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@ -216,6 +216,24 @@ There are following zero-dependency algorithms:
```ts ```ts
import { weierstrass } from '@noble/curves/abstract/weierstrass'; import { weierstrass } from '@noble/curves/abstract/weierstrass';
import { Fp } from '@noble/curves/abstract/modular'; // finite field for mod arithmetics
import { sha256 } from '@noble/hashes/sha256'; // 3rd-party sha256() of type utils.CHash
import { hmac } from '@noble/hashes/hmac'; // 3rd-party hmac() that will accept sha256()
import { concatBytes, randomBytes } from '@noble/hashes/utils'; // 3rd-party utilities
const secq256k1 = weierstrass({
// secq256k1: cycle of secp256k1 with Fp/N flipped.
// https://personaelabs.org/posts/spartan-ecdsa
// https://zcash.github.io/halo2/background/curves.html#cycles-of-curves
a: 0n,
b: 7n,
Fp: Fp(2n ** 256n - 432420386565659656852420866394968145599n),
n: 2n ** 256n - 2n ** 32n - 2n ** 9n - 2n ** 8n - 2n ** 7n - 2n ** 6n - 2n ** 4n - 1n,
Gx: 55066263022277343669578718895168534326250603453777594175500187360389116729240n,
Gy: 32670510020758816978083085130507043184471273380659243275938904335757337482424n,
hash: sha256,
hmac: (key: Uint8Array, ...msgs: Uint8Array[]) => hmac(sha256, key, concatBytes(...msgs)),
randomBytes,
});
``` ```
Short Weierstrass curve's formula is `y² = x³ + ax + b`. `weierstrass` Short Weierstrass curve's formula is `y² = x³ + ax + b`. `weierstrass`
@ -247,6 +265,29 @@ type CHash = {
6. Have `toAffine()` and `x` / `y` getters which convert to 2d xy affine coordinates 6. Have `toAffine()` and `x` / `y` getters which convert to 2d xy affine coordinates
```ts ```ts
// `weierstrass()` returns `CurveFn`
type SignOpts = { lowS?: boolean; prehash?: boolean; extraEntropy: boolean | Uint8Array };
type CurveFn = {
CURVE: ReturnType<typeof validateOpts>;
getPublicKey: (privateKey: PrivKey, isCompressed?: boolean) => Uint8Array;
getSharedSecret: (privateA: PrivKey, publicB: Hex, isCompressed?: boolean) => Uint8Array;
sign: (msgHash: Hex, privKey: PrivKey, opts?: SignOpts) => SignatureType;
verify: (
signature: Hex | SignatureType,
msgHash: Hex,
publicKey: Hex,
opts?: { lowS?: boolean; prehash?: boolean }
) => boolean;
ProjectivePoint: ProjectivePointConstructor;
Signature: SignatureConstructor;
utils: {
normPrivateKeyToScalar: (key: PrivKey) => bigint;
isValidPrivateKey(key: PrivKey): boolean;
randomPrivateKey: () => Uint8Array;
precompute: (windowSize?: number, point?: ProjPointType<bigint>) => ProjPointType<bigint>;
};
};
// T is usually bigint, but can be something else like complex numbers in BLS curves // T is usually bigint, but can be something else like complex numbers in BLS curves
interface ProjPointType<T> extends Group<ProjPointType<T>> { interface ProjPointType<T> extends Group<ProjPointType<T>> {
readonly px: T; readonly px: T;
@ -298,29 +339,9 @@ type SignatureConstructor = {
}; };
``` ```
Example implementing [secq256k1](https://personaelabs.org/posts/spartan-ecdsa) More examples:
[cycle](https://zcash.github.io/halo2/background/curves.html#cycles-of-curves)
of secp256k1 with Fp/N flipped.
