1412 lines
62 KiB
JavaScript
1412 lines
62 KiB
JavaScript
/**
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* Short Weierstrass curve methods. The formula is: y² = x³ + ax + b.
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*
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* ### Design rationale for types
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*
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* * Interaction between classes from different curves should fail:
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* `k256.Point.BASE.add(p256.Point.BASE)`
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* * For this purpose we want to use `instanceof` operator, which is fast and works during runtime
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* * Different calls of `curve()` would return different classes -
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* `curve(params) !== curve(params)`: if somebody decided to monkey-patch their curve,
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* it won't affect others
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*
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* TypeScript can't infer types for classes created inside a function. Classes is one instance
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* of nominative types in TypeScript and interfaces only check for shape, so it's hard to create
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* unique type for every function call.
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*
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* We can use generic types via some param, like curve opts, but that would:
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* 1. Enable interaction between `curve(params)` and `curve(params)` (curves of same params)
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* which is hard to debug.
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* 2. Params can be generic and we can't enforce them to be constant value:
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* if somebody creates curve from non-constant params,
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* it would be allowed to interact with other curves with non-constant params
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*
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* @todo https://www.typescriptlang.org/docs/handbook/release-notes/typescript-2-7.html#unique-symbol
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* @module
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*/
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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
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import { hmac as nobleHmac } from '@noble/hashes/hmac.js';
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import { ahash } from '@noble/hashes/utils.js';
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import { abignumber, abool, abytes, aInRange, asafenumber, bitLen, bitMask, bytesToHex, bytesToNumberBE, concatBytes, createHmacDrbg, hexToBytes, isBytes, numberToHexUnpadded, validateObject, randomBytes as wcRandomBytes, } from "../utils.js";
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import { createCurveFields, createKeygen, mulEndoUnsafe, negateCt, normalizeZ, wNAF, } from "./curve.js";
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import { FpInvertBatch, FpIsSquare, getMinHashLength, mapHashToField, validateField, } from "./modular.js";
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// We construct the basis so `den` is always positive and equals `n`,
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// but the `num` sign depends on the basis, not on the secret value.
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// Exact half-way cases round away from zero, which keeps the split symmetric
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// around the reduced-basis boundaries used by endomorphism decomposition.
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const divNearest = (num, den) => (num + (num >= 0 ? den : -den) / _2n) / den;
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/** Splits scalar for GLV endomorphism. */
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export function _splitEndoScalar(k, basis, n) {
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// Split scalar into two such that part is ~half bits: `abs(part) < sqrt(N)`
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// Since part can be negative, we need to do this on point.
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// Callers must provide a reduced GLV basis whose vectors satisfy
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// `a + b * lambda ≡ 0 (mod n)`; this helper only sees the basis and `n`.
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// Reject unreduced scalars instead of silently treating them mod n.
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aInRange('scalar', k, _0n, n);
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// TODO: verifyScalar function which consumes lambda
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const [[a1, b1], [a2, b2]] = basis;
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const c1 = divNearest(b2 * k, n);
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const c2 = divNearest(-b1 * k, n);
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// |k1|/|k2| is < sqrt(N), but can be negative.
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// If we do `k1 mod N`, we'll get big scalar (`> sqrt(N)`): so, we do cheaper negation instead.
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let k1 = k - c1 * a1 - c2 * a2;
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let k2 = -c1 * b1 - c2 * b2;
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const k1neg = k1 < _0n;
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const k2neg = k2 < _0n;
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if (k1neg)
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k1 = -k1;
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if (k2neg)
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k2 = -k2;
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// Double check that resulting scalar less than half bits of N: otherwise wNAF will fail.
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// This should only happen on wrong bases.
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// Also, the math inside is complex enough that this guard is worth keeping.
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const MAX_NUM = bitMask(Math.ceil(bitLen(n) / 2)) + _1n; // Half bits of N
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if (k1 < _0n || k1 >= MAX_NUM || k2 < _0n || k2 >= MAX_NUM) {
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throw new Error('splitScalar (endomorphism): failed for k');
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}
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return { k1neg, k1, k2neg, k2 };
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}
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function validateSigFormat(format) {
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if (!['compact', 'recovered', 'der'].includes(format))
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throw new Error('Signature format must be "compact", "recovered", or "der"');
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return format;
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}
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function validateSigOpts(opts, def) {
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validateObject(opts);
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const optsn = {};
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// Normalize only the declared option subset from `def`; unknown keys are
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// intentionally ignored so shared / superset option bags stay valid here too.
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// `extraEntropy` stays an opaque payload until the signing path consumes it.
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for (let optName of Object.keys(def)) {
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// @ts-ignore
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optsn[optName] = opts[optName] === undefined ? def[optName] : opts[optName];
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}
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abool(optsn.lowS, 'lowS');
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abool(optsn.prehash, 'prehash');
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if (optsn.format !== undefined)
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validateSigFormat(optsn.format);
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return optsn;
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}
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/**
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* @param m - Error message.
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* @example
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* Throw a DER-specific error when signature parsing encounters invalid bytes.
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*
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* ```ts
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* new DERErr('bad der');
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* ```
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*/
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export class DERErr extends Error {
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constructor(m = '') {
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super(m);
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}
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}
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/**
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* ASN.1 DER encoding utilities. ASN is very complex & fragile. Format:
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*
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* [0x30 (SEQUENCE), bytelength, 0x02 (INTEGER), intLength, R, 0x02 (INTEGER), intLength, S]
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*
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* Docs: {@link https://letsencrypt.org/docs/a-warm-welcome-to-asn1-and-der/ | Let's Encrypt ASN.1 guide} and
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* {@link https://luca.ntop.org/Teaching/Appunti/asn1.html | Luca Deri's ASN.1 notes}.
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* @example
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* ASN.1 DER encoding utilities.
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*
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* ```ts
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* const der = DER.hexFromSig({ r: 1n, s: 2n });
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* ```
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*/
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export const DER = {
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// asn.1 DER encoding utils
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Err: DERErr,
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// Basic building block is TLV (Tag-Length-Value)
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_tlv: {
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encode: (tag, data) => {
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const { Err: E } = DER;
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asafenumber(tag, 'tag');
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if (tag < 0 || tag > 255)
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throw new E('tlv.encode: wrong tag');
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if (typeof data !== 'string')
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throw new TypeError('"data" expected string, got type=' + typeof data);
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// Internal helper: callers hand this already-validated hex payload, so we only enforce
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// byte alignment here instead of re-validating every nibble.
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if (data.length & 1)
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throw new E('tlv.encode: unpadded data');
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const dataLen = data.length / 2;
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const len = numberToHexUnpadded(dataLen);
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if ((len.length / 2) & 0b1000_0000)
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throw new E('tlv.encode: long form length too big');
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// length of length with long form flag
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const lenLen = dataLen > 127 ? numberToHexUnpadded((len.length / 2) | 0b1000_0000) : '';
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const t = numberToHexUnpadded(tag);
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return t + lenLen + len + data;
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},
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// v - value, l - left bytes (unparsed)
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decode(tag, data) {
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const { Err: E } = DER;
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data = abytes(data, undefined, 'DER data');
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let pos = 0;
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if (tag < 0 || tag > 255)
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throw new E('tlv.encode: wrong tag');
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if (data.length < 2 || data[pos++] !== tag)
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throw new E('tlv.decode: wrong tlv');
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const first = data[pos++];
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// First bit of first length byte is the short/long form flag.
