PHP WebShell
Текущая директория: /usr/lib/node_modules/bitgo/node_modules/micro-eth-signer/node_modules/@noble/curves/src/abstract
Просмотр файла: weierstrass.ts
/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
// Short Weierstrass curve. The formula is: y² = x³ + ax + b
import * as mod from './modular.js';
import * as ut from './utils.js';
import { CHash, Hex, PrivKey, ensureBytes } from './utils.js';
import { Group, GroupConstructor, wNAF, BasicCurve, validateBasic, AffinePoint } from './curve.js';
export type { AffinePoint };
type HmacFnSync = (key: Uint8Array, ...messages: Uint8Array[]) => Uint8Array;
type EndomorphismOpts = {
beta: bigint;
splitScalar: (k: bigint) => { k1neg: boolean; k1: bigint; k2neg: boolean; k2: bigint };
};
export type BasicWCurve<T> = BasicCurve<T> & {
// Params: a, b
a: T;
b: T;
// Optional params
allowedPrivateKeyLengths?: readonly number[]; // for P521
wrapPrivateKey?: boolean; // bls12-381 requires mod(n) instead of rejecting keys >= n
endo?: EndomorphismOpts; // Endomorphism options for Koblitz curves
// When a cofactor != 1, there can be an effective methods to:
// 1. Determine whether a point is torsion-free
isTorsionFree?: (c: ProjConstructor<T>, point: ProjPointType<T>) => boolean;
// 2. Clear torsion component
clearCofactor?: (c: ProjConstructor<T>, point: ProjPointType<T>) => ProjPointType<T>;
};
type Entropy = Hex | true;
export type SignOpts = { lowS?: boolean; extraEntropy?: Entropy; prehash?: boolean };
export type VerOpts = { lowS?: boolean; prehash?: boolean };
/**
* ### 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
*/
// Instance for 3d XYZ points
export interface ProjPointType<T> extends Group<ProjPointType<T>> {
readonly px: T;
readonly py: T;
readonly pz: T;
get x(): T;
get y(): T;
multiply(scalar: bigint): ProjPointType<T>;
toAffine(iz?: T): AffinePoint<T>;
isTorsionFree(): boolean;
clearCofactor(): ProjPointType<T>;
assertValidity(): void;
hasEvenY(): boolean;
toRawBytes(isCompressed?: boolean): Uint8Array;
toHex(isCompressed?: boolean): string;
multiplyUnsafe(scalar: bigint): ProjPointType<T>;
multiplyAndAddUnsafe(Q: ProjPointType<T>, a: bigint, b: bigint): ProjPointType<T> | undefined;
_setWindowSize(windowSize: number): void;
}
// Static methods for 3d XYZ points
export interface ProjConstructor<T> extends GroupConstructor<ProjPointType<T>> {
new (x: T, y: T, z: T): ProjPointType<T>;
fromAffine(p: AffinePoint<T>): ProjPointType<T>;
fromHex(hex: Hex): ProjPointType<T>;
fromPrivateKey(privateKey: PrivKey): ProjPointType<T>;
normalizeZ(points: ProjPointType<T>[]): ProjPointType<T>[];
}
export type CurvePointsType<T> = BasicWCurve<T> & {
// Bytes
fromBytes?: (bytes: Uint8Array) => AffinePoint<T>;
toBytes?: (c: ProjConstructor<T>, point: ProjPointType<T>, isCompressed: boolean) => Uint8Array;
};
function validatePointOpts<T>(curve: CurvePointsType<T>) {
const opts = validateBasic(curve);
ut.validateObject(
opts,
{
a: 'field',
b: 'field',
},
{
allowedPrivateKeyLengths: 'array',
wrapPrivateKey: 'boolean',
isTorsionFree: 'function',
clearCofactor: 'function',
allowInfinityPoint: 'boolean',
fromBytes: 'function',
toBytes: 'function',
}
);
const { endo, Fp, a } = opts;
if (endo) {
if (!Fp.eql(a, Fp.ZERO)) {
throw new Error('Endomorphism can only be defined for Koblitz curves that have a=0');
}
if (
typeof endo !== 'object' ||
typeof endo.beta !== 'bigint' ||
typeof endo.splitScalar !== 'function'
) {
throw new Error('Expected endomorphism with beta: bigint and splitScalar: function');
}
}
return Object.freeze({ ...opts } as const);
}
export type CurvePointsRes<T> = {
CURVE: ReturnType<typeof validatePointOpts<T>>;
ProjectivePoint: ProjConstructor<T>;
normPrivateKeyToScalar: (key: PrivKey) => bigint;
weierstrassEquation: (x: T) => T;
isWithinCurveOrder: (num: bigint) => boolean;
};
// ASN.1 DER encoding utilities
const { bytesToNumberBE: b2n, hexToBytes: h2b } = ut;
export const DER = {
// asn.1 DER encoding utils
Err: class DERErr extends Error {
constructor(m = '') {
super(m);
}
},
_parseInt(data: Uint8Array): { d: bigint; l: Uint8Array } {
const { Err: E } = DER;
if (data.length < 2 || data[0] !== 0x02) throw new E('Invalid signature integer tag');
const len = data[1];
const res = data.subarray(2, len + 2);
if (!len || res.length !== len) throw new E('Invalid signature integer: wrong 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)
if (res[0] & 0b10000000) throw new E('Invalid signature integer: negative');
if (res[0] === 0x00 && !(res[1] & 0b10000000))
throw new E('Invalid signature integer: unnecessary leading zero');