```typescript ```typescript
import { weierstrass } from '@noble/curves/abstract/weierstrass';
import { Fp } from '@noble/curves/abstract/modular'; // finite field, mod arithmetics done over it
import { sha256 } from '@noble/hashes/sha256'; // 3rd-party sha256() of type utils.CHash
import { hmac } from '@noble/hashes/hmac'; // 3rd-party hmac() that will accept sha256()
import { concatBytes, randomBytes } from '@noble/hashes/utils'; // 3rd-party utilities
const secq256k1 = weierstrass({
// secq256k1: cycle of secp256k1 with Fp/N flipped.
a: 0n,
b: 7n,
Fp: Fp(2n ** 256n - 432420386565659656852420866394968145599n),
n: 2n ** 256n - 2n ** 32n - 2n ** 9n - 2n ** 8n - 2n ** 7n - 2n ** 6n - 2n ** 4n - 1n,
Gx: 55066263022277343669578718895168534326250603453777594175500187360389116729240n,
Gy: 32670510020758816978083085130507043184471273380659243275938904335757337482424n,
hash: sha256,
hmac: (key: Uint8Array, ...msgs: Uint8Array[]) => hmac(sha256, key, concatBytes(...msgs)),
randomBytes,
});
// All curves expose same generic interface. // All curves expose same generic interface.
const priv = secq256k1.utils.randomPrivateKey(); const priv = secq256k1.utils.randomPrivateKey();
secq256k1.getPublicKey(priv); // Convert private key to public. secq256k1.getPublicKey(priv); // Convert private key to public.
@ -346,78 +367,13 @@ const fast = secq256k1.utils.precompute(8, Point.fromHex(someonesPubKey));
fast.multiply(privKey); // much faster ECDH now fast.multiply(privKey); // much faster ECDH now
``` ```
`weierstrass()` returns `CurveFn`:
```ts
type SignOpts = { lowS?: boolean; prehash?: boolean; extraEntropy: boolean | Uint8Array };
type CurveFn = {
CURVE: ReturnType<typeof validateOpts>;
getPublicKey: (privateKey: PrivKey, isCompressed?: boolean) => Uint8Array;
getSharedSecret: (privateA: PrivKey, publicB: Hex, isCompressed?: boolean) => Uint8Array;
sign: (msgHash: Hex, privKey: PrivKey, opts?: SignOpts) => SignatureType;
verify: (
signature: Hex | SignatureType,
msgHash: Hex,
publicKey: Hex,
opts?: { lowS?: boolean; prehash?: boolean }
) => boolean;
ProjectivePoint: ProjectivePointConstructor;
Signature: SignatureConstructor;
utils: {
normPrivateKeyToScalar: (key: PrivKey) => bigint;
isValidPrivateKey(key: PrivKey): boolean;
randomPrivateKey: () => Uint8Array;
precompute: (windowSize?: number, point?: ProjPointType<bigint>) => ProjPointType<bigint>;
};
};
```
### abstract/edwards: Twisted Edwards curve ### abstract/edwards: Twisted Edwards curve
Twisted Edwards curve's formula is `ax² + y² = 1 + dx²y²`. You must specify `a`, `d`, field `Fp`, order `n`, cofactor `h`
and coordinates `Gx`, `Gy` of generator point.
For EdDSA signatures, `hash` param required. `adjustScalarBytes` which instructs how to change private scalars could be specified.
