Improve modular math
This commit is contained in:
Paul Miller
parent
cc2c84f040
commit
12da04a2bb
@ -1,10 +1,11 @@
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/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
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// TODO: remove circular imports
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import * as utils from './utils.js';
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// Utilities for modular arithmetics and finite fields
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// prettier-ignore
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const _0n = BigInt(0), _1n = BigInt(1), _2n = BigInt(2), _3n = BigInt(3);
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// prettier-ignore
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const _4n = BigInt(4), _5n = BigInt(5), _7n = BigInt(7), _8n = BigInt(8);
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const _4n = BigInt(4), _5n = BigInt(5), _8n = BigInt(8);
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// prettier-ignore
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const _9n = BigInt(9), _16n = BigInt(16);
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@ -66,26 +67,68 @@ export function invert(number: bigint, modulo: bigint): bigint {
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return mod(x, modulo);
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}
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/**
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* Calculates Legendre symbol (a | p), which denotes the value of a^((p-1)/2) (mod p).
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* * (a | p) ≡ 1 if a is a square (mod p)
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* * (a | p) ≡ -1 if a is not a square (mod p)
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* * (a | p) ≡ 0 if a ≡ 0 (mod p)
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*/
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export function legendre(num: bigint, fieldPrime: bigint): bigint {
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return pow(num, (fieldPrime - _1n) / _2n, fieldPrime);
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// Tonelli-Shanks algorithm
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// https://eprint.iacr.org/2012/685.pdf (page 12)
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export function tonelliShanks(P: bigint) {
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// Legendre constant: used to calculate Legendre symbol (a | p),
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// which denotes the value of a^((p-1)/2) (mod p).
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// (a | p) ≡ 1 if a is a square (mod p)
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// (a | p) ≡ -1 if a is not a square (mod p)
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// (a | p) ≡ 0 if a ≡ 0 (mod p)
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const legendreC = (P - _1n) / _2n;
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let Q: bigint, S: number, Z: bigint;
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// Step 1: By factoring out powers of 2 from p - 1,
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// find q and s such that p - 1 = q2s with q odd
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for (Q = P - _1n, S = 0; Q % _2n === _0n; Q /= _2n, S++);
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// Step 2: Select a non-square z such that (z | p) ≡ -1 and set c ≡ zq
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for (Z = _2n; Z < P && pow(Z, legendreC, P) !== P - _1n; Z++);
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// Fast-path
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if (S === 1) {
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const p1div4 = (P + _1n) / _4n;
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return function tonelliFast<T>(Fp: Field<T>, n: T) {
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const root = Fp.pow(n, p1div4);
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if (!Fp.equals(Fp.square(root), n)) throw new Error('Cannot find square root');
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return root;
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};
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}
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// Slow-path
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const Q1div2 = (Q + _1n) / _2n;
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return function tonelliSlow<T>(Fp: Field<T>, n: T): T {
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// Step 0: Check that n is indeed a square: (n | p) must be ≡ 1
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if (Fp.pow(n, legendreC) !== Fp.ONE) throw new Error('Cannot find square root');
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let s = S;
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let c = pow(Z, Q, P);
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let r = Fp.pow(n, Q1div2);
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let t = Fp.pow(n, Q);
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let t2 = Fp.ZERO;
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while (!Fp.equals(Fp.sub(t, Fp.ONE), Fp.ZERO)) {
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t2 = Fp.square(t);
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let i;
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for (i = 1; i < s; i++) {
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// stop if t2-1 == 0
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if (Fp.equals(Fp.sub(t2, Fp.ONE), Fp.ZERO)) break;
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// t2 *= t2
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t2 = Fp.square(t2);
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}
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let b = pow(c, BigInt(1 << (s - i - 1)), P);
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r = Fp.mul(r, b);
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c = mod(b * b, P);
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t = Fp.mul(t, c);
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s = i;
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}
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return r;
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};
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}
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/**
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* Calculates square root of a number in a finite field.