return { d: b2n(res), l: data.subarray(len + 2) }; // d is data, l is left
},
toSig(hex: string | Uint8Array): { r: bigint; s: bigint } {
// parse DER signature
const { Err: E } = DER;
const data = typeof hex === 'string' ? h2b(hex) : hex;
if (!ut.isBytes(data)) throw new Error('ui8a expected');
let l = data.length;
if (l < 2 || data[0] != 0x30) throw new E('Invalid signature tag');
if (data[1] !== l - 2) throw new E('Invalid signature: incorrect length');
const { d: r, l: sBytes } = DER._parseInt(data.subarray(2));
const { d: s, l: rBytesLeft } = DER._parseInt(sBytes);
if (rBytesLeft.length) throw new E('Invalid signature: left bytes after parsing');
return { r, s };
},
hexFromSig(sig: { r: bigint; s: bigint }): string {
// Add leading zero if first byte has negative bit enabled. More details in '_parseInt'
const slice = (s: string): string => (Number.parseInt(s[0], 16) & 0b1000 ? '00' + s : s);
const h = (num: number | bigint) => {
const hex = num.toString(16);
return hex.length & 1 ? `0${hex}` : hex;
};
const s = slice(h(sig.s));
const r = slice(h(sig.r));
const shl = s.length / 2;
const rhl = r.length / 2;
const sl = h(shl);
const rl = h(rhl);
return `30${h(rhl + shl + 4)}02${rl}${r}02${sl}${s}`;
},
};
// 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);
export function weierstrassPoints<T>(opts: CurvePointsType<T>): CurvePointsRes<T> {
const CURVE = validatePointOpts(opts);
const { Fp } = CURVE; // All curves has same field / group length as for now, but they can differ
const toBytes =
CURVE.toBytes ||
((_c: ProjConstructor<T>, point: ProjPointType<T>, _isCompressed: boolean) => {
const a = point.toAffine();
return ut.concatBytes(Uint8Array.from([0x04]), Fp.toBytes(a.x), Fp.toBytes(a.y));
});
const fromBytes =
CURVE.fromBytes ||
((bytes: Uint8Array) => {
// const head = bytes[0];
const tail = bytes.subarray(1);
// if (head !== 0x04) throw new Error('Only non-compressed encoding is supported');
const x = Fp.fromBytes(tail.subarray(0, Fp.BYTES));
const y = Fp.fromBytes(tail.subarray(Fp.BYTES, 2 * Fp.BYTES));
return { x, y };
});
/**
* y² = x³ + ax + b: Short weierstrass curve formula
* @returns y²
*/
function weierstrassEquation(x: T): T {
const { a, b } = CURVE;
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
}
// Validate whether the passed curve params are valid.
// We check if curve equation works for generator point.
// `assertValidity()` won't work: `isTorsionFree()` is not available at this point in bls12-381.
// ProjectivePoint class has not been initialized yet.
if (!Fp.eql(Fp.sqr(CURVE.Gy), weierstrassEquation(CURVE.Gx)))
throw new Error('bad generator point: equation left != right');
// Valid group elements reside in range 1..n-1
function isWithinCurveOrder(num: bigint): boolean {
return typeof num === 'bigint' && _0n < num && num < CURVE.n;
}
function assertGE(num: bigint) {
if (!isWithinCurveOrder(num)) throw new Error('Expected valid bigint: 0 < bigint < curve.n');
}
// Validates if priv key is valid and converts it to bigint.
// Supports options allowedPrivateKeyLengths and wrapPrivateKey.
function normPrivateKeyToScalar(key: PrivKey): bigint {
const { allowedPrivateKeyLengths: lengths, nByteLength, wrapPrivateKey, n } = CURVE;
if (lengths && typeof key !== 'bigint') {
if (ut.isBytes(key)) key = ut.bytesToHex(key);
// Normalize to hex string, pad. E.g. P521 would norm 130-132 char hex to 132-char bytes
if (typeof key !== 'string' || !lengths.includes(key.length)) throw new Error('Invalid key');
key = key.padStart(nByteLength * 2, '0');
}
let num: bigint;
try {
num =
typeof key === 'bigint'
? key
: ut.bytesToNumberBE(ensureBytes('private key', key, nByteLength));
} catch (error) {
throw new Error(`private key must be ${nByteLength} bytes, hex or bigint, not ${typeof key}`);
}
if (wrapPrivateKey) num = mod.mod(num, n); // disabled by default, enabled for BLS
assertGE(num); // num in range [1..N-1]
return num;
}
const pointPrecomputes = new Map<Point, Point[]>();
function assertPrjPoint(other: unknown) {
if (!(other instanceof Point)) throw new Error('ProjectivePoint expected');
}
/**
* 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 implements ProjPointType<T> {
static readonly BASE = new Point(CURVE.Gx, CURVE.Gy, Fp.ONE);
static readonly ZERO = new Point(Fp.ZERO, Fp.ONE, Fp.ZERO);
constructor(
readonly px: T,
readonly py: T,
readonly pz: T
) {
if (px == null || !Fp.isValid(px)) throw new Error('x required');
if (py == null || !Fp.isValid(py)) throw new Error('y required');
if (pz == null || !Fp.isValid(pz)) throw new Error('z required');
}
// Does not validate if the point is on-curve.
