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