('prehash', opts.prehash);
}
export class DERErr extends Error {
constructor(m = '') {
super(m);
}
}
/**
* ASN.1 DER encoding utilities. ASN is very complex & fragile. Format:
*
* [0x30 (SEQUENCE), bytelength, 0x02 (INTEGER), intLength, R, 0x02 (INTEGER), intLength, S]
*
* Docs: https://letsencrypt.org/docs/a-warm-welcome-to-asn1-and-der/, https://luca.ntop.org/Teaching/Appunti/asn1.html
*/
export const DER = {
// asn.1 DER encoding utils
Err: DERErr,
// Basic building block is TLV (Tag-Length-Value)
_tlv: {
encode: (tag, data) => {
const { Err: E } = DER;
if (tag < 0 || tag > 256)
throw new E('tlv.encode: wrong tag');
if (data.length & 1)
throw new E('tlv.encode: unpadded data');
const dataLen = data.length / 2;
const len = numberToHexUnpadded(dataLen);
if ((len.length / 2) & 128)
throw new E('tlv.encode: long form length too big');
// length of length with long form flag
const lenLen = dataLen > 127 ? numberToHexUnpadded((len.length / 2) | 128) : '';
const t = numberToHexUnpadded(tag);
return t + lenLen + len + data;
},
// v - value, l - left bytes (unparsed)
decode(tag, data) {
const { Err: E } = DER;
let pos = 0;
if (tag < 0 || tag > 256)
throw new E('tlv.encode: wrong tag');
if (data.length < 2 || data[pos++] !== tag)
throw new E('tlv.decode: wrong tlv');
const first = data[pos++];
const isLong = !!(first & 128); // First bit of first length byte is flag for short/long form
let length = 0;
if (!isLong)
length = first;
else {
// Long form: [longFlag(1bit), lengthLength(7bit), length (BE)]
const lenLen = first & 127;
if (!lenLen)
throw new E('tlv.decode(long): indefinite length not supported');
if (lenLen > 4)
throw new E('tlv.decode(long): byte length is too big'); // this will overflow u32 in js
const lengthBytes = data.subarray(pos, pos + lenLen);
if (lengthBytes.length !== lenLen)
throw new E('tlv.decode: length bytes not complete');
if (lengthBytes[0] === 0)
throw new E('tlv.decode(long): zero leftmost byte');
for (const b of lengthBytes)
length = (length << 8) | b;
pos += lenLen;
if (length < 128)
throw new E('tlv.decode(long): not minimal encoding');
}
const v = data.subarray(pos, pos + length);
if (v.length !== length)
throw new E('tlv.decode: wrong value length');
return { v, l: data.subarray(pos + length) };
},
},
// https://crypto.stackexchange.com/a/57734 Leftmost bit of first byte is 'negative' flag,
// since we always use positive integers here. It must always be empty:
// - add zero byte if exists
// - if next byte doesn't have a flag, leading zero is not allowed (minimal encoding)
_int: {
encode(num) {
const { Err: E } = DER;
if (num < _0n)
throw new E('integer: negative integers are not allowed');
let hex = numberToHexUnpadded(num);
// Pad with zero byte if negative flag is present
if (Number.parseInt(hex[0], 16) & 0b1000)
hex = '00' + hex;
if (hex.length & 1)
throw new E('unexpected DER parsing assertion: unpadded hex');
return hex;
},
decode(data) {
const { Err: E } = DER;
if (data[0] & 128)
throw new E('invalid signature integer: negative');
if (data[0] === 0x00 && !(data[1] & 128))
throw new E('invalid signature integer: unnecessary leading zero');
return bytesToNumberBE(data);
},
},
toSig(hex) {
// parse DER signature
const { Err: E, _int: int, _tlv: tlv } = DER;
const data = ensureBytes('signature', hex);
const { v: seqBytes, l: seqLeftBytes } = tlv.decode(0x30, data);
if (seqLeftBytes.length)
throw new E('invalid signature: left bytes after parsing');
const { v: rBytes, l: rLeftBytes } = tlv.decode(0x02, seqBytes);
const { v: sBytes, l: sLeftBytes } = tlv.decode(0x02, rLeftBytes);
if (sLeftBytes.length)
throw new E('invalid signature: left bytes after parsing');
return { r: int.decode(rBytes), s: int.decode(sBytes) };
},
hexFromSig(sig) {
const { _tlv: tlv, _int: int } = DER;
const rs = tlv.encode(0x02, int.encode(sig.r));
const ss = tlv.encode(0x02, int.encode(sig.s));
const seq = rs + ss;
return tlv.encode(0x30, seq);
},
};
// Be friendly to bad ECMAScript parsers by not using bigint literals
// prettier-ignore
const _0n = BigInt(0), _1n = BigInt(1), _2n = BigInt(2), _3n = BigInt(3), _4n = BigInt(4);
// TODO: remove
export function _legacyHelperEquat(Fp, a, b) {
/**
* y² = x³ + ax + b: Short weierstrass curve formula. Takes x, returns y².
* @returns y²
*/
function weierstrassEquation(x) {
const x2 = Fp.sqr(x); // x * x
const x3 = Fp.mul(x2, x); // x² * x
return Fp.add(Fp.add(x3, Fp.mul(x, a)), b); // x³ + a * x + b
}
return weierstrassEquation;
}
export function _legacyHelperNormPriv(Fn, allowedPrivateKeyLengths, wrapPrivateKey) {
const { BYTES: expected } = Fn;
// Validates if priv key is valid and converts it to bigint.