// Use fromHex instead, or call assertValidity() later.
static fromAffine(p: AffinePoint<T>): Point {
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');
const is0 = (i: T) => Fp.eql(i, Fp.ZERO);
// fromAffine(x:0, y:0) would produce (x:0, y:0, z:1), but we need (x:0, y:1, z:0)
if (is0(x) && is0(y)) return Point.ZERO;
return new Point(x, y, Fp.ONE);
}
get x(): T {
return this.toAffine().x;
}
get y(): T {
return this.toAffine().y;
}
/**
* Takes a bunch of Projective Points but executes only one
* inversion on all of them. Inversion is very slow operation,
* so this improves performance massively.
* Optimization: converts a list of projective points to a list of identical points with Z=1.
*/
static normalizeZ(points: Point[]): Point[] {
const toInv = Fp.invertBatch(points.map((p) => p.pz));
return points.map((p, i) => p.toAffine(toInv[i])).map(Point.fromAffine);
}
/**
* Converts hash string or Uint8Array to Point.
* @param hex short/long ECDSA hex
*/
static fromHex(hex: Hex): Point {
const P = Point.fromAffine(fromBytes(ensureBytes('pointHex', hex)));
P.assertValidity();
return P;
}
// Multiplies generator point by privateKey.
static fromPrivateKey(privateKey: PrivKey) {
return Point.BASE.multiply(normPrivateKeyToScalar(privateKey));
}
// 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);
}
// A point on curve is valid if it conforms to equation.
assertValidity(): void {
if (this.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 wrong representation of ZERO and is always invalid.
if (CURVE.allowInfinityPoint && !Fp.is0(this.py)) return;
throw new Error('bad point: ZERO');
}
// Some 3rd-party test vectors require different wording between here & `fromCompressedHex`
const { x, y } = this.toAffine();
// Check if x, y are valid field elements
if (!Fp.isValid(x) || !Fp.isValid(y)) throw new Error('bad point: x or y not FE');
const left = Fp.sqr(y); // y²
const right = weierstrassEquation(x); // x³ + ax + b
if (!Fp.eql(left, right)) throw new Error('bad point: equation left != right');
if (!this.isTorsionFree()) throw new Error('bad point: not in prime-order subgroup');
}
hasEvenY(): boolean {
const { y } = this.toAffine();
if (Fp.isOdd) return !Fp.isOdd(y);
throw new Error("Field doesn't support isOdd");
}
/**
* Compare one point to another.
*/
equals(other: Point): boolean {
assertPrjPoint(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(): Point {
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: Point): Point {
assertPrjPoint(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: Point) {
return this.add(other.negate());
}
private is0() {
return this.equals(Point.ZERO);
}
private wNAF(n: bigint): { p: Point; f: Point } {
return wnaf.wNAFCached(this, pointPrecomputes, n, (comp: Point[]) => {
const toInv = Fp.invertBatch(comp.map((p) => p.pz));
return comp.map((p, i) => p.toAffine(toInv[i])).map(Point.fromAffine);
});
}
/**
* 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(n: bigint): Point {
const I = Point.ZERO;
if (n === _0n) return I;
assertGE(n); // Will throw on 0
if (n === _1n) return this;
const { endo } = CURVE;
if (!endo) return wnaf.unsafeLadder(this, n);
// Apply endomorphism
let { k1neg, k1, k2neg, k2 } = endo.splitScalar(n);
let k1p = I;
let k2p = I;
let d: Point = this;
while (k1 > _0n || k2 > _0n) {
if (k1 & _1n) k1p = k1p.add(d);
if (k2 & _1n) k2p = k2p.add(d);
d = d.double();
k1 >>= _1n;
k2 >>= _1n;
}
if (k1neg) k1p = k1p.negate();
if (k2neg) k2p = k2p.negate();
k2p = new Point(Fp.mul(k2p.px, endo.beta), k2p.py, k2p.pz);
return k1p.add(k2p);
}
/**
* 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: bigint): Point {
assertGE(scalar);
let n = scalar;
let point: Point, fake: Point; // Fake point is used to const-time mult
const { endo } = CURVE;
if (endo) {
const { k1neg, k1, k2neg, k2 } = endo.splitScalar(n);
let { p: k1p, f: f1p } = this.wNAF(k1);
let { p: k2p, f: f2p } = this.wNAF(k2);
k1p = wnaf.constTimeNegate(k1neg, k1p);
k2p = wnaf.constTimeNegate(k2neg, k2p);
k2p = new Point(Fp.mul(k2p.px, endo.beta), k2p.py, k2p.pz);
point = k1p.add(k2p);
fake = f1p.add(f2p);
} else {
const { p, f } = this.wNAF(n);
point = p;
fake = f;
}
// Normalize `z` for both points, but return only real one
return Point.normalizeZ([point, fake])[0];
}
/**
* Efficiently calculate `aP + bQ`. Unsafe, can expose private key, if used incorrectly.