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const isLong = !!(first & 0b1000_0000);
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let length = 0;
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if (!isLong)
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length = first;
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else {
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// Long form: [longFlag(1bit), lengthLength(7bit), length (BE)]
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const lenLen = first & 0b0111_1111;
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if (!lenLen)
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throw new E('tlv.decode(long): indefinite length not supported');
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// This would overflow u32 in JS.
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if (lenLen > 4)
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throw new E('tlv.decode(long): byte length is too big');
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const lengthBytes = data.subarray(pos, pos + lenLen);
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if (lengthBytes.length !== lenLen)
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throw new E('tlv.decode: length bytes not complete');
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if (lengthBytes[0] === 0)
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throw new E('tlv.decode(long): zero leftmost byte');
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for (const b of lengthBytes)
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length = (length << 8) | b;
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pos += lenLen;
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if (length < 128)
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throw new E('tlv.decode(long): not minimal encoding');
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}
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const v = data.subarray(pos, pos + length);
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if (v.length !== length)
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throw new E('tlv.decode: wrong value length');
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return { v, l: data.subarray(pos + length) };
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},
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},
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// https://crypto.stackexchange.com/a/57734 Leftmost bit of first byte is 'negative' flag,
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// since we always use positive integers here. It must always be empty:
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// - add zero byte if exists
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// - if next byte doesn't have a flag, leading zero is not allowed (minimal encoding)
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_int: {
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encode(num) {
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const { Err: E } = DER;
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abignumber(num);
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if (num < _0n)
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throw new E('integer: negative integers are not allowed');
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let hex = numberToHexUnpadded(num);
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// Pad with zero byte if negative flag is present
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if (Number.parseInt(hex[0], 16) & 0b1000)
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hex = '00' + hex;
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if (hex.length & 1)
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throw new E('unexpected DER parsing assertion: unpadded hex');
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return hex;
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},
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decode(data) {
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const { Err: E } = DER;
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if (data.length < 1)
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throw new E('invalid signature integer: empty');
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if (data[0] & 0b1000_0000)
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throw new E('invalid signature integer: negative');
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// Single-byte zero `00` is the canonical DER INTEGER encoding for zero.
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if (data.length > 1 && data[0] === 0x00 && !(data[1] & 0b1000_0000))
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throw new E('invalid signature integer: unnecessary leading zero');
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return bytesToNumberBE(data);
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},
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},
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toSig(bytes) {
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// parse DER signature
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const { Err: E, _int: int, _tlv: tlv } = DER;
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const data = abytes(bytes, undefined, 'signature');
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const { v: seqBytes, l: seqLeftBytes } = tlv.decode(0x30, data);
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if (seqLeftBytes.length)
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throw new E('invalid signature: left bytes after parsing');
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const { v: rBytes, l: rLeftBytes } = tlv.decode(0x02, seqBytes);
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const { v: sBytes, l: sLeftBytes } = tlv.decode(0x02, rLeftBytes);
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if (sLeftBytes.length)
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throw new E('invalid signature: left bytes after parsing');
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return { r: int.decode(rBytes), s: int.decode(sBytes) };
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},
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hexFromSig(sig) {
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const { _tlv: tlv, _int: int } = DER;
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const rs = tlv.encode(0x02, int.encode(sig.r));
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const ss = tlv.encode(0x02, int.encode(sig.s));
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const seq = rs + ss;
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return tlv.encode(0x30, seq);
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},
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};
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Object.freeze(DER._tlv);
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Object.freeze(DER._int);
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Object.freeze(DER);
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// Be friendly to bad ECMAScript parsers by not using bigint literals
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// prettier-ignore
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const _0n = /* @__PURE__ */ BigInt(0), _1n = /* @__PURE__ */ BigInt(1), _2n = /* @__PURE__ */ BigInt(2), _3n = /* @__PURE__ */ BigInt(3), _4n = /* @__PURE__ */ BigInt(4);
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/**
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* Creates weierstrass Point constructor, based on specified curve options.
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*
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* See {@link WeierstrassOpts}.
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* @param params - Curve parameters. See {@link WeierstrassOpts}.
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* @param extraOpts - Optional helpers and overrides. See {@link WeierstrassExtraOpts}.
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* @returns Weierstrass point constructor.
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* @throws If the curve parameters, overrides, or point codecs are invalid. {@link Error}
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*
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* @example
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* Construct a point type from explicit Weierstrass curve parameters.
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*
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* ```js
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* const opts = {
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* p: 0xfffffffffffffffffffffffffffffffeffffac73n,
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* n: 0x100000000000000000001b8fa16dfab9aca16b6b3n,
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* h: 1n,
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* a: 0n,
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* b: 7n,
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* Gx: 0x3b4c382ce37aa192a4019e763036f4f5dd4d7ebbn,
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* Gy: 0x938cf935318fdced6bc28286531733c3f03c4feen,
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* };
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* const secp160k1_Point = weierstrass(opts);
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* ```
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*/
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export function weierstrass(params, extraOpts = {}) {
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const validated = createCurveFields('weierstrass', params, extraOpts);
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const Fp = validated.Fp;
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const Fn = validated.Fn;
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let CURVE = validated.CURVE;
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const { h: cofactor, n: CURVE_ORDER } = CURVE;
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validateObject(extraOpts, {}, {
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allowInfinityPoint: 'boolean',
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clearCofactor: 'function',
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isTorsionFree: 'function',
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fromBytes: 'function',
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toBytes: 'function',
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endo: 'object',
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});
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// Snapshot constructor-time flags whose later mutation would otherwise change
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// validity semantics of an already-built point type.
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const { endo, allowInfinityPoint } = extraOpts;
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if (endo) {
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// validateObject(endo, { beta: 'bigint', splitScalar: 'function' });
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if (!Fp.is0(CURVE.a) || typeof endo.beta !== 'bigint' || !Array.isArray(endo.basises)) {
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throw new Error('invalid endo: expected "beta": bigint and "basises": array');
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}
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}
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const lengths = getWLengths(Fp, Fn);
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function assertCompressionIsSupported() {
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if (!Fp.isOdd)
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throw new Error('compression is not supported: Field does not have .isOdd()');
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}
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// Implements IEEE P1363 point encoding
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function pointToBytes(_c, point, isCompressed) {
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// SEC 1 v2.0 §2.3.3 encodes infinity as the single octet 0x00. Only curves
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// that opt into infinity as a public point value should expose that byte form.