function normPrivateKeyToScalar(key) {
let num;
if (typeof key === 'bigint') {
num = key;
}
else {
let bytes = ensureBytes('private key', key);
if (allowedPrivateKeyLengths) {
if (!allowedPrivateKeyLengths.includes(bytes.length * 2))
throw new Error('invalid private key');
const padded = new Uint8Array(expected);
padded.set(bytes, padded.length - bytes.length);
bytes = padded;
}
try {
num = Fn.fromBytes(bytes);
}
catch (error) {
throw new Error(`invalid private key: expected ui8a of size ${expected}, got ${typeof key}`);
}
}
if (wrapPrivateKey)
num = Fn.create(num); // disabled by default, enabled for BLS
if (!Fn.isValidNot0(num))
throw new Error('invalid private key: out of range [1..N-1]');
return num;
}
return normPrivateKeyToScalar;
}
export function weierstrassN(CURVE, curveOpts = {}) {
const { Fp, Fn } = _createCurveFields('weierstrass', CURVE, curveOpts);
const { h: cofactor, n: CURVE_ORDER } = CURVE;
_validateObject(curveOpts, {}, {
allowInfinityPoint: 'boolean',
clearCofactor: 'function',
isTorsionFree: 'function',
fromBytes: 'function',
toBytes: 'function',
endo: 'object',
wrapPrivateKey: 'boolean',
});
const { endo } = curveOpts;
if (endo) {
// validateObject(endo, { beta: 'bigint', splitScalar: 'function' });
if (!Fp.is0(CURVE.a) ||
typeof endo.beta !== 'bigint' ||
typeof endo.splitScalar !== 'function') {
throw new Error('invalid endo: expected "beta": bigint and "splitScalar": function');
}
}
function assertCompressionIsSupported() {
if (!Fp.isOdd)
throw new Error('compression is not supported: Field does not have .isOdd()');
}
// Implements IEEE P1363 point encoding
function pointToBytes(_c, point, isCompressed) {
const { x, y } = point.toAffine();
const bx = Fp.toBytes(x);
abool('isCompressed', isCompressed);
if (isCompressed) {
assertCompressionIsSupported();
const hasEvenY = !Fp.isOdd(y);
return concatBytes(pprefix(hasEvenY), bx);
}
else {
return concatBytes(Uint8Array.of(0x04), bx, Fp.toBytes(y));
}
}
function pointFromBytes(bytes) {
abytes(bytes);
const L = Fp.BYTES;
const LC = L + 1; // length compressed, e.g. 33 for 32-byte field
const LU = 2 * L + 1; // length uncompressed, e.g. 65 for 32-byte field
const length = bytes.length;
const head = bytes[0];
const tail = bytes.subarray(1);
// No actual validation is done here: use .assertValidity()
if (length === LC && (head === 0x02 || head === 0x03)) {
const x = Fp.fromBytes(tail);
if (!Fp.isValid(x))
throw new Error('bad point: is not on curve, wrong x');
const y2 = weierstrassEquation(x); // y² = x³ + ax + b
let y;
try {
y = Fp.sqrt(y2); // y = y² ^ (p+1)/4
}
catch (sqrtError) {
const err = sqrtError instanceof Error ? ': ' + sqrtError.message : '';
throw new Error('bad point: is not on curve, sqrt error' + err);
}
assertCompressionIsSupported();
const isYOdd = Fp.isOdd(y); // (y & _1n) === _1n;
const isHeadOdd = (head & 1) === 1; // ECDSA-specific
if (isHeadOdd !== isYOdd)
y = Fp.neg(y);
return { x, y };
}
else if (length === LU && head === 0x04) {
// TODO: more checks
const x = Fp.fromBytes(tail.subarray(L * 0, L * 1));
const y = Fp.fromBytes(tail.subarray(L * 1, L * 2));
if (!isValidXY(x, y))
throw new Error('bad point: is not on curve');
return { x, y };
}
else {
throw new Error(`bad point: got length ${length}, expected compressed=${LC} or uncompressed=${LU}`);
}
}
const toBytes = curveOpts.toBytes || pointToBytes;
const fromBytes = curveOpts.fromBytes || pointFromBytes;
const weierstrassEquation = _legacyHelperEquat(Fp, CURVE.a, CURVE.b);
// TODO: move top-level
/** Checks whether equation holds for given x, y: y² == x³ + ax + b */
function isValidXY(x, y) {
const left = Fp.sqr(y); // y²
const right = weierstrassEquation(x); // x³ + ax + b
return Fp.eql(left, right);
}
// Validate whether the passed curve params are valid.
// Test 1: equation y² = x³ + ax + b should work for generator point.
if (!isValidXY(CURVE.Gx, CURVE.Gy))
throw new Error('bad curve params: generator point');
// Test 2: discriminant Δ part should be non-zero: 4a³ + 27b² != 0.
// Guarantees curve is genus-1, smooth (non-singular).
const _4a3 = Fp.mul(Fp.pow(CURVE.a, _3n), _4n);
const _27b2 = Fp.mul(Fp.sqr(CURVE.b), BigInt(27));
if (Fp.is0(Fp.add(_4a3, _27b2)))
throw new Error('bad curve params: a or b');
/** Asserts coordinate is valid: 0 <= n < Fp.ORDER. */
function acoord(title, n, banZero = false) {
if (!Fp.isValid(n) || (banZero && Fp.is0(n)))
throw new Error(`bad point coordinate ${title}`);
return n;
}
function aprjpoint(other) {
if (!(other instanceof Point))
throw new Error('ProjectivePoint expected');
}
// Memoized toAffine / validity check. They are heavy. Points are immutable.
// Converts Projective point to affine (x, y) coordinates.
// Can accept precomputed Z^-1 - for example, from invertBatch.
// (X, Y, Z) ∋ (x=X/Z, y=Y/Z)
const toAffineMemo = memoized((p, iz) => {
const { px: x, py: y, pz: z } = p;
// Fast-path for normalized points
if (Fp.eql(z, Fp.ONE))
return { x, y };
const is0 = p.is0();
// If invZ was 0, we return zero point. However we still want to execute
// all operations, so we replace invZ with a random number, 1.
if (iz == null)
iz = is0 ? Fp.ONE : Fp.inv(z);
const ax = Fp.mul(x, iz);
const ay = Fp.mul(y, iz);
const zz = Fp.mul(z, iz);
if (is0)
return { x: Fp.ZERO, y: Fp.ZERO };
if (!Fp.eql(zz, Fp.ONE))
throw new Error('invZ was invalid');
return { x: ax, y: ay };
});
// NOTE: on exception this will crash 'cached' and no value will be set.