* Not using Strauss-Shamir trick: precomputation tables are faster.
* The trick could be useful if both P and Q are not G (not in our case).
* @returns non-zero affine point
*/
multiplyAndAddUnsafe(Q: Point, a: bigint, b: bigint): Point | undefined {
const G = Point.BASE; // No Strauss-Shamir trick: we have 10% faster G precomputes
const mul = (
P: Point,
a: bigint // Select faster multiply() method
) => (a === _0n || a === _1n || !P.equals(G) ? P.multiplyUnsafe(a) : P.multiply(a));
const sum = mul(this, a).add(mul(Q, b));
return sum.is0() ? undefined : sum;
}
// 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)
toAffine(iz?: T): AffinePoint<T> {
const { px: x, py: y, pz: z } = this;
const is0 = this.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 };
}
isTorsionFree(): boolean {
const { h: cofactor, isTorsionFree } = CURVE;
if (cofactor === _1n) return true; // No subgroups, always torsion-free
if (isTorsionFree) return isTorsionFree(Point, this);
throw new Error('isTorsionFree() has not been declared for the elliptic curve');
}
clearCofactor(): Point {
const { h: cofactor, clearCofactor } = CURVE;
if (cofactor === _1n) return this; // Fast-path
if (clearCofactor) return clearCofactor(Point, this) as Point;
return this.multiplyUnsafe(CURVE.h);
}
toRawBytes(isCompressed = true): Uint8Array {
this.assertValidity();
return toBytes(Point, this, isCompressed);
}
toHex(isCompressed = true): string {
return ut.bytesToHex(this.toRawBytes(isCompressed));
}
}
const _bits = CURVE.nBitLength;
const wnaf = wNAF(Point, CURVE.endo ? Math.ceil(_bits / 2) : _bits);
// Validate if generator point is on curve
return {
CURVE,
ProjectivePoint: Point as ProjConstructor<T>,
normPrivateKeyToScalar,
weierstrassEquation,
isWithinCurveOrder,
};
}
// Instance
export interface SignatureType {
readonly r: bigint;
readonly s: bigint;
readonly recovery?: number;
assertValidity(): void;
addRecoveryBit(recovery: number): RecoveredSignatureType;
hasHighS(): boolean;
normalizeS(): SignatureType;
recoverPublicKey(msgHash: Hex): ProjPointType<bigint>;
toCompactRawBytes(): Uint8Array;
toCompactHex(): string;
// DER-encoded
toDERRawBytes(isCompressed?: boolean): Uint8Array;
toDERHex(isCompressed?: boolean): string;
}
export type RecoveredSignatureType = SignatureType & {
readonly recovery: number;
};
// Static methods
export type SignatureConstructor = {
new (r: bigint, s: bigint): SignatureType;
fromCompact(hex: Hex): SignatureType;
fromDER(hex: Hex): SignatureType;
};
type SignatureLike = { r: bigint; s: bigint };
export type PubKey = Hex | ProjPointType<bigint>;
export type CurveType = BasicWCurve<bigint> & {
hash: CHash; // CHash not FHash because we need outputLen for DRBG
hmac: HmacFnSync;
randomBytes: (bytesLength?: number) => Uint8Array;
lowS?: boolean;
bits2int?: (bytes: Uint8Array) => bigint;
bits2int_modN?: (bytes: Uint8Array) => bigint;
};
function validateOpts(curve: CurveType) {
const opts = validateBasic(curve);
ut.validateObject(
opts,
{
hash: 'hash',
hmac: 'function',
randomBytes: 'function',
},
{
bits2int: 'function',
bits2int_modN: 'function',
lowS: 'boolean',
}
);
return Object.freeze({ lowS: true, ...opts } as const);
}
export type CurveFn = {
CURVE: ReturnType<typeof validateOpts>;
getPublicKey: (privateKey: PrivKey, isCompressed?: boolean) => Uint8Array;
getSharedSecret: (privateA: PrivKey, publicB: Hex, isCompressed?: boolean) => Uint8Array;
sign: (msgHash: Hex, privKey: PrivKey, opts?: SignOpts) => RecoveredSignatureType;
verify: (signature: Hex | SignatureLike, msgHash: Hex, publicKey: Hex, opts?: VerOpts) => boolean;
ProjectivePoint: ProjConstructor<bigint>;
Signature: SignatureConstructor;
utils: {
normPrivateKeyToScalar: (key: PrivKey) => bigint;
isValidPrivateKey(privateKey: PrivKey): boolean;
randomPrivateKey: () => Uint8Array;
precompute: (windowSize?: number, point?: ProjPointType<bigint>) => ProjPointType<bigint>;
};
};
export function weierstrass(curveDef: CurveType): CurveFn {
const CURVE = validateOpts(curveDef) as ReturnType<typeof validateOpts>;
