noble-curves/README.md

noble-curves

Minimal, auditable JS implementation of elliptic curve cryptography.

• Short Weierstrass, Edwards, Montgomery curves
• ECDSA, EdDSA, Schnorr, BLS signature schemes, ECDH key agreement
• hash to curve for encoding or hashing an arbitrary string to a point on an elliptic curve
• Poseidon ZK-friendly hash
• 🏎 Ultra-fast, hand-optimized for caveats of JS engines
• 🔍 Unique tests ensure correctness. Wycheproof vectors included
• 🔻 Tree-shaking-friendly: there is no entry point, which ensures small size of your app

Package consists of two parts:

1. abstract/ directory specifies zero-dependency EC algorithms
2. root directory utilizes one dependency @noble/hashes and provides ready-to-use:
   • NIST curves secp192r1/P192, secp224r1/P224, secp256r1/P256, secp384r1/P384, secp521r1/P521
   • SECG curve secp256k1
   • pairing-friendly curves bls12-381, bn254
   • ed25519/curve25519/x25519/ristretto, edwards448/curve448/x448 RFC7748 / RFC8032 / ZIP215 stuff

Curves incorporate work from previous noble packages (secp256k1, ed25519), which had security audits and were developed from 2019 to 2022. Check out Upgrading section if you've used them before.

This library belongs to noble crypto

noble-crypto — high-security, easily auditable set of contained cryptographic libraries and tools.

• Protection against supply chain attacks
• Easily auditable TypeScript/JS code
• Supported in all major browsers and stable node.js versions
• All releases are signed with PGP keys
• Check out homepage & all libraries: curves (secp256k1, ed25519), hashes

Usage

Use NPM in node.js / browser, or include single file from GitHub's releases page:

npm install @noble/curves

The library does not have an entry point. It allows you to select specific primitives and drop everything else. If you only want to use secp256k1, just use the library with rollup or other bundlers. This is done to make your bundles tiny. All curves:

import { secp256k1 } from '@noble/curves/secp256k1';
import { ed25519, ed25519ph, ed25519ctx, x25519, RistrettoPoint } from '@noble/curves/ed25519';
import { ed448, ed448ph, ed448ctx, x448 } from '@noble/curves/ed448';
import { p256 } from '@noble/curves/p256';
import { p384 } from '@noble/curves/p384';
import { p521 } from '@noble/curves/p521';
import { pallas, vesta } from '@noble/curves/pasta';
import * as stark from '@noble/curves/stark';
import { bls12_381 } from '@noble/curves/bls12-381';
import { bn254 } from '@noble/curves/bn';
import { jubjub } from '@noble/curves/jubjub';

Every curve can be used in the following way:

import { secp256k1 } from '@noble/curves/secp256k1'; // Common.js and ECMAScript Modules (ESM)

const key = secp256k1.utils.randomPrivateKey();
const pub = secp256k1.getPublicKey(key);
const msg = new Uint8Array(32).fill(1);
const sig = secp256k1.sign(msg, key);
// weierstrass curves should use extraEntropy: https://moderncrypto.org/mail-archive/curves/2017/000925.html
const sigImprovedSecurity = secp256k1.sign(msg, key, { extraEntropy: true });
secp256k1.verify(sig, msg, pub) === true;
// secp, p*, pasta curves allow pub recovery
sig.recoverPublicKey(msg) === pub;
const someonesPub = secp256k1.getPublicKey(secp256k1.utils.randomPrivateKey());
const shared = secp256k1.getSharedSecret(key, someonesPub);

To define a custom curve, check out docs below.