// Otherwise true will be return
const assertValidMemo = memoized((p) => {
if (p.is0()) {
// (0, 1, 0) aka ZERO is invalid in most contexts.
// In BLS, ZERO can be serialized, so we allow it.
// (0, 0, 0) is invalid representation of ZERO.
if (curveOpts.allowInfinityPoint && !Fp.is0(p.py))
return;
throw new Error('bad point: ZERO');
}
// Some 3rd-party test vectors require different wording between here & `fromCompressedHex`
const { x, y } = p.toAffine();
if (!Fp.isValid(x) || !Fp.isValid(y))
throw new Error('bad point: x or y not field elements');
if (!isValidXY(x, y))
throw new Error('bad point: equation left != right');
if (!p.isTorsionFree())
throw new Error('bad point: not in prime-order subgroup');
return true;
});
function finishEndo(endoBeta, k1p, k2p, k1neg, k2neg) {
k2p = new Point(Fp.mul(k2p.px, endoBeta), k2p.py, k2p.pz);
k1p = negateCt(k1neg, k1p);
k2p = negateCt(k2neg, k2p);
return k1p.add(k2p);
}
/**
* Projective Point works in 3d / projective (homogeneous) coordinates:(X, Y, Z) ∋ (x=X/Z, y=Y/Z).
* Default Point works in 2d / affine coordinates: (x, y).
* We're doing calculations in projective, because its operations don't require costly inversion.
*/
class Point {
/** Does NOT validate if the point is valid. Use `.assertValidity()`. */
constructor(px, py, pz) {
this.px = acoord('x', px);
this.py = acoord('y', py, true);
this.pz = acoord('z', pz);
Object.freeze(this);
}
/** Does NOT validate if the point is valid. Use `.assertValidity()`. */
static fromAffine(p) {
const { x, y } = p || {};
if (!p || !Fp.isValid(x) || !Fp.isValid(y))
throw new Error('invalid affine point');
if (p instanceof Point)
throw new Error('projective point not allowed');
// (0, 0) would've produced (0, 0, 1) - instead, we need (0, 1, 0)
if (Fp.is0(x) && Fp.is0(y))
return Point.ZERO;
return new Point(x, y, Fp.ONE);
}
get x() {
return this.toAffine().x;
}
get y() {
return this.toAffine().y;
}
static normalizeZ(points) {
return normalizeZ(Point, 'pz', points);
}
static fromBytes(bytes) {
abytes(bytes);
return Point.fromHex(bytes);
}
/** Converts hash string or Uint8Array to Point. */
static fromHex(hex) {
const P = Point.fromAffine(fromBytes(ensureBytes('pointHex', hex)));
P.assertValidity();
return P;
}
/** Multiplies generator point by privateKey. */
static fromPrivateKey(privateKey) {
const normPrivateKeyToScalar = _legacyHelperNormPriv(Fn, curveOpts.allowedPrivateKeyLengths, curveOpts.wrapPrivateKey);
return Point.BASE.multiply(normPrivateKeyToScalar(privateKey));
}
/** Multiscalar Multiplication */
static msm(points, scalars) {
return pippenger(Point, Fn, points, scalars);
}
/**
*
* @param windowSize
* @param isLazy true will defer table computation until the first multiplication
* @returns
*/
precompute(windowSize = 8, isLazy = true) {
wnaf.setWindowSize(this, windowSize);
if (!isLazy)
this.multiply(_3n); // random number
return this;
}
/** "Private method", don't use it directly */
_setWindowSize(windowSize) {
this.precompute(windowSize);
}
// TODO: return `this`
/** A point on curve is valid if it conforms to equation. */
assertValidity() {
assertValidMemo(this);
}
hasEvenY() {
const { y } = this.toAffine();
if (!Fp.isOdd)
throw new Error("Field doesn't support isOdd");
return !Fp.isOdd(y);
}
/** Compare one point to another. */
equals(other) {
aprjpoint(other);
const { px: X1, py: Y1, pz: Z1 } = this;
const { px: X2, py: Y2, pz: Z2 } = other;
const U1 = Fp.eql(Fp.mul(X1, Z2), Fp.mul(X2, Z1));
const U2 = Fp.eql(Fp.mul(Y1, Z2), Fp.mul(Y2, Z1));
return U1 && U2;
}
/** Flips point to one corresponding to (x, -y) in Affine coordinates. */
negate() {
return new Point(this.px, Fp.neg(this.py), this.pz);
}
// Renes-Costello-Batina exception-free doubling formula.
// There is 30% faster Jacobian formula, but it is not complete.
// https://eprint.iacr.org/2015/1060, algorithm 3
// Cost: 8M + 3S + 3*a + 2*b3 + 15add.
double() {
const { a, b } = CURVE;
const b3 = Fp.mul(b, _3n);
const { px: X1, py: Y1, pz: Z1 } = this;
let X3 = Fp.ZERO, Y3 = Fp.ZERO, Z3 = Fp.ZERO; // prettier-ignore
let t0 = Fp.mul(X1, X1); // step 1
let t1 = Fp.mul(Y1, Y1);
let t2 = Fp.mul(Z1, Z1);
let t3 = Fp.mul(X1, Y1);
t3 = Fp.add(t3, t3); // step 5
Z3 = Fp.mul(X1, Z1);
Z3 = Fp.add(Z3, Z3);
X3 = Fp.mul(a, Z3);
Y3 = Fp.mul(b3, t2);
Y3 = Fp.add(X3, Y3); // step 10
X3 = Fp.sub(t1, Y3);
Y3 = Fp.add(t1, Y3);
Y3 = Fp.mul(X3, Y3);
X3 = Fp.mul(t3, X3);
Z3 = Fp.mul(b3, Z3); // step 15
t2 = Fp.mul(a, t2);
t3 = Fp.sub(t0, t2);
t3 = Fp.mul(a, t3);
t3 = Fp.add(t3, Z3);
Z3 = Fp.add(t0, t0); // step 20
t0 = Fp.add(Z3, t0);
t0 = Fp.add(t0, t2);
t0 = Fp.mul(t0, t3);
Y3 = Fp.add(Y3, t0);
t2 = Fp.mul(Y1, Z1); // step 25
t2 = Fp.add(t2, t2);
t0 = Fp.mul(t2, t3);
X3 = Fp.sub(X3, t0);
Z3 = Fp.mul(t2, t1);
Z3 = Fp.add(Z3, Z3); // step 30
Z3 = Fp.add(Z3, Z3);
return new Point(X3, Y3, Z3);
}
// Renes-Costello-Batina exception-free addition formula.