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if (allowInfinityPoint && point.is0())
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return Uint8Array.of(0);
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const { x, y } = point.toAffine();
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const bx = Fp.toBytes(x);
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abool(isCompressed, 'isCompressed');
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if (isCompressed) {
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assertCompressionIsSupported();
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const hasEvenY = !Fp.isOdd(y);
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return concatBytes(pprefix(hasEvenY), bx);
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}
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else {
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return concatBytes(Uint8Array.of(0x04), bx, Fp.toBytes(y));
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}
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}
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function pointFromBytes(bytes) {
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abytes(bytes, undefined, 'Point');
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const { publicKey: comp, publicKeyUncompressed: uncomp } = lengths; // e.g. for 32-byte: 33, 65
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const length = bytes.length;
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const head = bytes[0];
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const tail = bytes.subarray(1);
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if (allowInfinityPoint && length === 1 && head === 0x00)
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return { x: Fp.ZERO, y: Fp.ZERO };
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// SEC 1 v2.0 §2.3.4 decodes 0x00 as infinity, but §3.2.2 public-key validation
|
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// rejects infinity. We therefore keep 0x00 rejected by default because callers
|
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// reuse this parser as the strict public-key boundary, and only admit it when
|
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// the curve explicitly opts into infinity as a public point value. secp256k1
|
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// crosstests show OpenSSL raw point codecs accept 0x00 too.
|
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// No actual validation is done here: use .assertValidity()
|
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if (length === comp && (head === 0x02 || head === 0x03)) {
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const x = Fp.fromBytes(tail);
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if (!Fp.isValid(x))
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throw new Error('bad point: is not on curve, wrong x');
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const y2 = weierstrassEquation(x); // y² = x³ + ax + b
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let y;
|
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try {
|
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y = Fp.sqrt(y2); // y = y² ^ (p+1)/4
|
||
}
|
||
catch (sqrtError) {
|
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const err = sqrtError instanceof Error ? ': ' + sqrtError.message : '';
|
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throw new Error('bad point: is not on curve, sqrt error' + err);
|
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}
|
||
assertCompressionIsSupported();
|
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const evenY = Fp.isOdd(y);
|
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const evenH = (head & 1) === 1; // ECDSA-specific
|
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if (evenH !== evenY)
|
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y = Fp.neg(y);
|
||
return { x, y };
|
||
}
|
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else if (length === uncomp && head === 0x04) {
|
||
// TODO: more checks
|
||
const L = Fp.BYTES;
|
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const x = Fp.fromBytes(tail.subarray(0, L));
|
||
const y = Fp.fromBytes(tail.subarray(L, L * 2));
|
||
if (!isValidXY(x, y))
|
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throw new Error('bad point: is not on curve');
|
||
return { x, y };
|
||
}
|
||
else {
|
||
throw new Error(`bad point: got length ${length}, expected compressed=${comp} or uncompressed=${uncomp}`);
|
||
}
|
||
}
|
||
const encodePoint = extraOpts.toBytes === undefined ? pointToBytes : extraOpts.toBytes;
|
||
const decodePoint = extraOpts.fromBytes === undefined ? pointFromBytes : extraOpts.fromBytes;
|
||
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, CURVE.a)), CURVE.b); // x³ + a * x + 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);
|
||
}
|
||
// Keep constructor-time generator validation cheap: callers are responsible for supplying the
|
||
// correct prime-order base point, while eager subgroup checks here would slow heavy module imports.
|
||
// 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('Weierstrass Point expected');
|
||
}
|
||
function splitEndoScalarN(k) {
|
||
if (!endo || !endo.basises)
|
||
throw new Error('no endo');
|
||
return _splitEndoScalar(k, endo.basises, Fn.ORDER);
|
||
}
|
||
function finishEndo(endoBeta, k1p, k2p, k1neg, k2neg) {
|
||
k2p = new Point(Fp.mul(k2p.X, endoBeta), k2p.Y, k2p.Z);
|
||
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 {
|
||
// base / generator point
|
||
static BASE = new Point(CURVE.Gx, CURVE.Gy, Fp.ONE);
|
||
// zero / infinity / identity point
|
||
static ZERO = new Point(Fp.ZERO, Fp.ONE, Fp.ZERO); // 0, 1, 0
|
||
// math field
|
||
static Fp = Fp;
|
||
// scalar field
|
||
static Fn = Fn;
|
||
X;
|
||
Y;
|
||
Z;
|
||
/** Does NOT validate if the point is valid. Use `.assertValidity()`. */
|
||
constructor(X, Y, Z) {
|
||
this.X = acoord('x', X);
|
||
// This is not just about ZERO / infinity: ambient curves can have real
|
||
// finite points with y=0. Those points are 2-torsion, so they cannot lie
|
||
// in the odd prime-order subgroups this point type is meant to represent.
|
||
this.Y = acoord('y', Y, true);
|
||
this.Z = acoord('z', Z);
|
||
Object.freeze(this);
|
||
}
|
||
static CURVE() {
|
||
return CURVE;
|
||
}
|
||
/** 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);
|
||
}
|
||
static fromBytes(bytes) {
|
||
const P = Point.fromAffine(decodePoint(abytes(bytes, undefined, 'point')));
|
||
P.assertValidity();
|
||
return P;
|
||
}
|
||
static fromHex(hex) {
|
||
return Point.fromBytes(hexToBytes(hex));
|
||
}
|
||
get x() {
|
||
return this.toAffine().x;
|
||
}
|
||
get y() {
|
||
return this.toAffine().y;
|
||
}
|
||
/**
|
||
*
|
||
* @param windowSize
|
||
* @param isLazy - true will defer table computation until the first multiplication
|
||
* @returns
|
||
*/
|
||
precompute(windowSize = 8, isLazy = true) {
|
||
wnaf.createCache(this, windowSize);
|
||
if (!isLazy)
|
||
this.multiply(_3n); // random number
|
||
return this;
|
||
}
|
||
// TODO: return `this`
|
||
/** A point on curve is valid if it conforms to equation. */
|
||
assertValidity() {
|
||
const p = this;
|
||
if (p.is0()) {
|
||
// (0, 1, 0) aka ZERO is invalid in most contexts.
|
||
// In BLS, ZERO can be serialized, so we allow it.
|
||
// Keep the accepted infinity encoding canonical: projective-equivalent (X, Y, 0) points
|
||
// like (1, 1, 0) compare equal to ZERO, but only (0, 1, 0) should pass this guard.
|
||
if (extraOpts.allowInfinityPoint && Fp.is0(p.X) && Fp.eql(p.Y, Fp.ONE) && Fp.is0(p.Z))
|
||
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');
|
||
}
|
||
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 { X: X1, Y: Y1, Z: Z1 } = this;
|
||
const { X: X2, Y: Y2, Z: 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.X, Fp.neg(this.Y), this.Z);
|
||
}
|
||
// 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 { X: X1, Y: Y1, Z: 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 { X: X1, Y: Y1, Z: Z1 } = this;
|
||
const { X: X2, Y: Y2, Z: 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) {
|
||
// Validate before calling `negate()` so wrong inputs fail with the point guard
|
||
// instead of leaking a foreign `negate()` error.
|
||
aprjpoint(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 } = extraOpts;
|
||
// Keep the subgroup-scalar contract strict instead of reducing 0 / n to ZERO.
|
||
// In key/signature-style callers, those values usually mean broken hash/scalar plumbing,
|
||
// and failing closed is safer than silently producing the identity point.
|
||
if (!Fn.isValidNot0(scalar))
|
||
throw new RangeError('invalid scalar: out of range'); // 0 is invalid
|
||
let point, fake; // Fake point is used to const-time mult
|
||
const mul = (n) => wnaf.cached(this, n, (p) => normalizeZ(Point, p));
|
||
/** See docs for {@link EndomorphismOpts} */
|
||
if (endo) {
|
||
const { k1neg, k1, k2neg, k2 } = splitEndoScalarN(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 normalizeZ(Point, [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 secret key e.g. sig verification, which works over *public* keys.
|
||
*/
|
||
multiplyUnsafe(scalar) {
|
||
const { endo } = extraOpts;
|
||
const p = this;
|
||
const sc = scalar;
|
||
// Public-scalar callers may need 0, but n and larger values stay rejected here too.