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* √a mod P
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*/
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// TODO: rewrite as generic Fp function && remove bls versions
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export function sqrt(number: bigint, modulo: bigint): bigint {
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// prettier-ignore
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const n = number;
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const P = modulo;
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const p1div4 = (P + _1n) / _4n;
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export function FpSqrt(P: bigint) {
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// NOTE: different algorithms can give different roots, it is up to user to decide which one they want.
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// For example there is FpSqrtOdd/FpSqrtEven to choice root based on oddness (used for hash-to-curve).
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// P ≡ 3 (mod 4)
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// √n = n^((P+1)/4)
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@ -94,48 +137,54 @@ export function sqrt(number: bigint, modulo: bigint): bigint {
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// const ORDER =
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// 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaaabn;
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// const NUM = 72057594037927816n;
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// TODO: fix sqrtMod in secp256k1
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const root = pow(n, p1div4, P);
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if (mod(root * root, modulo) !== number) throw new Error('Cannot find square root');
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const p1div4 = (P + _1n) / _4n;
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return function sqrt3mod4<T>(Fp: Field<T>, n: T) {
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const root = Fp.pow(n, p1div4);
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// Throw if root**2 != n
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if (!Fp.equals(Fp.square(root), n)) throw new Error('Cannot find square root');
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return root;
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};
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}
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// P ≡ 5 (mod 8)
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// Atkin algorithm for q ≡ 5 (mod 8), https://eprint.iacr.org/2012/685.pdf (page 10)
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if (P % _8n === _5n) {
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const n2 = mod(n * _2n, P);
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const v = pow(n2, (P - _5n) / _8n, P);
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const nv = mod(n * v, P);
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const i = mod(_2n * nv * v, P);
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const r = mod(nv * (i - _1n), P);
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return r;
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const c1 = (P - _5n) / _8n;
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return function sqrt5mod8<T>(Fp: Field<T>, n: T) {
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const n2 = Fp.mul(n, _2n);
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const v = Fp.pow(n2, c1);
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const nv = Fp.mul(n, v);
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const i = Fp.mul(Fp.mul(nv, _2n), v);
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const root = Fp.mul(nv, Fp.sub(i, Fp.ONE));
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if (!Fp.equals(Fp.square(root), n)) throw new Error('Cannot find square root');
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return root;
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};
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}
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// P ≡ 9 (mod 16)
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if (P % _16n === _9n) {
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// NOTE: tonelli is too slow for bls-Fp2 calculations even on start
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// Means we cannot use sqrt for constants at all!
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//
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// const c1 = Fp.sqrt(Fp.negate(Fp.ONE)); // 1. c1 = sqrt(-1) in F, i.e., (c1^2) == -1 in F
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// const c2 = Fp.sqrt(c1); // 2. c2 = sqrt(c1) in F, i.e., (c2^2) == c1 in F
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// const c3 = Fp.sqrt(Fp.negate(c1)); // 3. c3 = sqrt(-c1) in F, i.e., (c3^2) == -c1 in F
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// const c4 = (P + _7n) / _16n; // 4. c4 = (q + 7) / 16 # Integer arithmetic
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// sqrt = (x) => {
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// let tv1 = Fp.pow(x, c4); // 1. tv1 = x^c4
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// let tv2 = Fp.mul(c1, tv1); // 2. tv2 = c1 * tv1
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// const tv3 = Fp.mul(c2, tv1); // 3. tv3 = c2 * tv1
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// let tv4 = Fp.mul(c3, tv1); // 4. tv4 = c3 * tv1
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// const e1 = Fp.equals(Fp.square(tv2), x); // 5. e1 = (tv2^2) == x
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// const e2 = Fp.equals(Fp.square(tv3), x); // 6. e2 = (tv3^2) == x
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// tv1 = Fp.cmov(tv1, tv2, e1); // 7. tv1 = CMOV(tv1, tv2, e1) # Select tv2 if (tv2^2) == x
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// tv2 = Fp.cmov(tv4, tv3, e2); // 8. tv2 = CMOV(tv4, tv3, e2) # Select tv3 if (tv3^2) == x
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// const e3 = Fp.equals(Fp.square(tv2), x); // 9. e3 = (tv2^2) == x
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// return Fp.cmov(tv1, tv2, e3); // 10. z = CMOV(tv1, tv2, e3) # Select the sqrt from tv1 and tv2
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// }
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}