// There is 30% faster Jacobian formula, but it is not complete.
// https://eprint.iacr.org/2015/1060, algorithm 1
// Cost: 12M + 0S + 3*a + 3*b3 + 23add.
add(other) {
aprjpoint(other);
const { px: X1, py: Y1, pz: Z1 } = this;
const { px: X2, py: Y2, pz: Z2 } = other;
let X3 = Fp.ZERO, Y3 = Fp.ZERO, Z3 = Fp.ZERO; // prettier-ignore
const a = CURVE.a;
const b3 = Fp.mul(CURVE.b, _3n);
let t0 = Fp.mul(X1, X2); // step 1
let t1 = Fp.mul(Y1, Y2);
let t2 = Fp.mul(Z1, Z2);
let t3 = Fp.add(X1, Y1);
let t4 = Fp.add(X2, Y2); // step 5
t3 = Fp.mul(t3, t4);
t4 = Fp.add(t0, t1);
t3 = Fp.sub(t3, t4);
t4 = Fp.add(X1, Z1);
let t5 = Fp.add(X2, Z2); // step 10
t4 = Fp.mul(t4, t5);
t5 = Fp.add(t0, t2);
t4 = Fp.sub(t4, t5);
t5 = Fp.add(Y1, Z1);
X3 = Fp.add(Y2, Z2); // step 15
t5 = Fp.mul(t5, X3);
X3 = Fp.add(t1, t2);
t5 = Fp.sub(t5, X3);
Z3 = Fp.mul(a, t4);
X3 = Fp.mul(b3, t2); // step 20
Z3 = Fp.add(X3, Z3);
X3 = Fp.sub(t1, Z3);
Z3 = Fp.add(t1, Z3);
Y3 = Fp.mul(X3, Z3);
t1 = Fp.add(t0, t0); // step 25
t1 = Fp.add(t1, t0);
t2 = Fp.mul(a, t2);
t4 = Fp.mul(b3, t4);
t1 = Fp.add(t1, t2);
t2 = Fp.sub(t0, t2); // step 30
t2 = Fp.mul(a, t2);
t4 = Fp.add(t4, t2);
t0 = Fp.mul(t1, t4);
Y3 = Fp.add(Y3, t0);
t0 = Fp.mul(t5, t4); // step 35
X3 = Fp.mul(t3, X3);
X3 = Fp.sub(X3, t0);
t0 = Fp.mul(t3, t1);
Z3 = Fp.mul(t5, Z3);
Z3 = Fp.add(Z3, t0); // step 40
return new Point(X3, Y3, Z3);
}
subtract(other) {
return this.add(other.negate());
}
is0() {
return this.equals(Point.ZERO);
}
/**
* Constant time multiplication.
* Uses wNAF method. Windowed method may be 10% faster,
* but takes 2x longer to generate and consumes 2x memory.
* Uses precomputes when available.
* Uses endomorphism for Koblitz curves.
* @param scalar by which the point would be multiplied
* @returns New point
*/
multiply(scalar) {
const { endo } = curveOpts;
if (!Fn.isValidNot0(scalar))
throw new Error('invalid scalar: out of range'); // 0 is invalid
let point, fake; // Fake point is used to const-time mult
const mul = (n) => wnaf.wNAFCached(this, n, Point.normalizeZ);
/** See docs for {@link EndomorphismOpts} */
if (endo) {
const { k1neg, k1, k2neg, k2 } = endo.splitScalar(scalar);
const { p: k1p, f: k1f } = mul(k1);
const { p: k2p, f: k2f } = mul(k2);
fake = k1f.add(k2f);
point = finishEndo(endo.beta, k1p, k2p, k1neg, k2neg);
}
else {
const { p, f } = mul(scalar);
point = p;
fake = f;
}
// Normalize `z` for both points, but return only real one
return Point.normalizeZ([point, fake])[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, which works over *public* keys.
*/
multiplyUnsafe(sc) {
const { endo } = curveOpts;
const p = this;
if (!Fn.isValid(sc))
throw new Error('invalid scalar: out of range'); // 0 is valid
if (sc === _0n || p.is0())
return Point.ZERO;
if (sc === _1n)
return p; // fast-path
if (wnaf.hasPrecomputes(this))
return this.multiply(sc);
if (endo) {
const { k1neg, k1, k2neg, k2 } = endo.splitScalar(sc);
// `wNAFCachedUnsafe` is 30% slower
const { p1, p2 } = mulEndoUnsafe(Point, p, k1, k2);
return finishEndo(endo.beta, p1, p2, k1neg, k2neg);
}
else {
return wnaf.wNAFCachedUnsafe(p, sc);
}
}
multiplyAndAddUnsafe(Q, a, b) {
const sum = this.multiplyUnsafe(a).add(Q.multiplyUnsafe(b));
return sum.is0() ? undefined : sum;
}
/**
* Converts Projective point to affine (x, y) coordinates.
* @param invertedZ Z^-1 (inverted zero) - optional, precomputation is useful for invertBatch
*/
toAffine(invertedZ) {
return toAffineMemo(this, invertedZ);
}
/**
* Checks whether Point is free of torsion elements (is in prime subgroup).