|
||
// Reducing them mod n would turn bad caller input into an accidental identity point.
|
||
if (!Fn.isValid(sc))
|
||
throw new RangeError('invalid scalar: out of range'); // 0 is valid
|
||
if (sc === _0n || p.is0())
|
||
return Point.ZERO; // 0
|
||
if (sc === _1n)
|
||
return p; // 1
|
||
if (wnaf.hasCache(this))
|
||
return this.multiply(sc); // precomputes
|
||
// We don't have method for double scalar multiplication (aP + bQ):
|
||
// Even with using Strauss-Shamir trick, it's 35% slower than naïve mul+add.
|
||
if (endo) {
|
||
const { k1neg, k1, k2neg, k2 } = splitEndoScalarN(sc);
|
||
const { p1, p2 } = mulEndoUnsafe(Point, p, k1, k2); // 30% faster vs wnaf.unsafe
|
||
return finishEndo(endo.beta, p1, p2, k1neg, k2neg);
|
||
}
|
||
else {
|
||
return wnaf.unsafe(p, sc);
|
||
}
|
||
}
|
||
/**
|
||
* Converts Projective point to affine (x, y) coordinates.
|
||
* (X, Y, Z) ∋ (x=X/Z, y=Y/Z).
|
||
* @param invertedZ - Z^-1 (inverted zero) - optional, precomputation is useful for invertBatch
|
||
*/
|
||
toAffine(invertedZ) {
|
||
const p = this;
|
||
let iz = invertedZ;
|
||
const { X, Y, Z } = p;
|
||
// Fast-path for normalized points
|
||
if (Fp.eql(Z, Fp.ONE))
|
||
return { x: X, y: 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 x = Fp.mul(X, iz);
|
||
const y = 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, y };
|
||
}
|
||
/**
|
||
* Checks whether Point is free of torsion elements (is in prime subgroup).
|
||
* Always torsion-free for cofactor=1 curves.
|
||
*/
|
||
isTorsionFree() {
|
||
const { isTorsionFree } = extraOpts;
|
||
if (cofactor === _1n)
|
||
return true;
|
||
if (isTorsionFree)
|
||
return isTorsionFree(Point, this);
|
||
return wnaf.unsafe(this, CURVE_ORDER).is0();
|
||
}
|
||
clearCofactor() {
|
||
const { clearCofactor } = extraOpts;
|
||
if (cofactor === _1n)
|
||
return this; // Fast-path
|
||
if (clearCofactor)
|
||
return clearCofactor(Point, this);
|
||
// Default fallback assumes the cofactor fits the usual subgroup-scalar
|
||
// multiplyUnsafe() contract. Curves with larger / structured cofactors
|
||
// should define a clearCofactor override anyway (e.g. psi/Frobenius maps).
|
||
return this.multiplyUnsafe(cofactor);
|
||
}
|
||
isSmallOrder() {
|
||
if (cofactor === _1n)
|
||
return this.is0(); // Fast-path
|
||
return this.clearCofactor().is0();
|
||
}
|
||
toBytes(isCompressed = true) {
|
||
abool(isCompressed, 'isCompressed');
|
||
// Same policy as pointFromBytes(): keep ZERO out of the default byte surface because
|
||
// callers use these encodings as public keys, where SEC 1 validation rejects infinity.
|
||
this.assertValidity();
|
||
return encodePoint(Point, this, isCompressed);
|
||
}
|
||
toHex(isCompressed = true) {
|
||
return bytesToHex(this.toBytes(isCompressed));
|
||
}
|
||
toString() {
|
||
return `<Point ${this.is0() ? 'ZERO' : this.toHex()}>`;
|
||
}
|
||
}
|
||
const bits = Fn.BITS;
|
||
const wnaf = new wNAF(Point, extraOpts.endo ? Math.ceil(bits / 2) : bits);
|
||
// Tiny toy curves can have scalar fields narrower than 8 bits. Skip the
|
||
// eager W=8 cache there instead of rejecting an otherwise valid constructor.
|
||
if (bits >= 8)
|
||
Point.BASE.precompute(8); // Enable precomputes. Slows down first publicKey computation by 20ms.
|
||
Object.freeze(Point.prototype);
|
||
Object.freeze(Point);
|
||
return Point;
|
||
}
|
||
// Points start with byte 0x02 when y is even; otherwise 0x03
|
||
function pprefix(hasEvenY) {
|
||
return Uint8Array.of(hasEvenY ? 0x02 : 0x03);
|
||
}
|
||
/**
|
||
* 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.
|
||
* RFC 9380 expects callers to provide `v != 0`; this helper does not enforce it.
|
||
* @param Fp - Field implementation.
|
||
* @param Z - Simplified SWU map parameter.
|
||
* @returns Square-root ratio helper.
|
||
* @example
|
||
* Build the square-root ratio helper used by SWU map implementations.
|
||
*
|
||
* ```ts
|
||
* import { SWUFpSqrtRatio } from '@noble/curves/abstract/weierstrass.js';
|
||
* import { Field } from '@noble/curves/abstract/modular.js';
|
||
* const Fp = Field(17n);
|
||
* const sqrtRatio = SWUFpSqrtRatio(Fp, 3n);
|
||
* const out = sqrtRatio(4n, 1n);
|
||
* ```
|
||
*/
|
||
export function SWUFpSqrtRatio(Fp, Z) {
|
||
// Fail with the usual field-shape error before touching pow/cmov on malformed field shims.
|
||
const F = validateField(Fp);
|
||
// Generic implementation
|
||
const q = F.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 = F.pow(Z, c2); // 6. c6 = Z^c2
|
||
const c7 = F.pow(Z, (c2 + _1n) / _2n); // 7. c7 = Z^((c2 + 1) / 2)
|
||
// RFC 9380 Appendix F.2.1.1 defines sqrt_ratio(u, v) only for v != 0.
|
||
// We keep v=0 on the regular result path with isValid=false instead of
|
||
// throwing so the helper stays closer to the RFC's fixed control flow.
|
||
let sqrtRatio = (u, v) => {
|
||
let tv1 = c6; // 1. tv1 = c6
|
||
let tv2 = F.pow(v, c4); // 2. tv2 = v^c4
|
||
let tv3 = F.sqr(tv2); // 3. tv3 = tv2^2
|
||
tv3 = F.mul(tv3, v); // 4. tv3 = tv3 * v
|
||
let tv5 = F.mul(u, tv3); // 5. tv5 = u * tv3
|
||
tv5 = F.pow(tv5, c3); // 6. tv5 = tv5^c3
|
||
tv5 = F.mul(tv5, tv2); // 7. tv5 = tv5 * tv2
|
||
tv2 = F.mul(tv5, v); // 8. tv2 = tv5 * v
|
||
tv3 = F.mul(tv5, u); // 9. tv3 = tv5 * u
|
||
let tv4 = F.mul(tv3, tv2); // 10. tv4 = tv3 * tv2
|
||
tv5 = F.pow(tv4, c5); // 11. tv5 = tv4^c5
|
||
let isQR = F.eql(tv5, F.ONE); // 12. isQR = tv5 == 1
|
||
tv2 = F.mul(tv3, c7); // 13. tv2 = tv3 * c7
|
||
tv5 = F.mul(tv4, tv1); // 14. tv5 = tv4 * tv1
|
||
tv3 = F.cmov(tv2, tv3, isQR); // 15. tv3 = CMOV(tv2, tv3, isQR)
|
||
tv4 = F.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 = F.pow(tv4, tv5); // 20. tv5 = tv4^tv5
|
||
const e1 = F.eql(tvv5, F.ONE); // 21. e1 = tv5 == 1
|
||
tv2 = F.mul(tv3, tv1); // 22. tv2 = tv3 * tv1
|
||
tv1 = F.mul(tv1, tv1); // 23. tv1 = tv1 * tv1
|
||
tvv5 = F.mul(tv4, tv1); // 24. tv5 = tv4 * tv1
|
||
tv3 = F.cmov(tv2, tv3, e1); // 25. tv3 = CMOV(tv2, tv3, e1)
|
||
tv4 = F.cmov(tvv5, tv4, e1); // 26. tv4 = CMOV(tv5, tv4, e1)
|
||
}
|
||
// RFC 9380 Appendix F.2.1.1 defines sqrt_ratio(u, v) for v != 0.