const { Fp, n: CURVE_ORDER } = CURVE;
const compressedLen = Fp.BYTES + 1; // e.g. 33 for 32
const uncompressedLen = 2 * Fp.BYTES + 1; // e.g. 65 for 32
function isValidFieldElement(num: bigint): boolean {
return _0n < num && num < Fp.ORDER; // 0 is banned since it's not invertible FE
}
function modN(a: bigint) {
return mod.mod(a, CURVE_ORDER);
}
function invN(a: bigint) {
return mod.invert(a, CURVE_ORDER);
}
const {
ProjectivePoint: Point,
normPrivateKeyToScalar,
weierstrassEquation,
isWithinCurveOrder,
} = weierstrassPoints({
...CURVE,
toBytes(_c, point, isCompressed: boolean): Uint8Array {
const a = point.toAffine();
const x = Fp.toBytes(a.x);
const cat = ut.concatBytes;
if (isCompressed) {
return cat(Uint8Array.from([point.hasEvenY() ? 0x02 : 0x03]), x);
} else {
return cat(Uint8Array.from([0x04]), x, Fp.toBytes(a.y));
}
},
fromBytes(bytes: Uint8Array) {
const len = bytes.length;
const head = bytes[0];
const tail = bytes.subarray(1);
// this.assertValidity() is done inside of fromHex
if (len === compressedLen && (head === 0x02 || head === 0x03)) {
const x = ut.bytesToNumberBE(tail);
if (!isValidFieldElement(x)) throw new Error('Point is not on curve');
const y2 = weierstrassEquation(x); // y² = x³ + ax + b
let y = Fp.sqrt(y2); // y = y² ^ (p+1)/4
const isYOdd = (y & _1n) === _1n;
// ECDSA
const isHeadOdd = (head & 1) === 1;
if (isHeadOdd !== isYOdd) y = Fp.neg(y);
return { x, y };
} else if (len === uncompressedLen && head === 0x04) {
const x = Fp.fromBytes(tail.subarray(0, Fp.BYTES));
const y = Fp.fromBytes(tail.subarray(Fp.BYTES, 2 * Fp.BYTES));
return { x, y };
} else {
throw new Error(
`Point of length ${len} was invalid. Expected ${compressedLen} compressed bytes or ${uncompressedLen} uncompressed bytes`
);
}
},
});
const numToNByteStr = (num: bigint): string =>
ut.bytesToHex(ut.numberToBytesBE(num, CURVE.nByteLength));
function isBiggerThanHalfOrder(number: bigint) {
const HALF = CURVE_ORDER >> _1n;
return number > HALF;
}
function normalizeS(s: bigint) {
return isBiggerThanHalfOrder(s) ? modN(-s) : s;
}
// slice bytes num
const slcNum = (b: Uint8Array, from: number, to: number) => ut.bytesToNumberBE(b.slice(from, to));
/**
* ECDSA signature with its (r, s) properties. Supports DER & compact representations.
*/
class Signature implements SignatureType {
constructor(
readonly r: bigint,
readonly s: bigint,
readonly recovery?: number
) {
this.assertValidity();
}
// pair (bytes of r, bytes of s)
static fromCompact(hex: Hex) {
const l = CURVE.nByteLength;
hex = ensureBytes('compactSignature', hex, l * 2);
return new Signature(slcNum(hex, 0, l), slcNum(hex, l, 2 * l));
}
// DER encoded ECDSA signature
// https://bitcoin.stackexchange.com/questions/57644/what-are-the-parts-of-a-bitcoin-transaction-input-script
static fromDER(hex: Hex) {
const { r, s } = DER.toSig(ensureBytes('DER', hex));
return new Signature(r, s);
}
assertValidity(): void {
// can use assertGE here
if (!isWithinCurveOrder(this.r)) throw new Error('r must be 0 < r < CURVE.n');
if (!isWithinCurveOrder(this.s)) throw new Error('s must be 0 < s < CURVE.n');
}
addRecoveryBit(recovery: number): RecoveredSignature {
return new Signature(this.r, this.s, recovery) as RecoveredSignature;
}
recoverPublicKey(msgHash: Hex): typeof Point.BASE {
const { r, s, recovery: rec } = this;
const h = bits2int_modN(ensureBytes('msgHash', msgHash)); // Truncate hash
if (rec == null || ![0, 1, 2, 3].includes(rec)) throw new Error('recovery id invalid');
const radj = rec === 2 || rec === 3 ? r + CURVE.n : r;
if (radj >= Fp.ORDER) throw new Error('recovery id 2 or 3 invalid');
const prefix = (rec & 1) === 0 ? '02' : '03';
const R = Point.fromHex(prefix + numToNByteStr(radj));
const ir = invN(radj); // r^-1
const u1 = modN(-h * ir); // -hr^-1
const u2 = modN(s * ir); // sr^-1
const Q = Point.BASE.multiplyAndAddUnsafe(R, u1, u2); // (sr^-1)R-(hr^-1)G = -(hr^-1)G + (sr^-1)
if (!Q) throw new Error('point at infinify'); // unsafe is fine: no priv data leaked
Q.assertValidity();
return Q;
}
// Signatures should be low-s, to prevent malleability.