* Always torsion-free for cofactor=1 curves.
*/
isTorsionFree() {
const { isTorsionFree } = curveOpts;
if (cofactor === _1n)
return true;
if (isTorsionFree)
return isTorsionFree(Point, this);
return wnaf.wNAFCachedUnsafe(this, CURVE_ORDER).is0();
}
clearCofactor() {
const { clearCofactor } = curveOpts;
if (cofactor === _1n)
return this; // Fast-path
if (clearCofactor)
return clearCofactor(Point, this);
return this.multiplyUnsafe(cofactor);
}
toBytes(isCompressed = true) {
abool('isCompressed', isCompressed);
this.assertValidity();
return toBytes(Point, this, isCompressed);
}
/** @deprecated use `toBytes` */
toRawBytes(isCompressed = true) {
return this.toBytes(isCompressed);
}
toHex(isCompressed = true) {
return bytesToHex(this.toBytes(isCompressed));
}
toString() {
return `<Point ${this.is0() ? 'ZERO' : this.toHex()}>`;
}
}
// base / generator point
Point.BASE = new Point(CURVE.Gx, CURVE.Gy, Fp.ONE);
// zero / infinity / identity point
Point.ZERO = new Point(Fp.ZERO, Fp.ONE, Fp.ZERO); // 0, 1, 0
// fields
Point.Fp = Fp;
Point.Fn = Fn;
const bits = Fn.BITS;
const wnaf = wNAF(Point, curveOpts.endo ? Math.ceil(bits / 2) : bits);
return Point;
}
// _legacyWeierstrass
/** @deprecated use `weierstrassN` */
export function weierstrassPoints(c) {
const { CURVE, curveOpts } = _weierstrass_legacy_opts_to_new(c);
const Point = weierstrassN(CURVE, curveOpts);
return _weierstrass_new_output_to_legacy(c, Point);
}
// Points start with byte 0x02 when y is even; otherwise 0x03
function pprefix(hasEvenY) {
return Uint8Array.of(hasEvenY ? 0x02 : 0x03);
}
export function ecdsa(Point, ecdsaOpts, curveOpts = {}) {
_validateObject(ecdsaOpts, { hash: 'function' }, {
hmac: 'function',
lowS: 'boolean',
randomBytes: 'function',
bits2int: 'function',
bits2int_modN: 'function',
});
const randomBytes_ = ecdsaOpts.randomBytes || randomBytes;
const hmac_ = ecdsaOpts.hmac ||
((key, ...msgs) => hmac(ecdsaOpts.hash, key, concatBytes(...msgs)));
const { Fp, Fn } = Point;
const { ORDER: CURVE_ORDER, BITS: fnBits } = Fn;
function isBiggerThanHalfOrder(number) {
const HALF = CURVE_ORDER >> _1n;
return number > HALF;
}
function normalizeS(s) {
return isBiggerThanHalfOrder(s) ? Fn.neg(s) : s;
}
function aValidRS(title, num) {
if (!Fn.isValidNot0(num))
throw new Error(`invalid signature ${title}: out of range 1..CURVE.n`);
}
/**
* ECDSA signature with its (r, s) properties. Supports DER & compact representations.
*/
class Signature {
constructor(r, s, recovery) {
aValidRS('r', r); // r in [1..N-1]
aValidRS('s', s); // s in [1..N-1]
this.r = r;
this.s = s;
if (recovery != null)
this.recovery = recovery;
Object.freeze(this);
}
// pair (bytes of r, bytes of s)
static fromCompact(hex) {
const L = Fn.BYTES;
const b = ensureBytes('compactSignature', hex, L * 2);
return new Signature(Fn.fromBytes(b.subarray(0, L)), Fn.fromBytes(b.subarray(L, L * 2)));
}
// DER encoded ECDSA signature
// https://bitcoin.stackexchange.com/questions/57644/what-are-the-parts-of-a-bitcoin-transaction-input-script
static fromDER(hex) {
const { r, s } = DER.toSig(ensureBytes('DER', hex));
return new Signature(r, s);
}
/**
* @todo remove
* @deprecated
*/
assertValidity() { }
addRecoveryBit(recovery) {
return new Signature(this.r, this.s, recovery);
}
// ProjPointType<bigint>
recoverPublicKey(msgHash) {
const FIELD_ORDER = Fp.ORDER;
const { r, s, recovery: rec } = this;
if (rec == null || ![0, 1, 2, 3].includes(rec))
throw new Error('recovery id invalid');
// ECDSA recovery is hard for cofactor > 1 curves.
// In sign, `r = q.x mod n`, and here we recover q.x from r.
// While recovering q.x >= n, we need to add r+n for cofactor=1 curves.
// However, for cofactor>1, r+n may not get q.x:
// r+n*i would need to be done instead where i is unknown.
// To easily get i, we either need to:
// a. increase amount of valid recid values (4, 5...); OR
// b. prohibit non-prime-order signatures (recid > 1).
const hasCofactor = CURVE_ORDER * _2n < FIELD_ORDER;
if (hasCofactor && rec > 1)
throw new Error('recovery id is ambiguous for h>1 curve');
const radj = rec === 2 || rec === 3 ? r + CURVE_ORDER : r;
if (!Fp.isValid(radj))
throw new Error('recovery id 2 or 3 invalid');
const x = Fp.toBytes(radj);
const R = Point.fromHex(concatBytes(pprefix((rec & 1) === 0), x));
const ir = Fn.inv(radj); // r^-1
const h = bits2int_modN(ensureBytes('msgHash', msgHash)); // Truncate hash
const u1 = Fn.create(-h * ir); // -hr^-1
const u2 = Fn.create(s * ir); // sr^-1
// (sr^-1)R-(hr^-1)G = -(hr^-1)G + (sr^-1). unsafe is fine: there is no private data.