|
||
// When u = 0 and v != 0, u / v = 0 is square and the computed root is
|
||
// still 0, so widen only the final flag and keep the full control flow.
|
||
return { isValid: !F.is0(v) && (isQR || F.is0(u)), value: tv3 };
|
||
};
|
||
if (F.ORDER % _4n === _3n) {
|
||
// sqrt_ratio_3mod4(u, v)
|
||
const c1 = (F.ORDER - _3n) / _4n; // 1. c1 = (q - 3) / 4 # Integer arithmetic
|
||
const c2 = F.sqrt(F.neg(Z)); // 2. c2 = sqrt(-Z)
|
||
sqrtRatio = (u, v) => {
|
||
let tv1 = F.sqr(v); // 1. tv1 = v^2
|
||
const tv2 = F.mul(u, v); // 2. tv2 = u * v
|
||
tv1 = F.mul(tv1, tv2); // 3. tv1 = tv1 * tv2
|
||
let y1 = F.pow(tv1, c1); // 4. y1 = tv1^c1
|
||
y1 = F.mul(y1, tv2); // 5. y1 = y1 * tv2
|
||
const y2 = F.mul(y1, c2); // 6. y2 = y1 * c2
|
||
const tv3 = F.mul(F.sqr(y1), v); // 7. tv3 = y1^2; 8. tv3 = tv3 * v
|
||
const isQR = F.eql(tv3, u); // 9. isQR = tv3 == u
|
||
let y = F.cmov(y2, y1, isQR); // 10. y = CMOV(y2, y1, isQR)
|
||
return { isValid: !F.is0(v) && 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
|
||
* See {@link https://www.rfc-editor.org/rfc/rfc9380#section-6.6.2 | RFC 9380 section 6.6.2}.
|
||
* @param Fp - Field implementation.
|
||
* @param opts - SWU parameters:
|
||
* - `A`: Curve parameter `A`.
|
||
* - `B`: Curve parameter `B`.
|
||
* - `Z`: Simplified SWU map parameter.
|
||
* @returns Deterministic map-to-curve function.
|
||
* @throws If the SWU parameters are invalid or the field lacks the required helpers. {@link Error}
|
||
* @example
|
||
* Map one field element to a Weierstrass curve point with the SWU recipe.
|
||
*
|
||
* ```ts
|
||
* import { mapToCurveSimpleSWU } from '@noble/curves/abstract/weierstrass.js';
|
||
* import { Field } from '@noble/curves/abstract/modular.js';
|
||
* const Fp = Field(17n);
|
||
* const map = mapToCurveSimpleSWU(Fp, { A: 1n, B: 2n, Z: 3n });
|
||
* const point = map(5n);
|
||
* ```
|
||
*/
|
||
export function mapToCurveSimpleSWU(Fp, opts) {
|
||
const F = validateField(Fp);
|
||
const { A, B, Z } = opts;
|
||
if (!F.isValidNot0(A) || !F.isValidNot0(B) || !F.isValid(Z))
|
||
throw new Error('mapToCurveSimpleSWU: invalid opts');
|
||
// RFC 9380 §6.6.2 and Appendix H.2 require:
|
||
// 1. Z is non-square in F
|
||
// 2. Z != -1 in F
|
||
// 3. g(x) - Z is irreducible over F
|
||
// 4. g(B / (Z * A)) is square in F
|
||
// We can enforce 1, 2, and 4 with the current field API.
|
||
// Criterion 3 is not checked here because generic `IField<T>` does not expose
|
||
// polynomial-ring / irreducibility operations, and this helper is used for
|
||
// both prime and extension fields.
|
||
if (F.eql(Z, F.neg(F.ONE)) || FpIsSquare(F, Z))
|
||
throw new Error('mapToCurveSimpleSWU: invalid opts');
|
||
// RFC 9380 Appendix H.2 criterion 4: g(B / (Z * A)) is square in F.
|
||
// x = B / (Z * A)
|
||
const x = F.mul(B, F.inv(F.mul(Z, A)));
|
||
// g(x) = x^3 + A*x + B
|
||
const gx = F.add(F.add(F.mul(F.sqr(x), x), F.mul(A, x)), B);
|
||
if (!FpIsSquare(F, gx))
|
||
throw new Error('mapToCurveSimpleSWU: invalid opts');
|
||
const sqrtRatio = SWUFpSqrtRatio(F, Z);
|
||
if (!F.isOdd)
|
||
throw new Error('Field does not have .isOdd()');
|
||
// Input: u, an element of F.
|
||
// Output: (x, y), a point on E.
|
||
return (u) => {
|
||
// prettier-ignore
|
||
let tv1, tv2, tv3, tv4, tv5, tv6, x, y;
|
||
tv1 = F.sqr(u); // 1. tv1 = u^2
|
||
tv1 = F.mul(tv1, Z); // 2. tv1 = Z * tv1
|
||
tv2 = F.sqr(tv1); // 3. tv2 = tv1^2
|
||
tv2 = F.add(tv2, tv1); // 4. tv2 = tv2 + tv1
|
||
tv3 = F.add(tv2, F.ONE); // 5. tv3 = tv2 + 1
|
||
tv3 = F.mul(tv3, B); // 6. tv3 = B * tv3
|
||
tv4 = F.cmov(Z, F.neg(tv2), !F.eql(tv2, F.ZERO)); // 7. tv4 = CMOV(Z, -tv2, tv2 != 0)
|
||
tv4 = F.mul(tv4, A); // 8. tv4 = A * tv4
|
||
tv2 = F.sqr(tv3); // 9. tv2 = tv3^2
|
||
tv6 = F.sqr(tv4); // 10. tv6 = tv4^2
|
||
tv5 = F.mul(tv6, A); // 11. tv5 = A * tv6
|
||
tv2 = F.add(tv2, tv5); // 12. tv2 = tv2 + tv5
|
||
tv2 = F.mul(tv2, tv3); // 13. tv2 = tv2 * tv3
|
||
tv6 = F.mul(tv6, tv4); // 14. tv6 = tv6 * tv4
|
||
tv5 = F.mul(tv6, B); // 15. tv5 = B * tv6
|
||
tv2 = F.add(tv2, tv5); // 16. tv2 = tv2 + tv5
|
||
x = F.mul(tv1, tv3); // 17. x = tv1 * tv3
|
||
const { isValid, value } = sqrtRatio(tv2, tv6); // 18. (is_gx1_square, y1) = sqrt_ratio(tv2, tv6)
|
||
y = F.mul(tv1, u); // 19. y = tv1 * u -> Z * u^3 * y1
|
||
y = F.mul(y, value); // 20. y = y * y1
|
||
x = F.cmov(x, tv3, isValid); // 21. x = CMOV(x, tv3, is_gx1_square)
|
||
y = F.cmov(y, value, isValid); // 22. y = CMOV(y, y1, is_gx1_square)
|
||
const e1 = F.isOdd(u) === F.isOdd(y); // 23. e1 = sgn0(u) == sgn0(y)
|
||
y = F.cmov(F.neg(y), y, e1); // 24. y = CMOV(-y, y, e1)
|
||
const tv4_inv = FpInvertBatch(F, [tv4], true)[0];
|
||
x = F.mul(x, tv4_inv); // 25. x = x / tv4
|
||
return { x, y };
|
||
};
|
||
}
|
||
function getWLengths(Fp, Fn) {
|
||
return {
|
||
secretKey: Fn.BYTES,
|
||
publicKey: 1 + Fp.BYTES,
|
||
publicKeyUncompressed: 1 + 2 * Fp.BYTES,
|
||
publicKeyHasPrefix: true,
|
||
// Raw compact `(r || s)` signature width; DER and recovered signatures use
|
||
// different lengths outside this helper.