**Edwards points:**
1. Exported as `ExtendedPoint`
2. Represented in extended coordinates: (x, y, z, t) ∋ (x=x/z, y=y/z)
3. Use complete exception-free formulas for addition and doubling
4. Can be decoded/encoded from/to Uint8Array / hex strings using `ExtendedPoint.fromHex` and `ExtendedPoint#toRawBytes()`
5. Have `assertValidity()` which checks for being on-curve
6. Have `toAffine()` and `x` / `y` getters which convert to 2d xy affine coordinates
7. Have `isTorsionFree()`, `clearCofactor()` and `isSmallOrder()` utilities to handle torsions
```ts
interface ExtPointType extends Group<ExtPointType> {
readonly ex: bigint;
readonly ey: bigint;
readonly ez: bigint;
readonly et: bigint;
assertValidity(): void;
multiply(scalar: bigint): ExtPointType;
multiplyUnsafe(scalar: bigint): ExtPointType;
isSmallOrder(): boolean;
isTorsionFree(): boolean;
clearCofactor(): ExtPointType;
toAffine(iz?: bigint): AffinePoint<bigint>;
}
// Static methods of Extended Point with coordinates in X, Y, Z, T
interface ExtPointConstructor extends GroupConstructor<ExtPointType> {
new (x: bigint, y: bigint, z: bigint, t: bigint): ExtPointType;
fromAffine(p: AffinePoint<bigint>): ExtPointType;
fromHex(hex: Hex): ExtPointType;
fromPrivateKey(privateKey: Hex): ExtPointType;
}
```
Example implementing edwards25519:
```ts ```ts
import { twistedEdwards } from '@noble/curves/abstract/edwards'; import { twistedEdwards } from '@noble/curves/abstract/edwards';
import { Fp } from '@noble/curves/abstract/modular'; import { Fp } from '@noble/curves/abstract/modular';
import { sha512 } from '@noble/hashes/sha512'; import { sha512 } from '@noble/hashes/sha512';
import { randomBytes } from '@noble/hashes/utils';
const fp = Fp(2n ** 255n - 19n); const fp = Fp(2n ** 255n - 19n);
const ed25519 = twistedEdwards({ const ed25519 = twistedEdwards({
@ -440,9 +396,23 @@ const ed25519 = twistedEdwards({
} as const); } as const);
``` ```
`twistedEdwards()` returns `CurveFn` of following type: Twisted Edwards curve's formula is `ax² + y² = 1 + dx²y²`. You must specify `a`, `d`, field `Fp`, order `n`, cofactor `h`
and coordinates `Gx`, `Gy` of generator point.
For EdDSA signatures, `hash` param required. `adjustScalarBytes` which instructs how to change private scalars could be specified.
**Edwards points:**
1. Exported as `ExtendedPoint`
2. Represented in extended coordinates: (x, y, z, t) ∋ (x=x/z, y=y/z)
3. Use complete exception-free formulas for addition and doubling
4. Can be decoded/encoded from/to Uint8Array / hex strings using `ExtendedPoint.fromHex` and `ExtendedPoint#toRawBytes()`
5. Have `assertValidity()` which checks for being on-curve
6. Have `toAffine()` and `x` / `y` getters which convert to 2d xy affine coordinates
7. Have `isTorsionFree()`, `clearCofactor()` and `isSmallOrder()` utilities to handle torsions
```ts ```ts
// `twistedEdwards()` returns `CurveFn` of following type:
type CurveFn = { type CurveFn = {
CURVE: ReturnType<typeof validateOpts>; CURVE: ReturnType<typeof validateOpts>;
getPublicKey: (privateKey: Hex) => Uint8Array; getPublicKey: (privateKey: Hex) => Uint8Array;
@ -460,15 +430,31 @@ type CurveFn = {
}; };
}; };
}; };
interface ExtPointType extends Group<ExtPointType> {
readonly ex: bigint;
readonly ey: bigint;
readonly ez: bigint;
readonly et: bigint;
assertValidity(): void;
multiply(scalar: bigint): ExtPointType;
multiplyUnsafe(scalar: bigint): ExtPointType;
isSmallOrder(): boolean;
isTorsionFree(): boolean;
clearCofactor(): ExtPointType;
toAffine(iz?: bigint): AffinePoint<bigint>;
}
// Static methods of Extended Point with coordinates in X, Y, Z, T
interface ExtPointConstructor extends GroupConstructor<ExtPointType> {
new (x: bigint, y: bigint, z: bigint, t: bigint): ExtPointType;
fromAffine(p: AffinePoint<bigint>): ExtPointType;
fromHex(hex: Hex): ExtPointType;
fromPrivateKey(privateKey: Hex): ExtPointType;
}
``` ```
### abstract/montgomery: Montgomery curve ### abstract/montgomery: Montgomery curve
The module contains methods for x-only ECDH on Curve25519 / Curve448 from RFC7748.