const Q = Point.BASE.multiplyUnsafe(u1).add(R.multiplyUnsafe(u2));
if (Q.is0())
throw new Error('point at infinify');
Q.assertValidity();
return Q;
}
// Signatures should be low-s, to prevent malleability.
hasHighS() {
return isBiggerThanHalfOrder(this.s);
}
normalizeS() {
return this.hasHighS() ? new Signature(this.r, Fn.neg(this.s), this.recovery) : this;
}
toBytes(format) {
if (format === 'compact')
return concatBytes(Fn.toBytes(this.r), Fn.toBytes(this.s));
if (format === 'der')
return hexToBytes(DER.hexFromSig(this));
throw new Error('invalid format');
}
// DER-encoded
toDERRawBytes() {
return this.toBytes('der');
}
toDERHex() {
return bytesToHex(this.toBytes('der'));
}
// padded bytes of r, then padded bytes of s
toCompactRawBytes() {
return this.toBytes('compact');
}
toCompactHex() {
return bytesToHex(this.toBytes('compact'));
}
}
const normPrivateKeyToScalar = _legacyHelperNormPriv(Fn, curveOpts.allowedPrivateKeyLengths, curveOpts.wrapPrivateKey);
const utils = {
isValidPrivateKey(privateKey) {
try {
normPrivateKeyToScalar(privateKey);
return true;
}
catch (error) {
return false;
}
},
normPrivateKeyToScalar: normPrivateKeyToScalar,
/**
* Produces cryptographically secure private key from random of size
* (groupLen + ceil(groupLen / 2)) with modulo bias being negligible.
*/
randomPrivateKey: () => {
const n = CURVE_ORDER;
return mapHashToField(randomBytes_(getMinHashLength(n)), n);
},
precompute(windowSize = 8, point = Point.BASE) {
return point.precompute(windowSize, false);
},
};
/**
* Computes public key for a private key. Checks for validity of the private key.
* @param privateKey private key
* @param isCompressed whether to return compact (default), or full key
* @returns Public key, full when isCompressed=false; short when isCompressed=true
*/
function getPublicKey(privateKey, isCompressed = true) {
return Point.fromPrivateKey(privateKey).toBytes(isCompressed);
}
/**
* Quick and dirty check for item being public key. Does not validate hex, or being on-curve.
*/
function isProbPub(item) {
if (typeof item === 'bigint')
return false;
if (item instanceof Point)
return true;
const arr = ensureBytes('key', item);
const length = arr.length;
const L = Fp.BYTES;
const LC = L + 1; // e.g. 33 for 32
const LU = 2 * L + 1; // e.g. 65 for 32
if (curveOpts.allowedPrivateKeyLengths || Fn.BYTES === LC) {
return undefined;
}
else {
return length === LC || length === LU;
}
}
/**
* ECDH (Elliptic Curve Diffie Hellman).
* Computes shared public key from private key and public key.
* Checks: 1) private key validity 2) shared key is on-curve.
* Does NOT hash the result.
* @param privateA private key
* @param publicB different public key
* @param isCompressed whether to return compact (default), or full key
* @returns shared public key
*/
function getSharedSecret(privateA, publicB, isCompressed = true) {
if (isProbPub(privateA) === true)
throw new Error('first arg must be private key');
if (isProbPub(publicB) === false)
throw new Error('second arg must be public key');
const b = Point.fromHex(publicB); // check for being on-curve
return b.multiply(normPrivateKeyToScalar(privateA)).toBytes(isCompressed);
}
// RFC6979: ensure ECDSA msg is X bytes and < N. RFC suggests optional truncating via bits2octets.
// FIPS 186-4 4.6 suggests the leftmost min(nBitLen, outLen) bits, which matches bits2int.
// bits2int can produce res>N, we can do mod(res, N) since the bitLen is the same.
// int2octets can't be used; pads small msgs with 0: unacceptatble for trunc as per RFC vectors
const bits2int = ecdsaOpts.bits2int ||
function (bytes) {
// Our custom check "just in case", for protection against DoS
if (bytes.length > 8192)
throw new Error('input is too large');
// For curves with nBitLength % 8 !== 0: bits2octets(bits2octets(m)) !== bits2octets(m)
// for some cases, since bytes.length * 8 is not actual bitLength.
const num = bytesToNumberBE(bytes); // check for == u8 done here
const delta = bytes.length * 8 - fnBits; // truncate to nBitLength leftmost bits
return delta > 0 ? num >> BigInt(delta) : num;
};
const bits2int_modN = ecdsaOpts.bits2int_modN ||
function (bytes) {
return Fn.create(bits2int(bytes)); // can't use bytesToNumberBE here
};
// NOTE: pads output with zero as per spec
const ORDER_MASK = bitMask(fnBits);
/**
* Converts to bytes. Checks if num in `[0..ORDER_MASK-1]` e.g.: `[0..2^256-1]`.
*/
function int2octets(num) {
// IMPORTANT: the check ensures working for case `Fn.BYTES != Fn.BITS * 8`
aInRange('num < 2^' + fnBits, num, _0n, ORDER_MASK);
return Fn.toBytes(num);
}
// Steps A, D of RFC6979 3.2
// Creates RFC6979 seed; converts msg/privKey to numbers.
// Used only in sign, not in verify.
// NOTE: we cannot assume here that msgHash has same amount of bytes as curve order,
// this will be invalid at least for P521. Also it can be bigger for P224 + SHA256
function prepSig(msgHash, privateKey, opts = defaultSigOpts) {
if (['recovered', 'canonical'].some((k) => k in opts))
throw new Error('sign() legacy options not supported');
const { hash } = ecdsaOpts;
let { lowS, prehash, extraEntropy: ent } = opts; // generates low-s sigs by default
if (lowS == null)
lowS = true; // RFC6979 3.2: we skip step A, because we already provide hash
msgHash = ensureBytes('msgHash', msgHash);
validateSigVerOpts(opts);
if (prehash)
msgHash = ensureBytes('prehashed msgHash', hash(msgHash));
// We can't later call bits2octets, since nested bits2int is broken for curves
// with fnBits % 8 !== 0. Because of that, we unwrap it here as int2octets call.