|
||
signature: 2 * Fn.BYTES,
|
||
};
|
||
}
|
||
/**
|
||
* Sometimes users only need getPublicKey, getSharedSecret, and secret key handling.
|
||
* This helper ensures no signature functionality is present. Less code, smaller bundle size.
|
||
* @param Point - Weierstrass point constructor.
|
||
* @param ecdhOpts - Optional randomness helpers:
|
||
* - `randomBytes` (optional): Optional RNG override.
|
||
* @returns ECDH helper namespace.
|
||
* @example
|
||
* Sometimes users only need getPublicKey, getSharedSecret, and secret key handling.
|
||
*
|
||
* ```ts
|
||
* import { ecdh } from '@noble/curves/abstract/weierstrass.js';
|
||
* import { p256 } from '@noble/curves/nist.js';
|
||
* const dh = ecdh(p256.Point);
|
||
* const alice = dh.keygen();
|
||
* const shared = dh.getSharedSecret(alice.secretKey, alice.publicKey);
|
||
* ```
|
||
*/
|
||
export function ecdh(Point, ecdhOpts = {}) {
|
||
const { Fn } = Point;
|
||
const randomBytes_ = ecdhOpts.randomBytes === undefined ? wcRandomBytes : ecdhOpts.randomBytes;
|
||
// Keep the advertised seed length aligned with mapHashToField(), which keeps a hard 16-byte
|
||
// minimum even on toy curves.
|
||
const lengths = Object.assign(getWLengths(Point.Fp, Fn), {
|
||
seed: Math.max(getMinHashLength(Fn.ORDER), 16),
|
||
});
|
||
function isValidSecretKey(secretKey) {
|
||
try {
|
||
const num = Fn.fromBytes(secretKey);
|
||
return Fn.isValidNot0(num);
|
||
}
|
||
catch (error) {
|
||
return false;
|
||
}
|
||
}
|
||
function isValidPublicKey(publicKey, isCompressed) {
|
||
const { publicKey: comp, publicKeyUncompressed } = lengths;
|
||
try {
|
||
const l = publicKey.length;
|
||
if (isCompressed === true && l !== comp)
|
||
return false;
|
||
if (isCompressed === false && l !== publicKeyUncompressed)
|
||
return false;
|
||
return !!Point.fromBytes(publicKey);
|
||
}
|
||
catch (error) {
|
||
return false;
|
||
}
|
||
}
|
||
/**
|
||
* Produces cryptographically secure secret key from random of size
|
||
* (groupLen + ceil(groupLen / 2)) with modulo bias being negligible.
|
||
*/
|
||
function randomSecretKey(seed) {
|
||
seed = seed === undefined ? randomBytes_(lengths.seed) : seed;
|
||
return mapHashToField(abytes(seed, lengths.seed, 'seed'), Fn.ORDER);
|
||
}
|
||
/**
|
||
* Computes public key for a secret key. Checks for validity of the secret key.
|
||
* @param isCompressed - whether to return compact (default), or full key
|
||
* @returns Public key, full when isCompressed=false; short when isCompressed=true
|
||
*/
|
||
function getPublicKey(secretKey, isCompressed = true) {
|
||
return Point.BASE.multiply(Fn.fromBytes(secretKey)).toBytes(isCompressed);
|
||
}
|
||
/**
|
||
* Quick and dirty check for item being public key. Does not validate hex, or being on-curve.
|
||
*/
|
||
function isProbPub(item) {
|
||
const { secretKey, publicKey, publicKeyUncompressed } = lengths;
|
||
const allowedLengths = Fn._lengths;
|
||
if (!isBytes(item))
|
||
return undefined;
|
||
const l = abytes(item, undefined, 'key').length;
|
||
const isPub = l === publicKey || l === publicKeyUncompressed;
|
||
const isSec = l === secretKey || !!allowedLengths?.includes(l);
|
||
// P-521 accepts both 65- and 66-byte secret keys, so overlapping lengths stay ambiguous.
|
||
if (isPub && isSec)
|
||
return undefined;
|
||
return isPub;
|
||
}
|
||
/**
|
||
* ECDH (Elliptic Curve Diffie Hellman).
|
||
* Computes encoded shared point from secret key A and public key B.
|
||
* Checks: 1) secret key validity 2) shared key is on-curve.
|
||
* Does NOT hash the result or expose the SEC 1 x-coordinate-only `z`.
|
||
* Returns the encoded shared point on purpose: callers that need `x_P`
|
||
* can derive it from the encoded point, but `x_P` alone cannot recover the
|
||
* point/parity back.
|
||
* This helper only exposes the fully validated public-key path, not cofactor DH.
|
||
* @param isCompressed - whether to return compact (default), or full key
|
||
* @returns shared point encoding
|
||
*/
|
||
function getSharedSecret(secretKeyA, publicKeyB, isCompressed = true) {
|
||
if (isProbPub(secretKeyA) === true)
|
||
throw new Error('first arg must be private key');
|
||
if (isProbPub(publicKeyB) === false)
|
||
throw new Error('second arg must be public key');
|
||
const s = Fn.fromBytes(secretKeyA);
|
||
const b = Point.fromBytes(publicKeyB); // checks for being on-curve
|
||
return b.multiply(s).toBytes(isCompressed);
|
||
}
|
||
const utils = {
|
||
isValidSecretKey,
|
||
isValidPublicKey,
|
||
randomSecretKey,
|
||
};
|
||
const keygen = createKeygen(randomSecretKey, getPublicKey);
|
||
Object.freeze(utils);
|
||
Object.freeze(lengths);
|
||
return Object.freeze({ getPublicKey, getSharedSecret, keygen, Point, utils, lengths });
|
||
}
|
||
/**
|
||
* Creates ECDSA signing interface for given elliptic curve `Point` and `hash` function.
|
||
*
|
||
* @param Point - created using {@link weierstrass} function
|
||
* @param hash - used for 1) message prehash-ing 2) k generation in `sign`, using hmac_drbg(hash)
|
||
* @param ecdsaOpts - rarely needed, see {@link ECDSAOpts}:
|
||
* - `lowS`: Default low-S policy.