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// Other cases: Tonelli-Shanks algorithm
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if (legendre(n, P) !== _1n) throw new Error('Cannot find square root');
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let q: bigint, s: number, z: bigint;
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for (q = P - _1n, s = 0; q % _2n === _0n; q /= _2n, s++);
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if (s === 1) return pow(n, p1div4, P);
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for (z = _2n; z < P && legendre(z, P) !== P - _1n; z++);
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let c = pow(z, q, P);
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let r = pow(n, (q + _1n) / _2n, P);
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let t = pow(n, q, P);
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let t2 = _0n;
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while (mod(t - _1n, P) !== _0n) {
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t2 = mod(t * t, P);
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let i;
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for (i = 1; i < s; i++) {
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if (mod(t2 - _1n, P) === _0n) break;
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t2 = mod(t2 * t2, P);
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}
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let b = pow(c, BigInt(1 << (s - i - 1)), P);
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r = mod(r * b, P);
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c = mod(b * b, P);
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t = mod(t * c, P);
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s = i;
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}
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return r;
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return tonelliShanks(P);
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}
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// Little-endian check for first LE bit (last BE bit);
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@ -176,6 +225,7 @@ export interface Field<T> {
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// Optional
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// Should be same as sgn0 function in https://datatracker.ietf.org/doc/draft-irtf-cfrg-hash-to-curve/
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// NOTE: sgn0 is 'negative in LE', which is same as odd. And negative in LE is kinda strange definition anyway.
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isOdd?(num: T): boolean; // Odd instead of even since we have it for Fp2
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legendre?(num: T): T;
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pow(lhs: T, power: bigint): T;
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@ -246,21 +296,31 @@ export function FpDiv<T>(f: Field<T>, lhs: T, rhs: T | bigint): T {
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return f.mul(lhs, typeof rhs === 'bigint' ? invert(rhs, f.ORDER) : f.invert(rhs));
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}
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// This function returns True whenever the value x is a square in the field F.
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export function FpIsSquare<T>(f: Field<T>) {
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const legendreConst = (f.ORDER - _1n) / _2n; // Integer arithmetic
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return (x: T): boolean => {
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const p = f.pow(x, legendreConst);
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return f.equals(p, f.ZERO) || f.equals(p, f.ONE);
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};
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}
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// NOTE: very fragile, always bench. Major performance points:
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// - NonNormalized ops
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// - Object.freeze
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// - same shape of object (don't add/remove keys)
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type FpField = Field<bigint> & Required<Pick<Field<bigint>, 'isOdd'>>;
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export function Fp(
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ORDER: bigint,
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bitLen?: number,
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isLE = false,
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redef: Partial<Field<bigint>> = {}
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): Readonly<Field<bigint>> {
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): Readonly<FpField> {
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if (ORDER <= _0n) throw new Error(`Expected Fp ORDER > 0, got ${ORDER}`);
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const { nBitLength: BITS, nByteLength: BYTES } = utils.nLength(ORDER, bitLen);
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if (BYTES > 2048) throw new Error('Field lengths over 2048 bytes are not supported');
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const sqrtP = (num: bigint) => sqrt(num, ORDER);
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const f: Field<bigint> = Object.freeze({
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const sqrtP = FpSqrt(ORDER);
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const f: Readonly<FpField> = Object.freeze({
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ORDER,
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BITS,
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BYTES,
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@ -292,7 +352,7 @@ export function Fp(
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mulN: (lhs, rhs) => lhs * rhs,
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invert: (num) => invert(num, ORDER),
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sqrt: redef.sqrt || sqrtP,
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sqrt: redef.sqrt || ((n) => sqrtP(f, n)),
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invertBatch: (lst) => FpInvertBatch(f, lst),
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// TODO: do we really need constant cmov?
