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Просмотр файла: curve.ts
/**
* Methods for elliptic curve multiplication by scalars.
* Contains wNAF, pippenger
* @module
*/
/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { type IField, nLength, validateField } from './modular.ts';
import { bitLen, bitMask, validateObject } from './utils.ts';
const _0n = BigInt(0);
const _1n = BigInt(1);
export type AffinePoint<T> = {
x: T;
y: T;
} & { z?: never; t?: never };
export interface Group<T extends Group<T>> {
double(): T;
negate(): T;
add(other: T): T;
subtract(other: T): T;
equals(other: T): boolean;
multiply(scalar: bigint): T;
}
export type GroupConstructor<T> = {
BASE: T;
ZERO: T;
};
export type Mapper<T> = (i: T[]) => T[];
function constTimeNegate<T extends Group<T>>(condition: boolean, item: T): T {
const neg = item.negate();
return condition ? neg : item;
}
function validateW(W: number, bits: number) {
if (!Number.isSafeInteger(W) || W <= 0 || W > bits)
throw new Error('invalid window size, expected [1..' + bits + '], got W=' + W);
}
/** Internal wNAF opts for specific W and scalarBits */
export type WOpts = {
windows: number;
windowSize: number;
mask: bigint;
maxNumber: number;
shiftBy: bigint;
};
function calcWOpts(W: number, scalarBits: number): WOpts {
validateW(W, scalarBits);
const windows = Math.ceil(scalarBits / W) + 1; // W=8 33. Not 32, because we skip zero
const windowSize = 2 ** (W - 1); // W=8 128. Not 256, because we skip zero
const maxNumber = 2 ** W; // W=8 256
const mask = bitMask(W); // W=8 255 == mask 0b11111111
const shiftBy = BigInt(W); // W=8 8
return { windows, windowSize, mask, maxNumber, shiftBy };
}
function calcOffsets(n: bigint, window: number, wOpts: WOpts) {
const { windowSize, mask, maxNumber, shiftBy } = wOpts;
let wbits = Number(n & mask); // extract W bits.
let nextN = n >> shiftBy; // shift number by W bits.
// What actually happens here:
// const highestBit = Number(mask ^ (mask >> 1n));
// let wbits2 = wbits - 1; // skip zero
// if (wbits2 & highestBit) { wbits2 ^= Number(mask); // (~);
// split if bits > max: +224 => 256-32
if (wbits > windowSize) {
// we skip zero, which means instead of `>= size-1`, we do `> size`
wbits -= maxNumber; // -32, can be maxNumber - wbits, but then we need to set isNeg here.
nextN += _1n; // +256 (carry)
}
const offsetStart = window * windowSize;
const offset = offsetStart + Math.abs(wbits) - 1; // -1 because we skip zero
const isZero = wbits === 0; // is current window slice a 0?
const isNeg = wbits < 0; // is current window slice negative?
const isNegF = window % 2 !== 0; // fake random statement for noise
const offsetF = offsetStart; // fake offset for noise
return { nextN, offset, isZero, isNeg, isNegF, offsetF };
}
function validateMSMPoints(points: any[], c: any) {
if (!Array.isArray(points)) throw new Error('array expected');
points.forEach((p, i) => {
if (!(p instanceof c)) throw new Error('invalid point at index ' + i);
});
}
function validateMSMScalars(scalars: any[], field: any) {
if (!Array.isArray(scalars)) throw new Error('array of scalars expected');
scalars.forEach((s, i) => {
if (!field.isValid(s)) throw new Error('invalid scalar at index ' + i);
});
}
// Since points in different groups cannot be equal (different object constructor),
// we can have single place to store precomputes.
// Allows to make points frozen / immutable.
const pointPrecomputes = new WeakMap<any, any[]>();
const pointWindowSizes = new WeakMap<any, number>();
function getW(P: any): number {
return pointWindowSizes.get(P) || 1;
}
export type IWNAF<T extends Group<T>> = {
constTimeNegate: <T extends Group<T>>(condition: boolean, item: T) => T;
hasPrecomputes(elm: T): boolean;
unsafeLadder(elm: T, n: bigint, p?: T): T;
precomputeWindow(elm: T, W: number): Group<T>[];
getPrecomputes(W: number, P: T, transform: Mapper<T>): T[];
wNAF(W: number, precomputes: T[], n: bigint): { p: T; f: T };
wNAFUnsafe(W: number, precomputes: T[], n: bigint, acc?: T): T;
wNAFCached(P: T, n: bigint, transform: Mapper<T>): { p: T; f: T };
wNAFCachedUnsafe(P: T, n: bigint, transform: Mapper<T>, prev?: T): T;
setWindowSize(P: T, W: number): void;
};
/**
* Elliptic curve multiplication of Point by scalar. Fragile.