// const bits2octets = (bits) => int2octets(bits2int_modN(bits))
const h1int = bits2int_modN(msgHash);
const d = normPrivateKeyToScalar(privateKey); // validate private key, convert to bigint
const seedArgs = [int2octets(d), int2octets(h1int)];
// extraEntropy. RFC6979 3.6: additional k' (optional).
if (ent != null && ent !== false) {
// K = HMAC_K(V || 0x00 || int2octets(x) || bits2octets(h1) || k')
const e = ent === true ? randomBytes_(Fp.BYTES) : ent; // generate random bytes OR pass as-is
seedArgs.push(ensureBytes('extraEntropy', e)); // check for being bytes
}
const seed = concatBytes(...seedArgs); // Step D of RFC6979 3.2
const m = h1int; // NOTE: no need to call bits2int second time here, it is inside truncateHash!
// Converts signature params into point w r/s, checks result for validity.
// Can use scalar blinding b^-1(bm + bdr) where b ∈ [1,q−1] according to
// https://tches.iacr.org/index.php/TCHES/article/view/7337/6509. We've decided against it:
// a) dependency on CSPRNG b) 15% slowdown c) doesn't really help since bigints are not CT
function k2sig(kBytes) {
// RFC 6979 Section 3.2, step 3: k = bits2int(T)
// Important: all mod() calls here must be done over N
const k = bits2int(kBytes); // Cannot use fields methods, since it is group element
if (!Fn.isValidNot0(k))
return; // Valid scalars (including k) must be in 1..N-1
const ik = Fn.inv(k); // k^-1 mod n
const q = Point.BASE.multiply(k).toAffine(); // q = Gk
const r = Fn.create(q.x); // r = q.x mod n
if (r === _0n)
return;
const s = Fn.create(ik * Fn.create(m + r * d)); // Not using blinding here, see comment above
if (s === _0n)
return;
let recovery = (q.x === r ? 0 : 2) | Number(q.y & _1n); // recovery bit (2 or 3, when q.x > n)
let normS = s;
if (lowS && isBiggerThanHalfOrder(s)) {
normS = normalizeS(s); // if lowS was passed, ensure s is always
recovery ^= 1; // // in the bottom half of N
}
return new Signature(r, normS, recovery); // use normS, not s
}
return { seed, k2sig };
}
const defaultSigOpts = { lowS: ecdsaOpts.lowS, prehash: false };
const defaultVerOpts = { lowS: ecdsaOpts.lowS, prehash: false };
/**
* Signs message hash with a private key.
* ```
* sign(m, d, k) where
* (x, y) = G × k
* r = x mod n
* s = (m + dr)/k mod n
* ```
* @param msgHash NOT message. msg needs to be hashed to `msgHash`, or use `prehash`.
* @param privKey private key
* @param opts lowS for non-malleable sigs. extraEntropy for mixing randomness into k. prehash will hash first arg.
* @returns signature with recovery param
*/
function sign(msgHash, privKey, opts = defaultSigOpts) {
const { seed, k2sig } = prepSig(msgHash, privKey, opts); // Steps A, D of RFC6979 3.2.
const drbg = createHmacDrbg(ecdsaOpts.hash.outputLen, Fn.BYTES, hmac_);
return drbg(seed, k2sig); // Steps B, C, D, E, F, G
}
// Enable precomputes. Slows down first publicKey computation by 20ms.
Point.BASE.precompute(8);
/**
* Verifies a signature against message hash and public key.
* Rejects lowS signatures by default: to override,
* specify option `{lowS: false}`. Implements section 4.1.4 from https://www.secg.org/sec1-v2.pdf:
*
* ```
* verify(r, s, h, P) where
* U1 = hs^-1 mod n
* U2 = rs^-1 mod n
* R = U1⋅G - U2⋅P
* mod(R.x, n) == r
* ```
*/
function verify(signature, msgHash, publicKey, opts = defaultVerOpts) {
const sg = signature;
msgHash = ensureBytes('msgHash', msgHash);
publicKey = ensureBytes('publicKey', publicKey);
// Verify opts
validateSigVerOpts(opts);
const { lowS, prehash, format } = opts;
// TODO: remove
if ('strict' in opts)
throw new Error('options.strict was renamed to lowS');
if (format !== undefined && !['compact', 'der', 'js'].includes(format))
throw new Error('format must be "compact", "der" or "js"');
const isHex = typeof sg === 'string' || isBytes(sg);
const isObj = !isHex &&
!format &&
typeof sg === 'object' &&
sg !== null &&
typeof sg.r === 'bigint' &&
typeof sg.s === 'bigint';
if (!isHex && !isObj)
throw new Error('invalid signature, expected Uint8Array, hex string or Signature instance');
let _sig = undefined;
let P;
// deduce signature format
try {
// if (format === 'js') {
// if (sg != null && !isBytes(sg)) _sig = new Signature(sg.r, sg.s);
// } else if (format === 'compact') {
// _sig = Signature.fromCompact(sg);
// } else if (format === 'der') {
// _sig = Signature.fromDER(sg);
// } else {
// throw new Error('invalid format');
// }
if (isObj) {
if (format === undefined || format === 'js') {
_sig = new Signature(sg.r, sg.s);
}
else {
throw new Error('invalid format');
}
}
if (isHex) {
// TODO: remove this malleable check
// Signature can be represented in 2 ways: compact (2*Fn.BYTES) & DER (variable-length).