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// We don't have const-time bigints anyway, so probably will be not very useful
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@ -305,87 +365,18 @@ export function Fp(
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throw new Error(`Fp.fromBytes: expected ${BYTES}, got ${bytes.length}`);
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return isLE ? utils.bytesToNumberLE(bytes) : utils.bytesToNumberBE(bytes);
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},
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} as Field<bigint>);
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} as FpField);
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return Object.freeze(f);
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}
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// TODO: re-use in bls/generic sqrt for field/etc?
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// Something like sqrtUnsafe which always returns value, but sqrt throws exception if non-square
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// From draft-irtf-cfrg-hash-to-curve-16
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export function FpSqrt<T>(Fp: Field<T>) {
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// NOTE: it requires another sqrt for constant precomputes, but no need for roots of unity,
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// probably we can simply bls code using it
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const q = Fp.ORDER;
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const squareConst = (q - _1n) / _2n;
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// is_square(x) := { True, if x^((q - 1) / 2) is 0 or 1 in F;
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// { False, otherwise.
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let isSquare: (x: T) => boolean = (x) => {
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const p = Fp.pow(x, squareConst);
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return Fp.equals(p, Fp.ZERO) || Fp.equals(p, Fp.ONE);
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};
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// Constant-time Tonelli-Shanks algorithm
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let l = _0n;
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for (let o = q - _1n; o % _2n === _0n; o /= _2n) l += _1n;
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const c1 = l; // 1. c1, the largest integer such that 2^c1 divides q - 1.
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const c2 = (q - _1n) / _2n ** c1; // 2. c2 = (q - 1) / (2^c1) # Integer arithmetic
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const c3 = (c2 - _1n) / _2n; // 3. c3 = (c2 - 1) / 2 # Integer arithmetic
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// 4. c4, a non-square value in F
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// 5. c5 = c4^c2 in F
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let c4 = Fp.ONE;
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while (isSquare(c4)) c4 = Fp.add(c4, Fp.ONE);
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const c5 = Fp.pow(c4, c2);
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let sqrt: (x: T) => T = (x) => {
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let z = Fp.pow(x, c3); // 1. z = x^c3
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let t = Fp.square(z); // 2. t = z * z
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t = Fp.mul(t, x); // 3. t = t * x
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z = Fp.mul(z, x); // 4. z = z * x
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let b = t; // 5. b = t
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let c = c5; // 6. c = c5
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// 7. for i in (c1, c1 - 1, ..., 2):
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for (let i = c1; i > 1; i--) {
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// 8. for j in (1, 2, ..., i - 2):
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// 9. b = b * b
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for (let j = _1n; j < i - _1n; i++) b = Fp.square(b);
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const e = Fp.equals(b, Fp.ONE); // 10. e = b == 1
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const zt = Fp.mul(z, c); // 11. zt = z * c
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z = Fp.cmov(zt, z, e); // 12. z = CMOV(zt, z, e)
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c = Fp.square(c); // 13. c = c * c
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let tt = Fp.mul(t, c); // 14. tt = t * c
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t = Fp.cmov(tt, t, e); // 15. t = CMOV(tt, t, e)