hasHighS(): boolean {
return isBiggerThanHalfOrder(this.s);
}
normalizeS() {
return this.hasHighS() ? new Signature(this.r, modN(-this.s), this.recovery) : this;
}
// DER-encoded
toDERRawBytes() {
return ut.hexToBytes(this.toDERHex());
}
toDERHex() {
return DER.hexFromSig({ r: this.r, s: this.s });
}
// padded bytes of r, then padded bytes of s
toCompactRawBytes() {
return ut.hexToBytes(this.toCompactHex());
}
toCompactHex() {
return numToNByteStr(this.r) + numToNByteStr(this.s);
}
}
type RecoveredSignature = Signature & { recovery: number };
const utils = {
isValidPrivateKey(privateKey: PrivKey) {
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: (): Uint8Array => {
const length = mod.getMinHashLength(CURVE.n);
return mod.mapHashToField(CURVE.randomBytes(length), CURVE.n);
},
/**
* Creates precompute table for an arbitrary EC point. Makes point "cached".
* Allows to massively speed-up `point.multiply(scalar)`.
* @returns cached point
* @example
* const fast = utils.precompute(8, ProjectivePoint.fromHex(someonesPubKey));
* fast.multiply(privKey); // much faster ECDH now
*/
precompute(windowSize = 8, point = Point.BASE): typeof Point.BASE {
point._setWindowSize(windowSize);
point.multiply(BigInt(3)); // 3 is arbitrary, just need any number here
return point;
},
};
/**
* 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: PrivKey, isCompressed = true): Uint8Array {
return Point.fromPrivateKey(privateKey).toRawBytes(isCompressed);
}
/**
* Quick and dirty check for item being public key. Does not validate hex, or being on-curve.
*/
function isProbPub(item: PrivKey | PubKey): boolean {
const arr = ut.isBytes(item);
const str = typeof item === 'string';
const len = (arr || str) && (item as Hex).length;
if (arr) return len === compressedLen || len === uncompressedLen;
if (str) return len === 2 * compressedLen || len === 2 * uncompressedLen;
if (item instanceof Point) return true;
return false;
}
/**
* 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: PrivKey, publicB: Hex, isCompressed = true): Uint8Array {
if (isProbPub(privateA)) throw new Error('first arg must be private key');
if (!isProbPub(publicB)) throw new Error('second arg must be public key');
const b = Point.fromHex(publicB); // check for being on-curve
return b.multiply(normPrivateKeyToScalar(privateA)).toRawBytes(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 =
CURVE.bits2int ||
function (bytes: Uint8Array): bigint {
// For curves with nBitLength % 8 !== 0: bits2octets(bits2octets(m)) !== bits2octets(m)
// for some cases, since bytes.length * 8 is not actual bitLength.
const num = ut.bytesToNumberBE(bytes); // check for == u8 done here
const delta = bytes.length * 8 - CURVE.nBitLength; // truncate to nBitLength leftmost bits
return delta > 0 ? num >> BigInt(delta) : num;
};
const bits2int_modN =
CURVE.bits2int_modN ||
function (bytes: Uint8Array): bigint {
return modN(bits2int(bytes)); // can't use bytesToNumberBE here
};
// NOTE: pads output with zero as per spec
const ORDER_MASK = ut.bitMask(CURVE.nBitLength);
/**
* Converts to bytes. Checks if num in `[0..ORDER_MASK-1]` e.g.: `[0..2^256-1]`.
*/
function int2octets(num: bigint): Uint8Array {
if (typeof num !== 'bigint') throw new Error('bigint expected');
if (!(_0n <= num && num < ORDER_MASK))
throw new Error(`bigint expected < 2^${CURVE.nBitLength}`);
// works with order, can have different size than numToField!
return ut.numberToBytesBE(num, CURVE.nByteLength);
}
// 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 wrong at least for P521.
// Also it can be bigger for P224 + SHA256
function prepSig(msgHash: Hex, privateKey: PrivKey, opts = defaultSigOpts) {
if (['recovered', 'canonical'].some((k) => k in opts))
throw new Error('sign() legacy options not supported');
const { hash, randomBytes } = CURVE;
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);
if (prehash) msgHash = ensureBytes('prehashed msgHash', hash(msgHash));
// We can't later call bits2octets, since nested bits2int is broken for curves
// with nBitLength % 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) {
// 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 = ut.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.