API

• Overview
• abstract/edwards: Twisted Edwards curve
• abstract/montgomery: Montgomery curve
• abstract/weierstrass: Short Weierstrass curve
• abstract/hash-to-curve: Hashing strings to curve points
• abstract/poseidon: Poseidon hash
• abstract/modular
• abstract/utils

Overview

There are following zero-dependency abstract algorithms:

import { bls } from '@noble/curves/abstract/bls';
import { twistedEdwards } from '@noble/curves/abstract/edwards';
import { montgomery } from '@noble/curves/abstract/montgomery';
import { weierstrass } from '@noble/curves/abstract/weierstrass';
import * as mod from '@noble/curves/abstract/modular';
import * as utils from '@noble/curves/abstract/utils';

They allow to define a new curve in a few lines of code:

import { Field } from '@noble/curves/abstract/modular';
import { weierstrass } from '@noble/curves/abstract/weierstrass';
import { hmac } from '@noble/hashes/hmac';
import { sha256 } from '@noble/hashes/sha256';
import { concatBytes, randomBytes } from '@noble/hashes/utils';

// secq (NOT secp) 256k1: cycle of secp256k1 with Fp/N flipped.
// https://zcash.github.io/halo2/background/curves.html#cycles-of-curves
// https://personaelabs.org/posts/spartan-ecdsa
const secq256k1 = weierstrass({
  a: 0n,
  b: 7n,
  Fp: Field(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,
});

• To initialize new curve, you must specify its variables, order (number of points on curve), field prime (over which the modular division would be done)
• All curves expose same generic interface:
  • getPublicKey(), sign(), verify() functions
  • Point conforming to Group interface with add/multiply/double/negate/add/equals methods
  • CURVE object with curve variables like Gx, Gy, Fp (field), n (order)
  • utils object with randomPrivateKey(), mod(), invert() methods (mod CURVE.P)
• All arithmetics is done with JS bigints over finite fields, which is defined from modular sub-module
• Many features require hashing, which is not provided. @noble/hashes can be used for this purpose. Any other library must conform to the CHash interface:

  export type CHash = {
    (message: Uint8Array): Uint8Array;
    blockLen: number;
    outputLen: number;
    create(): any;
  };
  

• w-ary non-adjacent form (wNAF) method with constant-time adjustments is used for point multiplication. It is possible to enable precomputes for edwards & weierstrass curves. Precomputes are calculated once (takes ~20-40ms), after that most G base point multiplications: for example, getPublicKey(), sign() and similar methods - would be much faster. Use curve.utils.precompute() to adjust precomputation window size
• You could use optional special params to tune performance:
  • Fp({sqrt}) square root calculation, used for point decompression
  • endo endomorphism options for Koblitz curves

abstract/edwards: Twisted Edwards curve

Twisted Edwards curve's formula is: ax² + y² = 1 + dx²y².

• You must specify curve params a, d, field Fp, order n, cofactor h and coordinates Gx, Gy of generator point
• For EdDSA signatures, params hash is also required. adjustScalarBytes which instructs how to change private scalars could be specified

import { twistedEdwards } from '@noble/curves/abstract/edwards';
import { div } from '@noble/curves/abstract/modular';
import { sha512 } from '@noble/hashes/sha512';

const ed25519 = twistedEdwards({
  a: -1n,
  d: div(-121665n, 121666n, 2n ** 255n - 19n), // -121665n/121666n
  P: 2n ** 255n - 19n,
  n: 2n ** 252n + 27742317777372353535851937790883648493n,
  h: 8n,
  Gx: 15112221349535400772501151409588531511454012693041857206046113283949847762202n,
  Gy: 46316835694926478169428394003475163141307993866256225615783033603165251855960n,
  hash: sha512,
  randomBytes,
  adjustScalarBytes(bytes) {
    // optional in general, mandatory in ed25519
    bytes[0] &= 248;
    bytes[31] &= 127;
    bytes[31] |= 64;
    return bytes;
  },
} as const);
const key = ed25519.utils.randomPrivateKey();
const pub = ed25519.getPublicKey(key);
const msg = new TextEncoder().encode('hello world'); // strings not accepted, must be Uint8Array
const sig = ed25519.sign(msg, key);
ed25519.verify(sig, msg, pub) === true;

twistedEdwards() returns CurveFn of following type:

export type CurveFn = {
  CURVE: ReturnType<typeof validateOpts>;
  getPublicKey: (privateKey: PrivKey, isCompressed?: boolean) => Uint8Array;
  sign: (message: Hex, privateKey: Hex) => Uint8Array;
  verify: (sig: SigType, message: Hex, publicKey: PubKey) => boolean;
  Point: PointConstructor;
  ExtendedPoint: ExtendedPointConstructor;
  Signature: SignatureConstructor;
  utils: {
    randomPrivateKey: () => Uint8Array;
    getExtendedPublicKey: (key: PrivKey) => {
      head: Uint8Array;
      prefix: Uint8Array;
      scalar: bigint;
      point: PointType;
      pointBytes: Uint8Array;
    };
  };
};

abstract/montgomery: Montgomery curve

For now the module only contains methods for x-only ECDH on Curve25519 / Curve448 from RFC7748.

Proper Elliptic Curve Points are not implemented yet.

You must specify curve field, a24 special variable, montgomeryBits, nByteLength, and coordinate u of generator point.

import { montgomery } from '@noble/curves/abstract/montgomery';

const x25519 = montgomery({
  P: 2n ** 255n - 19n,
  a24: 121665n, // TODO: change to a
  montgomeryBits: 255,
  nByteLength: 32,
  Gu: '0900000000000000000000000000000000000000000000000000000000000000',

  // Optional params
  powPminus2: (x: bigint): bigint => {
    return mod.pow(x, P - 2, P);
  },
  adjustScalarBytes(bytes) {
    bytes[0] &= 248;
    bytes[31] &= 127;
    bytes[31] |= 64;
    return bytes;
  },
});

abstract/weierstrass: Short Weierstrass curve

Short Weierstrass curve's formula is: y² = x³ + ax + b. Uses deterministic ECDSA from RFC6979. You can also specify extraEntropy in sign().

• You must specify curve params: a, b, field Fp, order n, cofactor h and coordinates Gx, Gy of generator point
• For ECDSA, you must specify hash, hmac. It is also possible to recover keys from signatures
• For ECDH, use getSharedSecret(privKeyA, pubKeyB)
• Optional params are lowS (default value) and endo (endomorphism)

import { Fp } from '@noble/curves/abstract/modular';
import { weierstrass } from '@noble/curves/abstract/weierstrass'; // Short Weierstrass curve
import { sha256 } from '@noble/hashes/sha256';
import { hmac } from '@noble/hashes/hmac';
import { concatBytes, randomBytes } from '@noble/hashes/utils';

const secp256k1 = weierstrass({
  a: 0n,
  b: 7n,
  Fp: Fp(2n ** 256n - 2n ** 32n - 2n ** 9n - 2n ** 8n - 2n ** 7n - 2n ** 6n - 2n ** 4n - 1n),
  n: 2n ** 256n - 432420386565659656852420866394968145599n,
  Gx: 55066263022277343669578718895168534326250603453777594175500187360389116729240n,
  Gy: 32670510020758816978083085130507043184471273380659243275938904335757337482424n,
  hash: sha256,
  hmac: (k: Uint8Array, ...msgs: Uint8Array[]) => hmac(sha256, key, concatBytes(...msgs)),
  randomBytes,

  // Optional params
  h: 1n, // Cofactor
  lowS: true, // Allow only low-S signatures by default in sign() and verify()
  endo: {
    // Endomorphism options for Koblitz curve
    // Beta param
    beta: 0x7ae96a2b657c07106e64479eac3434e99cf0497512f58995c1396c28719501een,
    // Split scalar k into k1, k2
    splitScalar: (k: bigint) => {
      // return { k1neg: true, k1: 512n, k2neg: false, k2: 448n };
    },
  },
});