|
||
* - `hmac`: HMAC implementation used by RFC6979 DRBG.
|
||
* - `randomBytes`: Optional RNG override.
|
||
* - `bits2int`: Optional hash-to-int conversion override.
|
||
* - `bits2int_modN`: Optional hash-to-int-mod-n conversion override.
|
||
*
|
||
* @returns ECDSA helper namespace.
|
||
* @example
|
||
* Create an ECDSA signer/verifier bundle for one curve implementation.
|
||
*
|
||
* ```ts
|
||
* import { ecdsa } from '@noble/curves/abstract/weierstrass.js';
|
||
* import { p256 } from '@noble/curves/nist.js';
|
||
* import { sha256 } from '@noble/hashes/sha2.js';
|
||
* const p256ecdsa = ecdsa(p256.Point, sha256);
|
||
* const { secretKey, publicKey } = p256ecdsa.keygen();
|
||
* const msg = new TextEncoder().encode('hello noble');
|
||
* const sig = p256ecdsa.sign(msg, secretKey);
|
||
* const isValid = p256ecdsa.verify(sig, msg, publicKey);
|
||
* ```
|
||
*/
|
||
export function ecdsa(Point, hash, ecdsaOpts = {}) {
|
||
// Custom hash / bits2int hooks are treated as pure functions over validated caller-owned bytes.
|
||
const hash_ = hash;
|
||
ahash(hash_);
|
||
validateObject(ecdsaOpts, {}, {
|
||
hmac: 'function',
|
||
lowS: 'boolean',
|
||
randomBytes: 'function',
|
||
bits2int: 'function',
|
||
bits2int_modN: 'function',
|
||
});
|
||
ecdsaOpts = Object.assign({}, ecdsaOpts);
|
||
const randomBytes = ecdsaOpts.randomBytes === undefined ? wcRandomBytes : ecdsaOpts.randomBytes;
|
||
const hmac = ecdsaOpts.hmac === undefined
|
||
? (key, msg) => nobleHmac(hash_, key, msg)
|
||
: ecdsaOpts.hmac;
|
||
const { Fp, Fn } = Point;
|
||
const { ORDER: CURVE_ORDER, BITS: fnBits } = Fn;
|
||
const { keygen, getPublicKey, getSharedSecret, utils, lengths } = ecdh(Point, ecdsaOpts);
|
||
const defaultSigOpts = {
|
||
prehash: true,
|
||
lowS: typeof ecdsaOpts.lowS === 'boolean' ? ecdsaOpts.lowS : true,
|
||
format: 'compact',
|
||
extraEntropy: false,
|
||
};
|
||
// SEC 1 4.1.6 public-key recovery tries x = r + jn for j = 0..h. Our recovered-signature
|
||
// format only stores one overflow bit, so it can only distinguish q.x = r from q.x = r + n.
|
||
// A third lift would have the form q.x = r + 2n. Since valid ECDSA r is in 1..n-1, the
|
||
// smallest such lift is 1 + 2n, not 2n.
|
||
const hasLargeRecoveryLifts = CURVE_ORDER * _2n + _1n < Fp.ORDER;
|
||
function isBiggerThanHalfOrder(number) {
|
||
const HALF = CURVE_ORDER >> _1n;
|
||
return number > HALF;
|
||
}
|
||
function validateRS(title, num) {
|
||
if (!Fn.isValidNot0(num))
|
||
throw new Error(`invalid signature ${title}: out of range 1..Point.Fn.ORDER`);
|
||
return num;
|
||
}
|
||
function assertRecoverableCurve() {
|
||
// ECDSA recovery only supports curves where the current recovery id can distinguish
|
||
// q.x = r and q.x = r + n; larger lifts may need additional `r + n*i` branches.
|
||
// SEC 1 4.1.6 recovers candidates via x = r + jn, but this format only encodes j = 0 or 1.
|
||
// The next possible candidate is q.x = r + 2n, and its smallest valid value is 1 + 2n.
|
||
// To easily get i, we either need to:
|
||
// a. increase amount of valid recid values (4, 5...); OR
|
||
// b. prohibit recovered signatures for those curves.
|
||
if (hasLargeRecoveryLifts)
|
||
throw new Error('"recovered" sig type is not supported for cofactor >2 curves');
|
||
}
|
||
function validateSigLength(bytes, format) {
|
||
validateSigFormat(format);
|
||
const size = lengths.signature;
|
||
const sizer = format === 'compact' ? size : format === 'recovered' ? size + 1 : undefined;
|
||
return abytes(bytes, sizer);
|
||
}
|
||
/**
|
||
* ECDSA signature with its (r, s) properties. Supports compact, recovered & DER representations.
|
||
*/
|
||
class Signature {
|
||
r;
|
||
s;
|
||
recovery;
|
||
constructor(r, s, recovery) {
|
||
this.r = validateRS('r', r); // r in [1..N-1];
|
||
this.s = validateRS('s', s); // s in [1..N-1];
|
||
if (recovery != null) {
|
||
assertRecoverableCurve();
|
||
if (![0, 1, 2, 3].includes(recovery))
|
||
throw new Error('invalid recovery id');
|
||
this.recovery = recovery;
|
||
}
|
||
Object.freeze(this);
|
||
}
|
||
static fromBytes(bytes, format = defaultSigOpts.format) {
|
||
validateSigLength(bytes, format);
|
||
let recid;
|
||
if (format === 'der') {
|
||
const { r, s } = DER.toSig(abytes(bytes));
|
||
return new Signature(r, s);
|
||
}
|
||
if (format === 'recovered') {
|
||
recid = bytes[0];
|
||
format = 'compact';
|
||
bytes = bytes.subarray(1);
|
||
}
|
||
const L = lengths.signature / 2;
|
||
const r = bytes.subarray(0, L);
|
||
const s = bytes.subarray(L, L * 2);
|
||
return new Signature(Fn.fromBytes(r), Fn.fromBytes(s), recid);
|
||
}
|
||
static fromHex(hex, format) {
|
||
return this.fromBytes(hexToBytes(hex), format);
|
||
}
|
||
assertRecovery() {
|
||
const { recovery } = this;
|
||
if (recovery == null)
|
||
throw new Error('invalid recovery id: must be present');
|
||
return recovery;
|
||
}
|
||
addRecoveryBit(recovery) {
|
||
return new Signature(this.r, this.s, recovery);
|
||
}
|
||
// Unlike the top-level helper below, this method expects a digest that has
|
||
// already been hashed to the curve's message representative.
|
||
recoverPublicKey(messageHash) {
|
||
const { r, s } = this;
|
||
const recovery = this.assertRecovery();
|
||
const radj = recovery === 2 || recovery === 3 ? r + CURVE_ORDER : r;
|
||
if (!Fp.isValid(radj))
|
||
throw new Error('invalid recovery id: sig.r+curve.n != R.x');
|
||
const x = Fp.toBytes(radj);
|
||
const R = Point.fromBytes(concatBytes(pprefix((recovery & 1) === 0), x));
|
||
const ir = Fn.inv(radj); // r^-1
|
||
const h = bits2int_modN(abytes(messageHash, undefined, '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('invalid recovery: point at infinify');
|
||
Q.assertValidity();
|
||
return Q;
|
||
}
|
||
// Signatures should be low-s, to prevent malleability.