function k2sig(kBytes: Uint8Array): RecoveredSignature | undefined {
// RFC 6979 Section 3.2, step 3: k = bits2int(T)
const k = bits2int(kBytes); // Cannot use fields methods, since it is group element
if (!isWithinCurveOrder(k)) return; // Important: all mod() calls here must be done over N
const ik = invN(k); // k^-1 mod n
const q = Point.BASE.multiply(k).toAffine(); // q = Gk
const r = modN(q.x); // r = q.x mod n
if (r === _0n) return;
// 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
const s = modN(ik * modN(m + r * d)); // Not using blinding here
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) as RecoveredSignature; // use normS, not s
}
return { seed, k2sig };
}
const defaultSigOpts: SignOpts = { lowS: CURVE.lowS, prehash: false };
const defaultVerOpts: VerOpts = { lowS: CURVE.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: Hex, privKey: PrivKey, opts = defaultSigOpts): RecoveredSignature {
const { seed, k2sig } = prepSig(msgHash, privKey, opts); // Steps A, D of RFC6979 3.2.
const C = CURVE;
const drbg = ut.createHmacDrbg<RecoveredSignature>(C.hash.outputLen, C.nByteLength, C.hmac);
return drbg(seed, k2sig); // Steps B, C, D, E, F, G
}
// Enable precomputes. Slows down first publicKey computation by 20ms.
Point.BASE._setWindowSize(8);
// utils.precompute(8, ProjectivePoint.BASE)
/**
* 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: Hex | SignatureLike,
msgHash: Hex,
publicKey: Hex,
opts = defaultVerOpts
): boolean {
const sg = signature;
msgHash = ensureBytes('msgHash', msgHash);
publicKey = ensureBytes('publicKey', publicKey);
if ('strict' in opts) throw new Error('options.strict was renamed to lowS');
const { lowS, prehash } = opts;
let _sig: Signature | undefined = undefined;
let P: ProjPointType<bigint>;
try {
if (typeof sg === 'string' || ut.isBytes(sg)) {
// Signature can be represented in 2 ways: compact (2*nByteLength) & DER (variable-length).
// Since DER can also be 2*nByteLength bytes, we check for it first.
try {
_sig = Signature.fromDER(sg);
} catch (derError) {
if (!(derError instanceof DER.Err)) throw derError;
_sig = Signature.fromCompact(sg);
}
} else if (typeof sg === 'object' && typeof sg.r === 'bigint' && typeof sg.s === 'bigint') {
const { r, s } = sg;
_sig = new Signature(r, s);
} else {
throw new Error('PARSE');
}
P = Point.fromHex(publicKey);
} catch (error) {
if ((error as Error).message === 'PARSE')
throw new Error(`signature must be Signature instance, Uint8Array or hex string`);
return false;
}
if (lowS && _sig.hasHighS()) return false;
if (prehash) msgHash = CURVE.hash(msgHash);
const { r, s } = _sig;
const h = bits2int_modN(msgHash); // Cannot use fields methods, since it is group element
const is = invN(s); // s^-1
const u1 = modN(h * is); // u1 = hs^-1 mod n
const u2 = modN(r * is); // u2 = rs^-1 mod n
const R = Point.BASE.multiplyAndAddUnsafe(P, u1, u2)?.toAffine(); // R = u1⋅G + u2⋅P
if (!R) return false;
const v = modN(R.x);
return v === r;
}
return {
CURVE,
getPublicKey,
getSharedSecret,
sign,
verify,
ProjectivePoint: Point,
Signature,
utils,
};
}
/**
* 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<T>(Fp: mod.IField<T>, Z: T) {
// 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: T, v: T): { isValid: boolean; value: T } => {
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); // 9. tv3 = tv5 * u
let tv4 = Fp.mul(tv3, tv2); // 10. tv4 = tv3 * tv2
tv5 = Fp.pow(tv4, c5); // 11. tv5 = tv4^c5
let isQR = Fp.eql(tv5, Fp.ONE); // 12. isQR = tv5 == 1
tv2 = Fp.mul(tv3, c7); // 13. tv2 = tv3 * c7
tv5 = Fp.mul(tv4, tv1); // 14. tv5 = tv4 * tv1