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b = t; // 16. b = t
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}
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return z; // 17. return z
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};
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if (q % _4n === _3n) {
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const c1 = (q + _1n) / _4n; // 1. c1 = (q + 1) / 4 # Integer arithmetic
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sqrt = (x) => Fp.pow(x, c1);
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} else if (q % _8n === _5n) {
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const c1 = Fp.sqrt(Fp.negate(Fp.ONE)); // 1. c1 = sqrt(-1) in F, i.e., (c1^2) == -1 in F
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const c2 = (q + _3n) / _8n; // 2. c2 = (q + 3) / 8 # Integer arithmetic
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sqrt = (x) => {
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let tv1 = Fp.pow(x, c2); // 1. tv1 = x^c2
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let tv2 = Fp.mul(tv1, c1); // 2. tv2 = tv1 * c1
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let e = Fp.equals(Fp.square(tv1), x); // 3. e = (tv1^2) == x
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return Fp.cmov(tv2, tv1, e); // 4. z = CMOV(tv2, tv1, e)
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};
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} else if (Fp.ORDER % _16n === _9n) {
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const c1 = Fp.sqrt(Fp.negate(Fp.ONE)); // 1. c1 = sqrt(-1) in F, i.e., (c1^2) == -1 in F
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const c2 = Fp.sqrt(c1); // 2. c2 = sqrt(c1) in F, i.e., (c2^2) == c1 in F
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const c3 = Fp.sqrt(Fp.negate(c1)); // 3. c3 = sqrt(-c1) in F, i.e., (c3^2) == -c1 in F
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const c4 = (Fp.ORDER + _7n) / _16n; // 4. c4 = (q + 7) / 16 # Integer arithmetic
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sqrt = (x) => {
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let tv1 = Fp.pow(x, c4); // 1. tv1 = x^c4
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let tv2 = Fp.mul(c1, tv1); // 2. tv2 = c1 * tv1
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const tv3 = Fp.mul(c2, tv1); // 3. tv3 = c2 * tv1
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let tv4 = Fp.mul(c3, tv1); // 4. tv4 = c3 * tv1
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const e1 = Fp.equals(Fp.square(tv2), x); // 5. e1 = (tv2^2) == x
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const e2 = Fp.equals(Fp.square(tv3), x); // 6. e2 = (tv3^2) == x
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tv1 = Fp.cmov(tv1, tv2, e1); // 7. tv1 = CMOV(tv1, tv2, e1) # Select tv2 if (tv2^2) == x
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tv2 = Fp.cmov(tv4, tv3, e2); // 8. tv2 = CMOV(tv4, tv3, e2) # Select tv3 if (tv3^2) == x
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const e3 = Fp.equals(Fp.square(tv2), x); // 9. e3 = (tv2^2) == x
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return Fp.cmov(tv1, tv2, e3); // 10. z = CMOV(tv1, tv2, e3) # Select the sqrt from tv1 and tv2
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};
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}
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return { sqrt, isSquare };
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export function FpSqrtOdd<T>(Fp: Field<T>, elm: T) {
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if (!Fp.isOdd) throw new Error(`Field doesn't have isOdd`);
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const root = Fp.sqrt(elm);
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return Fp.isOdd(root) ? root : Fp.negate(root);
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||||
}
|
||||
|
||||
export function FpSqrtEven<T>(Fp: Field<T>, elm: T) {
|
||||
if (!Fp.isOdd) throw new Error(`Field doesn't have isOdd`);
|
||||
const root = Fp.sqrt(elm);
|
||||
return Fp.isOdd(root) ? Fp.negate(root) : root;
|
||||
}
|
||||
|
||||
@ -56,7 +56,9 @@ function sqrtMod(y: bigint): bigint {
|
||||
const b223 = (pow2(b220, _3n, P) * b3) % P;
|
||||
const t1 = (pow2(b223, _23n, P) * b22) % P;
|
||||
const t2 = (pow2(t1, _6n, P) * b2) % P;
|
||||
return pow2(t2, _2n, P);
|
||||
const root = pow2(t2, _2n, P);
|
||||
if (!Fp.equals(Fp.square(root), y)) throw new Error('Cannot find square root');
|
||||
return root;
|
||||
}
|
||||
|
||||
const Fp = Field(secp256k1P, undefined, undefined, { sqrt: sqrtMod });
|
||||
@ -152,7 +154,7 @@ export const secp256k1 = createCurve(
|
||||
p: Fp.ORDER,
|
||||
m: 1,
|
||||
k: 128,
|
||||
expand: true,
|
||||
expand: 'xmd',
|
||||
hash: sha256,
|
||||
},
|
||||
},
|
||||
|
||||
Loading…
Reference in New Issue
Block a user