* Scalars should always be less than curve order: this should be checked inside of a curve itself.
* Creates precomputation tables for fast multiplication:
* - private scalar is split by fixed size windows of W bits
* - every window point is collected from window's table & added to accumulator
* - since windows are different, same point inside tables won't be accessed more than once per calc
* - each multiplication is 'Math.ceil(CURVE_ORDER / 𝑊) + 1' point additions (fixed for any scalar)
* - +1 window is neccessary for wNAF
* - wNAF reduces table size: 2x less memory + 2x faster generation, but 10% slower multiplication
*
* @todo Research returning 2d JS array of windows, instead of a single window.
* This would allow windows to be in different memory locations
*/
export function wNAF<T extends Group<T>>(c: GroupConstructor<T>, bits: number): IWNAF<T> {
return {
constTimeNegate,
hasPrecomputes(elm: T) {
return getW(elm) !== 1;
},
// non-const time multiplication ladder
unsafeLadder(elm: T, n: bigint, p = c.ZERO) {
let d: T = elm;
while (n > _0n) {
if (n & _1n) p = p.add(d);
d = d.double();
n >>= _1n;
}
return p;
},
/**
* Creates a wNAF precomputation window. Used for caching.
* Default window size is set by `utils.precompute()` and is equal to 8.
* Number of precomputed points depends on the curve size:
* 2^(𝑊−1) * (Math.ceil(𝑛 / 𝑊) + 1), where:
* - 𝑊 is the window size
* - 𝑛 is the bitlength of the curve order.
* For a 256-bit curve and window size 8, the number of precomputed points is 128 * 33 = 4224.
* @param elm Point instance
* @param W window size
* @returns precomputed point tables flattened to a single array
*/
precomputeWindow(elm: T, W: number): Group<T>[] {
const { windows, windowSize } = calcWOpts(W, bits);
const points: T[] = [];
let p: T = elm;
let base = p;
for (let window = 0; window < windows; window++) {
base = p;
points.push(base);
// i=1, bc we skip 0
for (let i = 1; i < windowSize; i++) {
base = base.add(p);
points.push(base);
}
p = base.double();
}
return points;
},
/**
* Implements ec multiplication using precomputed tables and w-ary non-adjacent form.
* @param W window size
* @param precomputes precomputed tables
* @param n scalar (we don't check here, but should be less than curve order)
* @returns real and fake (for const-time) points
*/
wNAF(W: number, precomputes: T[], n: bigint): { p: T; f: T } {
// Smaller version:
// https://github.com/paulmillr/noble-secp256k1/blob/47cb1669b6e506ad66b35fe7d76132ae97465da2/index.ts#L502-L541
// TODO: check the scalar is less than group order?
// wNAF behavior is undefined otherwise. But have to carefully remove
// other checks before wNAF. ORDER == bits here.
// Accumulators
let p = c.ZERO;
let f = c.BASE;
// This code was first written with assumption that 'f' and 'p' will never be infinity point:
// since each addition is multiplied by 2 ** W, it cannot cancel each other. However,
// there is negate now: it is possible that negated element from low value
// would be the same as high element, which will create carry into next window.
// It's not obvious how this can fail, but still worth investigating later.
const wo = calcWOpts(W, bits);
for (let window = 0; window < wo.windows; window++) {
// (n === _0n) is handled and not early-exited. isEven and offsetF are used for noise
const { nextN, offset, isZero, isNeg, isNegF, offsetF } = calcOffsets(n, window, wo);
n = nextN;
if (isZero) {
// bits are 0: add garbage to fake point
// Important part for const-time getPublicKey: add random "noise" point to f.
f = f.add(constTimeNegate(isNegF, precomputes[offsetF]));
} else {
// bits are 1: add to result point
p = p.add(constTimeNegate(isNeg, precomputes[offset]));
}
}
// Return both real and fake points: JIT won't eliminate f.
// At this point there is a way to F be infinity-point even if p is not,
// which makes it less const-time: around 1 bigint multiply.
return { p, f };
},
/**
* Implements ec unsafe (non const-time) multiplication using precomputed tables and w-ary non-adjacent form.