tv3 = Fp.cmov(tv2, tv3, isQR); // 15. tv3 = CMOV(tv2, tv3, isQR)
tv4 = Fp.cmov(tv5, tv4, isQR); // 16. tv4 = CMOV(tv5, tv4, isQR)
// 17. for i in (c1, c1 - 1, ..., 2):
for (let i = c1; i > _1n; i--) {
let tv5 = i - _2n; // 18. tv5 = i - 2
tv5 = _2n << (tv5 - _1n); // 19. tv5 = 2^tv5
let tvv5 = Fp.pow(tv4, tv5); // 20. tv5 = tv4^tv5
const e1 = Fp.eql(tvv5, Fp.ONE); // 21. e1 = tv5 == 1
tv2 = Fp.mul(tv3, tv1); // 22. tv2 = tv3 * tv1
tv1 = Fp.mul(tv1, tv1); // 23. tv1 = tv1 * tv1
tvv5 = Fp.mul(tv4, tv1); // 24. tv5 = tv4 * tv1
tv3 = Fp.cmov(tv2, tv3, e1); // 25. tv3 = CMOV(tv2, tv3, e1)
tv4 = Fp.cmov(tvv5, tv4, e1); // 26. tv4 = CMOV(tv5, tv4, e1)
}
return { isValid: isQR, value: tv3 };
};
if (Fp.ORDER % _4n === _3n) {
// sqrt_ratio_3mod4(u, v)
const c1 = (Fp.ORDER - _3n) / _4n; // 1. c1 = (q - 3) / 4 # Integer arithmetic
const c2 = Fp.sqrt(Fp.neg(Z)); // 2. c2 = sqrt(-Z)
sqrtRatio = (u: T, v: T) => {
let tv1 = Fp.sqr(v); // 1. tv1 = v^2
const tv2 = Fp.mul(u, v); // 2. tv2 = u * v
tv1 = Fp.mul(tv1, tv2); // 3. tv1 = tv1 * tv2
let y1 = Fp.pow(tv1, c1); // 4. y1 = tv1^c1
y1 = Fp.mul(y1, tv2); // 5. y1 = y1 * tv2
const y2 = Fp.mul(y1, c2); // 6. y2 = y1 * c2
const tv3 = Fp.mul(Fp.sqr(y1), v); // 7. tv3 = y1^2; 8. tv3 = tv3 * v
const isQR = Fp.eql(tv3, u); // 9. isQR = tv3 == u
let y = Fp.cmov(y2, y1, isQR); // 10. y = CMOV(y2, y1, isQR)
return { isValid: isQR, value: y }; // 11. return (isQR, y) isQR ? y : y*c2
};
}
// No curves uses that
// if (Fp.ORDER % _8n === _5n) // sqrt_ratio_5mod8
return sqrtRatio;
}
/**
* Simplified Shallue-van de Woestijne-Ulas Method
* https://www.rfc-editor.org/rfc/rfc9380#section-6.6.2
*/
export function mapToCurveSimpleSWU<T>(
Fp: mod.IField<T>,
opts: {
A: T;
B: T;
Z: T;
}
) {
mod.validateField(Fp);
if (!Fp.isValid(opts.A) || !Fp.isValid(opts.B) || !Fp.isValid(opts.Z))
throw new Error('mapToCurveSimpleSWU: invalid opts');
const sqrtRatio = SWUFpSqrtRatio(Fp, opts.Z);
if (!Fp.isOdd) throw new Error('Fp.isOdd is not implemented!');
// Input: u, an element of F.
// Output: (x, y), a point on E.
return (u: T): { x: T; y: T } => {
// prettier-ignore
let tv1, tv2, tv3, tv4, tv5, tv6, x, y;
tv1 = Fp.sqr(u); // 1. tv1 = u^2
tv1 = Fp.mul(tv1, opts.Z); // 2. tv1 = Z * tv1
tv2 = Fp.sqr(tv1); // 3. tv2 = tv1^2
tv2 = Fp.add(tv2, tv1); // 4. tv2 = tv2 + tv1
tv3 = Fp.add(tv2, Fp.ONE); // 5. tv3 = tv2 + 1
tv3 = Fp.mul(tv3, opts.B); // 6. tv3 = B * tv3
tv4 = Fp.cmov(opts.Z, Fp.neg(tv2), !Fp.eql(tv2, Fp.ZERO)); // 7. tv4 = CMOV(Z, -tv2, tv2 != 0)
tv4 = Fp.mul(tv4, opts.A); // 8. tv4 = A * tv4
tv2 = Fp.sqr(tv3); // 9. tv2 = tv3^2
tv6 = Fp.sqr(tv4); // 10. tv6 = tv4^2
tv5 = Fp.mul(tv6, opts.A); // 11. tv5 = A * tv6
tv2 = Fp.add(tv2, tv5); // 12. tv2 = tv2 + tv5
tv2 = Fp.mul(tv2, tv3); // 13. tv2 = tv2 * tv3
tv6 = Fp.mul(tv6, tv4); // 14. tv6 = tv6 * tv4
tv5 = Fp.mul(tv6, opts.B); // 15. tv5 = B * tv6
tv2 = Fp.add(tv2, tv5); // 16. tv2 = tv2 + tv5
x = Fp.mul(tv1, tv3); // 17. x = tv1 * tv3
const { isValid, value } = sqrtRatio(tv2, tv6); // 18. (is_gx1_square, y1) = sqrt_ratio(tv2, tv6)
y = Fp.mul(tv1, u); // 19. y = tv1 * u -> Z * u^3 * y1
y = Fp.mul(y, value); // 20. y = y * y1
x = Fp.cmov(x, tv3, isValid); // 21. x = CMOV(x, tv3, is_gx1_square)
y = Fp.cmov(y, value, isValid); // 22. y = CMOV(y, y1, is_gx1_square)
const e1 = Fp.isOdd!(u) === Fp.isOdd!(y); // 23. e1 = sgn0(u) == sgn0(y)
y = Fp.cmov(Fp.neg(y), y, e1); // 24. y = CMOV(-y, y, e1)
x = Fp.div(x, tv4); // 25. x = x / tv4
return { x, y };
};
}
Выполнить команду
Для локальной разработки. Не используйте в интернете!