|
||
hasHighS() {
|
||
return isBiggerThanHalfOrder(this.s);
|
||
}
|
||
toBytes(format = defaultSigOpts.format) {
|
||
validateSigFormat(format);
|
||
if (format === 'der')
|
||
return hexToBytes(DER.hexFromSig(this));
|
||
const { r, s } = this;
|
||
const rb = Fn.toBytes(r);
|
||
const sb = Fn.toBytes(s);
|
||
if (format === 'recovered') {
|
||
assertRecoverableCurve();
|
||
return concatBytes(Uint8Array.of(this.assertRecovery()), rb, sb);
|
||
}
|
||
return concatBytes(rb, sb);
|
||
}
|
||
toHex(format) {
|
||
return bytesToHex(this.toBytes(format));
|
||
}
|
||
}
|
||
Object.freeze(Signature.prototype);
|
||
Object.freeze(Signature);
|
||
// 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 === undefined
|
||
? function bits2int_def(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;
|
||
}
|
||
: ecdsaOpts.bits2int;
|
||
const bits2int_modN = ecdsaOpts.bits2int_modN === undefined
|
||
? function bits2int_modN_def(bytes) {
|
||
return Fn.create(bits2int(bytes)); // can't use bytesToNumberBE here
|
||
}
|
||
: ecdsaOpts.bits2int_modN;
|
||
const ORDER_MASK = bitMask(fnBits);
|
||
// Pads output with zero as per spec.
|
||
/** Converts to bytes. Checks if num in `[0..ORDER_MASK-1]` e.g.: `[0..2^256-1]`. */
|
||
function int2octets(num) {
|
||
aInRange('num < 2^' + fnBits, num, _0n, ORDER_MASK);
|
||
return Fn.toBytes(num);
|
||
}
|
||
function validateMsgAndHash(message, prehash) {
|
||
abytes(message, undefined, 'message');
|
||
return (prehash ? abytes(hash_(message), undefined, 'prehashed message') : message);
|
||
}
|
||
/**
|
||
* Steps A, D of RFC6979 3.2.
|
||
* Creates RFC6979 seed; converts msg/privKey to numbers.
|
||
* Used only in sign, not in verify.
|
||
*
|
||
* Warning: we cannot assume here that message 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(message, secretKey, opts) {
|
||
const { lowS, prehash, extraEntropy } = validateSigOpts(opts, defaultSigOpts);
|
||
message = validateMsgAndHash(message, prehash); // RFC6979 3.2 A: h1 = H(m)
|
||
// 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(message);
|
||
const d = Fn.fromBytes(secretKey); // validate secret key, convert to bigint
|
||
if (!Fn.isValidNot0(d))
|
||
throw new Error('invalid private key');
|
||
const seedArgs = [int2octets(d), int2octets(h1int)];
|
||
// extraEntropy. RFC6979 3.6: additional k' (optional).
|
||
if (extraEntropy != null && extraEntropy !== false) {
|
||
// K = HMAC_K(V || 0x00 || int2octets(x) || bits2octets(h1) || k')
|
||
// gen random bytes OR pass as-is
|
||
const e = extraEntropy === true ? randomBytes(lengths.secretKey) : extraEntropy;
|
||
seedArgs.push(abytes(e, undefined, 'extraEntropy')); // check for being bytes
|
||
}
|
||
const seed = concatBytes(...seedArgs); // Step D of RFC6979 3.2
|
||
const m = h1int; // no need to call bits2int second time here, it is inside truncateHash!
|
||
// Converts signature params into point w r/s, checks result for validity.
|
||
// To transform k => Signature:
|
||
// q = k⋅G
|
||
// r = q.x mod n
|
||
// s = k^-1(m + rd) mod n
|
||
// 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 = k⋅G
|
||
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)); // s = k^-1(m + rd) mod n
|
||
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 = Fn.neg(s); // if lowS was passed, ensure s is always in the bottom half of N
|
||
recovery ^= 1;
|
||
}
|
||
return new Signature(r, normS, hasLargeRecoveryLifts ? undefined : recovery);
|
||
}
|
||
return { seed, k2sig };
|
||
}
|
||
/**
|
||
* Signs a message or message hash with a secret key.
|
||
* With the default `prehash: true`, raw message bytes are hashed internally;
|
||
* only `{ prehash: false }` expects a caller-supplied digest.
|
||
*
|
||
* ```
|
||
* sign(m, d) where
|
||
* k = rfc6979_hmac_drbg(m, d)
|
||
* (x, y) = G × k
|
||
* r = x mod n
|
||
* s = (m + dr) / k mod n
|
||
* ```
|
||
*/
|
||
function sign(message, secretKey, opts = {}) {
|
||
const { seed, k2sig } = prepSig(message, secretKey, opts); // Steps A, D of RFC6979 3.2.
|
||
const drbg = createHmacDrbg(hash_.outputLen, Fn.BYTES, hmac);
|
||
const sig = drbg(seed, k2sig); // Steps B, C, D, E, F, G
|
||
return sig.toBytes(opts.format);
|
||
}
|
||
/**
|
||
* Verifies a signature against message and public key.
|
||
* Rejects lowS signatures by default: see {@link ECDSAVerifyOpts}.
|
||
* 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, message, publicKey, opts = {}) {
|
||
const { lowS, prehash, format } = validateSigOpts(opts, defaultSigOpts);
|
||
publicKey = abytes(publicKey, undefined, 'publicKey');
|
||
message = validateMsgAndHash(message, prehash);
|
||
if (!isBytes(signature)) {
|
||
const end = signature instanceof Signature ? ', use sig.toBytes()' : '';
|
||
throw new Error('verify expects Uint8Array signature' + end);
|
||
}
|
||
validateSigLength(signature, format); // execute this twice because we want loud error
|
||
try {
|
||
const sig = Signature.fromBytes(signature, format);
|
||
const P = Point.fromBytes(publicKey);
|
||
if (lowS && sig.hasHighS())
|
||
return false;
|
||
const { r, s } = sig;
|
||
const h = bits2int_modN(message); // mod n, not mod p
|
||
const is = Fn.inv(s); // s^-1 mod n
|
||
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)); // u1⋅G + u2⋅P
|
||
if (R.is0())
|
||
return false;
|
||
const v = Fn.create(R.x); // v = r.x mod n
|
||
return v === r;
|
||
}
|
||
catch (e) {
|
||
return false;
|
||
}
|
||
}
|
||
function recoverPublicKey(signature, message, opts = {}) {
|
||
// Top-level recovery mirrors `sign()` / `verify()`: it hashes raw message
|
||
// bytes first unless the caller passes `{ prehash: false }`.
|
||
const { prehash } = validateSigOpts(opts, defaultSigOpts);
|
||
message = validateMsgAndHash(message, prehash);
|
||
return Signature.fromBytes(signature, 'recovered').recoverPublicKey(message).toBytes();
|
||
}
|
||
return Object.freeze({
|
||
keygen,
|
||
getPublicKey,
|
||
getSharedSecret,
|
||
utils,
|
||
lengths,
|
||
Point,
|
||
sign,
|
||
verify,
|
||
recoverPublicKey,
|
||
Signature,
|
||
hash: hash_,
|
||
});
|
||
}
|
||
//# sourceMappingURL=weierstrass.js.map
|