Proper Elliptic Curve Points are not implemented yet.
You must specify curve params `Fp`, `a`, `Gu` coordinate of u, `montgomeryBits` and `nByteLength`.
```typescript ```typescript
import { montgomery } from '@noble/curves/abstract/montgomery'; import { montgomery } from '@noble/curves/abstract/montgomery';
import { Fp } from '@noble/curves/abstract/modular'; import { Fp } from '@noble/curves/abstract/modular';
@ -489,6 +475,11 @@ const x25519 = montgomery({
}); });
``` ```
The module contains methods for x-only ECDH on Curve25519 / Curve448 from RFC7748.
Proper Elliptic Curve Points are not implemented yet.
You must specify curve params `Fp`, `a`, `Gu` coordinate of u, `montgomeryBits` and `nByteLength`.
### abstract/hash-to-curve: Hashing strings to curve points ### abstract/hash-to-curve: Hashing strings to curve points
The module allows to hash arbitrary strings to elliptic curve points. Implements [hash-to-curve v16](https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-16). The module allows to hash arbitrary strings to elliptic curve points. Implements [hash-to-curve v16](https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-16).
@ -529,19 +520,37 @@ function expand_message_xof(
): Uint8Array; ): Uint8Array;
``` ```
`hash_to_field(msg, count, options)` [(spec)](https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-5.3) `hash_to_field(msg, count, options)`
[(spec)](https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-5.3)
hashes arbitrary-length byte strings to a list of one or more elements of a finite field F. hashes arbitrary-length byte strings to a list of one or more elements of a finite field F.
- `msg` a byte string containing the message to hash
- `count` the number of elements of F to output
- `options` `{DST: string, p: bigint, m: number, k: number, expand: 'xmd' | 'xof', hash: H}`.
- `p` is field prime, m=field extension (1 for prime fields)
- `k` is security target in bits (e.g. 128).
- `expand` should be `xmd` for SHA2, SHA3, BLAKE; `xof` for SHAKE, BLAKE-XOF
- `hash` conforming to `utils.CHash` interface, with `outputLen` / `blockLen` props
- Returns `[u_0, ..., u_(count - 1)]`, a list of field elements.
```ts ```ts
/**
* * `DST` is a domain separation tag, defined in section 2.2.5
* * `p` characteristic of F, where F is a finite field of characteristic p and order q = p^m
* * `m` is extension degree (1 for prime fields)
* * `k` is the target security target in bits (e.g. 128), from section 5.1
* * `expand` is `xmd` (SHA2, SHA3, BLAKE) or `xof` (SHAKE, BLAKE-XOF)
* * `hash` conforming to `utils.CHash` interface, with `outputLen` / `blockLen` props
*/
type UnicodeOrBytes = string | Uint8Array;
type Opts = {
DST: UnicodeOrBytes;
p: bigint;
m: number;
k: number;
expand?: 'xmd' | 'xof';
hash: CHash;
};
/**
* Hashes arbitrary-length byte strings to a list of one or more elements of a finite field F
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-5.3
* @param msg a byte string containing the message to hash
* @param count the number of elements of F to output
* @param options `{DST: string, p: bigint, m: number, k: number, expand: 'xmd' | 'xof', hash: H}`, see above
* @returns [u_0, ..., u_(count - 1)], a list of field elements.
*/
function hash_to_field(msg: Uint8Array, count: number, options: Opts): bigint[][]; function hash_to_field(msg: Uint8Array, count: number, options: Opts): bigint[][];
``` ```