* @param W window size
* @param precomputes precomputed tables
* @param n scalar (we don't check here, but should be less than curve order)
* @param acc accumulator point to add result of multiplication
* @returns point
*/
wNAFUnsafe(W: number, precomputes: T[], n: bigint, acc: T = c.ZERO): T {
const wo = calcWOpts(W, bits);
for (let window = 0; window < wo.windows; window++) {
if (n === _0n) break; // Early-exit, skip 0 value
const { nextN, offset, isZero, isNeg } = calcOffsets(n, window, wo);
n = nextN;
if (isZero) {
// Window bits are 0: skip processing.
// Move to next window.
continue;
} else {
const item = precomputes[offset];
acc = acc.add(isNeg ? item.negate() : item); // Re-using acc allows to save adds in MSM
}
}
return acc;
},
getPrecomputes(W: number, P: T, transform: Mapper<T>): T[] {
// Calculate precomputes on a first run, reuse them after
let comp = pointPrecomputes.get(P);
if (!comp) {
comp = this.precomputeWindow(P, W) as T[];
if (W !== 1) pointPrecomputes.set(P, transform(comp));
}
return comp;
},
wNAFCached(P: T, n: bigint, transform: Mapper<T>): { p: T; f: T } {
const W = getW(P);
return this.wNAF(W, this.getPrecomputes(W, P, transform), n);
},
wNAFCachedUnsafe(P: T, n: bigint, transform: Mapper<T>, prev?: T): T {
const W = getW(P);
if (W === 1) return this.unsafeLadder(P, n, prev); // For W=1 ladder is ~x2 faster
return this.wNAFUnsafe(W, this.getPrecomputes(W, P, transform), n, prev);
},
// We calculate precomputes for elliptic curve point multiplication
// using windowed method. This specifies window size and
// stores precomputed values. Usually only base point would be precomputed.
setWindowSize(P: T, W: number) {
validateW(W, bits);
pointWindowSizes.set(P, W);
pointPrecomputes.delete(P);
},
};
}
/**
* Pippenger algorithm for multi-scalar multiplication (MSM, Pa + Qb + Rc + ...).
* 30x faster vs naive addition on L=4096, 10x faster than precomputes.
* For N=254bit, L=1, it does: 1024 ADD + 254 DBL. For L=5: 1536 ADD + 254 DBL.
* Algorithmically constant-time (for same L), even when 1 point + scalar, or when scalar = 0.
* @param c Curve Point constructor
* @param fieldN field over CURVE.N - important that it's not over CURVE.P
* @param points array of L curve points
* @param scalars array of L scalars (aka private keys / bigints)
*/
export function pippenger<T extends Group<T>>(
c: GroupConstructor<T>,
fieldN: IField<bigint>,
points: T[],
scalars: bigint[]
): T {
// If we split scalars by some window (let's say 8 bits), every chunk will only
// take 256 buckets even if there are 4096 scalars, also re-uses double.
// TODO:
// - https://eprint.iacr.org/2024/750.pdf
// - https://tches.iacr.org/index.php/TCHES/article/view/10287
// 0 is accepted in scalars
validateMSMPoints(points, c);
validateMSMScalars(scalars, fieldN);
const plength = points.length;
const slength = scalars.length;
if (plength !== slength) throw new Error('arrays of points and scalars must have equal length');
// if (plength === 0) throw new Error('array must be of length >= 2');
const zero = c.ZERO;
const wbits = bitLen(BigInt(plength));
let windowSize = 1; // bits
if (wbits > 12) windowSize = wbits - 3;
else if (wbits > 4) windowSize = wbits - 2;
else if (wbits > 0) windowSize = 2;
const MASK = bitMask(windowSize);
const buckets = new Array(Number(MASK) + 1).fill(zero); // +1 for zero array
const lastBits = Math.floor((fieldN.BITS - 1) / windowSize) * windowSize;
let sum = zero;
for (let i = lastBits; i >= 0; i -= windowSize) {
buckets.fill(zero);
for (let j = 0; j < slength; j++) {
const scalar = scalars[j];
const wbits = Number((scalar >> BigInt(i)) & MASK);
buckets[wbits] = buckets[wbits].add(points[j]);
}
let resI = zero; // not using this will do small speed-up, but will lose ct
// Skip first bucket, because it is zero
for (let j = buckets.length - 1, sumI = zero; j > 0; j--) {
sumI = sumI.add(buckets[j]);
resI = resI.add(sumI);
}
sum = sum.add(resI);
if (i !== 0) for (let j = 0; j < windowSize; j++) sum = sum.double();
}
return sum as T;
}
/**
* Precomputed multi-scalar multiplication (MSM, Pa + Qb + Rc + ...).