// Since DER can also be 2*Fn.BYTES bytes, we check for it first.
try {
if (format !== 'compact')
_sig = Signature.fromDER(sg);
}
catch (derError) {
if (!(derError instanceof DER.Err))
throw derError;
}
if (!_sig && format !== 'der')
_sig = Signature.fromCompact(sg);
}
P = Point.fromHex(publicKey);
}
catch (error) {
return false;
}
if (!_sig)
return false;
if (lowS && _sig.hasHighS())
return false;
// todo: optional.hash => hash
if (prehash)
msgHash = ecdsaOpts.hash(msgHash);
const { r, s } = _sig;
const h = bits2int_modN(msgHash); // Cannot use fields methods, since it is group element
const is = Fn.inv(s); // s^-1
const u1 = Fn.create(h * is); // u1 = hs^-1 mod n
const u2 = Fn.create(r * is); // u2 = rs^-1 mod n
const R = Point.BASE.multiplyUnsafe(u1).add(P.multiplyUnsafe(u2));
if (R.is0())
return false;
const v = Fn.create(R.x); // v = r.x mod n
return v === r;
}
// TODO: clarify API for cloning .clone({hash: sha512}) ? .createWith({hash: sha512})?
// const clone = (hash: CHash): ECDSA => ecdsa(Point, { ...ecdsaOpts, ...getHash(hash) }, curveOpts);
return Object.freeze({
getPublicKey,
getSharedSecret,
sign,
verify,
utils,
Point,
Signature,
});
}
function _weierstrass_legacy_opts_to_new(c) {
const CURVE = {
a: c.a,
b: c.b,
p: c.Fp.ORDER,
n: c.n,
h: c.h,
Gx: c.Gx,
Gy: c.Gy,
};
const Fp = c.Fp;
const Fn = Field(CURVE.n, c.nBitLength);
const curveOpts = {
Fp,
Fn,
allowedPrivateKeyLengths: c.allowedPrivateKeyLengths,
allowInfinityPoint: c.allowInfinityPoint,
endo: c.endo,
wrapPrivateKey: c.wrapPrivateKey,
isTorsionFree: c.isTorsionFree,
clearCofactor: c.clearCofactor,
fromBytes: c.fromBytes,
toBytes: c.toBytes,
};
return { CURVE, curveOpts };
}
function _ecdsa_legacy_opts_to_new(c) {
const { CURVE, curveOpts } = _weierstrass_legacy_opts_to_new(c);
const ecdsaOpts = {
hash: c.hash,
hmac: c.hmac,
randomBytes: c.randomBytes,
lowS: c.lowS,
bits2int: c.bits2int,
bits2int_modN: c.bits2int_modN,
};
return { CURVE, curveOpts, ecdsaOpts };
}
function _weierstrass_new_output_to_legacy(c, Point) {
const { Fp, Fn } = Point;
// TODO: remove
function isWithinCurveOrder(num) {
return inRange(num, _1n, Fn.ORDER);
}
const weierstrassEquation = _legacyHelperEquat(Fp, c.a, c.b);
const normPrivateKeyToScalar = _legacyHelperNormPriv(Fn, c.allowedPrivateKeyLengths, c.wrapPrivateKey);
return Object.assign({}, {
CURVE: c,
Point: Point,
ProjectivePoint: Point,
normPrivateKeyToScalar,
weierstrassEquation,
isWithinCurveOrder,
});
}
function _ecdsa_new_output_to_legacy(c, ecdsa) {
return Object.assign({}, ecdsa, {
ProjectivePoint: ecdsa.Point,
CURVE: c,
});
}
// _ecdsa_legacy
export function weierstrass(c) {
const { CURVE, curveOpts, ecdsaOpts } = _ecdsa_legacy_opts_to_new(c);
const Point = weierstrassN(CURVE, curveOpts);
const signs = ecdsa(Point, ecdsaOpts, curveOpts);
return _ecdsa_new_output_to_legacy(c, signs);
}
/**
* Implementation of the Shallue and van de Woestijne method for any weierstrass curve.
* TODO: check if there is a way to merge this with uvRatio in Edwards; move to modular.
* b = True and y = sqrt(u / v) if (u / v) is square in F, and
* b = False and y = sqrt(Z * (u / v)) otherwise.
* @param Fp
* @param Z
* @returns
*/
export function SWUFpSqrtRatio(Fp, Z) {
// Generic implementation
const q = Fp.ORDER;
let l = _0n;
for (let o = q - _1n; o % _2n === _0n; o /= _2n)
l += _1n;
const c1 = l; // 1. c1, the largest integer such that 2^c1 divides q - 1.
// We need 2n ** c1 and 2n ** (c1-1). We can't use **; but we can use <<.
// 2n ** c1 == 2n << (c1-1)
const _2n_pow_c1_1 = _2n << (c1 - _1n - _1n);
const _2n_pow_c1 = _2n_pow_c1_1 * _2n;
const c2 = (q - _1n) / _2n_pow_c1; // 2. c2 = (q - 1) / (2^c1) # Integer arithmetic
const c3 = (c2 - _1n) / _2n; // 3. c3 = (c2 - 1) / 2 # Integer arithmetic
const c4 = _2n_pow_c1 - _1n; // 4. c4 = 2^c1 - 1 # Integer arithmetic
const c5 = _2n_pow_c1_1; // 5. c5 = 2^(c1 - 1) # Integer arithmetic
const c6 = Fp.pow(Z, c2); // 6. c6 = Z^c2
const c7 = Fp.pow(Z, (c2 + _1n) / _2n); // 7. c7 = Z^((c2 + 1) / 2)
let sqrtRatio = (u, v) => {
let tv1 = c6; // 1. tv1 = c6
let tv2 = Fp.pow(v, c4); // 2. tv2 = v^c4
let tv3 = Fp.sqr(tv2); // 3. tv3 = tv2^2
tv3 = Fp.mul(tv3, v); // 4. tv3 = tv3 * v
let tv5 = Fp.mul(u, tv3); // 5. tv5 = u * tv3
tv5 = Fp.pow(tv5, c3); // 6. tv5 = tv5^c3
tv5 = Fp.mul(tv5, tv2); // 7. tv5 = tv5 * tv2
tv2 = Fp.mul(tv5, v); // 8. tv2 = tv5 * v
tv3 = Fp.mul(tv5, u)