* @param c Curve Point constructor
* @param fieldN field over CURVE.N - important that it's not over CURVE.P
* @param points array of L curve points
* @returns function which multiplies points with scaars
*/
export function precomputeMSMUnsafe<T extends Group<T>>(
c: GroupConstructor<T>,
fieldN: IField<bigint>,
points: T[],
windowSize: number
): (scalars: bigint[]) => T {
/**
* Performance Analysis of Window-based Precomputation
*
* Base Case (256-bit scalar, 8-bit window):
* - Standard precomputation requires:
* - 31 additions per scalar × 256 scalars = 7,936 ops
* - Plus 255 summary additions = 8,191 total ops
* Note: Summary additions can be optimized via accumulator
*
* Chunked Precomputation Analysis:
* - Using 32 chunks requires:
* - 255 additions per chunk
* - 256 doublings
* - Total: (255 × 32) + 256 = 8,416 ops
*
* Memory Usage Comparison:
* Window Size | Standard Points | Chunked Points
* ------------|-----------------|---------------
* 4-bit | 520 | 15
* 8-bit | 4,224 | 255
* 10-bit | 13,824 | 1,023
* 16-bit | 557,056 | 65,535
*
* Key Advantages:
* 1. Enables larger window sizes due to reduced memory overhead
* 2. More efficient for smaller scalar counts:
* - 16 chunks: (16 × 255) + 256 = 4,336 ops
* - ~2x faster than standard 8,191 ops
*
* Limitations:
* - Not suitable for plain precomputes (requires 256 constant doublings)
* - Performance degrades with larger scalar counts:
* - Optimal for ~256 scalars
* - Less efficient for 4096+ scalars (Pippenger preferred)
*/
validateW(windowSize, fieldN.BITS);
validateMSMPoints(points, c);
const zero = c.ZERO;
const tableSize = 2 ** windowSize - 1; // table size (without zero)
const chunks = Math.ceil(fieldN.BITS / windowSize); // chunks of item
const MASK = bitMask(windowSize);
const tables = points.map((p: T) => {
const res = [];
for (let i = 0, acc = p; i < tableSize; i++) {
res.push(acc);
acc = acc.add(p);
}
return res;
});
return (scalars: bigint[]): T => {
validateMSMScalars(scalars, fieldN);
if (scalars.length > points.length)
throw new Error('array of scalars must be smaller than array of points');
let res = zero;
for (let i = 0; i < chunks; i++) {
// No need to double if accumulator is still zero.
if (res !== zero) for (let j = 0; j < windowSize; j++) res = res.double();
const shiftBy = BigInt(chunks * windowSize - (i + 1) * windowSize);
for (let j = 0; j < scalars.length; j++) {
const n = scalars[j];
const curr = Number((n >> shiftBy) & MASK);
if (!curr) continue; // skip zero scalars chunks
res = res.add(tables[j][curr - 1]);
}
}
return res;
};
}
/**
* Generic BasicCurve interface: works even for polynomial fields (BLS): P, n, h would be ok.
* Though generator can be different (Fp2 / Fp6 for BLS).
*/
export type BasicCurve<T> = {
Fp: IField<T>; // Field over which we'll do calculations (Fp)
n: bigint; // Curve order, total count of valid points in the field
nBitLength?: number; // bit length of curve order
nByteLength?: number; // byte length of curve order
h: bigint; // cofactor. we can assign default=1, but users will just ignore it w/o validation
hEff?: bigint; // Number to multiply to clear cofactor
Gx: T; // base point X coordinate
Gy: T; // base point Y coordinate
allowInfinityPoint?: boolean; // bls12-381 requires it. ZERO point is valid, but invalid pubkey
};
export function validateBasic<FP, T>(
curve: BasicCurve<FP> & T
): Readonly<
{
readonly nBitLength: number;
readonly nByteLength: number;
} & BasicCurve<FP> &
T & {
p: bigint;
}
> {
validateField(curve.Fp);
validateObject(
curve,
{
n: 'bigint',
h: 'bigint',
Gx: 'field',
Gy: 'field',
},
{
nBitLength: 'isSafeInteger',
nByteLength: 'isSafeInteger',
}
);
// Set defaults
return Object.freeze({
...nLength(curve.n, curve.nBitLength),
...curve,
...{ p: curve.Fp.ORDER },
} as const);
}
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