// Usage
const key = secp256k1.utils.randomPrivateKey();
const pub = secp256k1.getPublicKey(key);
const msg = randomBytes(32);
const sig = secp256k1.sign(msg, key);
secp256k1.verify(sig, msg, pub); // true
sig.recoverPublicKey(msg); // == pub
const someonesPubkey = secp256k1.getPublicKey(secp256k1.utils.randomPrivateKey());
const shared = secp256k1.getSharedSecret(key, someonesPubkey);

weierstrass() returns CurveFn:

export 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 }
  ) => boolean;
  Point: PointConstructor;
  ProjectivePoint: ProjectivePointConstructor;
  Signature: SignatureConstructor;
  utils: {
    isValidPrivateKey(privateKey: PrivKey): boolean;
    hashToPrivateKey: (hash: Hex) => Uint8Array;
    randomPrivateKey: () => Uint8Array;
  };
};

abstract/hash-to-curve: Hashing strings to curve points

The module allows to hash arbitrary strings to elliptic curve points.

• expand_message_xmd (spec) produces a uniformly random byte string using a cryptographic hash function H that outputs b bits..

  function expand_message_xmd(
    msg: Uint8Array,
    DST: Uint8Array,
    lenInBytes: number,
    H: CHash
  ): Uint8Array;
  function expand_message_xof(
    msg: Uint8Array,
    DST: Uint8Array,
    lenInBytes: number,
    k: number,
    H: CHash
  ): Uint8Array;
  

• hash_to_field(msg, count, options) (spec) 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} _ Returns [u_0, ..., u_(count - 1)], a list of field elements.

  ```ts
  function hash_to_field(msg: Uint8Array, count: number, options: htfOpts): bigint[][];
  type htfOpts = {
    // DST: a domain separation tag
    // defined in section 2.2.5
    DST: string;
    // p: the characteristic of F
    //    where F is a finite field of characteristic p and order q = p^m
    p: bigint;
    // m: the extension degree of F, m >= 1
    //     where F is a finite field of characteristic p and order q = p^m
    m: number;
    // k: the target security level for the suite in bits
    // defined in section 5.1
    k: number;
    // option to use a message that has already been processed by
    // expand_message_xmd
    expand?: 'xmd' | 'xof';
    // Hash functions for: expand_message_xmd is appropriate for use with a
    // wide range of hash functions, including SHA-2, SHA-3, BLAKE2, and others.
    // BBS+ uses blake2: https://github.com/hyperledger/aries-framework-go/issues/2247
    // TODO: verify that hash is shake if expand==='xof' via types
    hash: CHash;
  };
  ```
  

abstract/poseidon: Poseidon hash

Implements Poseidon ZK-friendly hash.

There are many poseidon instances with different constants. We don't provide them, but we provide ability to specify them manually. For actual usage, check out stark curve source code.

import { poseidon } from '@noble/curves/abstract/poseidon';

type PoseidonOpts = {
  Fp: Field<bigint>;
  t: number;
  roundsFull: number;
  roundsPartial: number;
  sboxPower?: number;
  reversePartialPowIdx?: boolean; // Hack for stark
  mds: bigint[][];
  roundConstants: bigint[][];
};
const instance = poseidon(opts: PoseidonOpts);

abstract/modular

Modular arithmetics utilities.

import { Fp, mod, invert, div, invertBatch, sqrt } from '@noble/curves/abstract/modular';
const fp = Fp(2n ** 255n - 19n); // Finite field over 2^255-19
fp.mul(591n, 932n);
fp.pow(481n, 11024858120n);

// Generic non-FP utils are also available
mod(21n, 10n); // 21 mod 10 == 1n; fixed version of 21 % 10
invert(17n, 10n); // invert(17) mod 10; modular multiplicative inverse
div(5n, 17n, 10n); // 5/17 mod 10 == 5 * invert(17) mod 10; division
invertBatch([1n, 2n, 4n], 21n); // => [1n, 11n, 16n] in one inversion
sqrt(21n, 73n); // √21 mod 73; square root

abstract/utils

import * as utils from '@noble/curves/abstract/utils';

utils.bytesToHex(Uint8Array.from([0xde, 0xad, 0xbe, 0xef]));
utils.hexToBytes('deadbeef');
utils.hexToNumber();
utils.bytesToNumberBE(Uint8Array.from([0xde, 0xad, 0xbe, 0xef]));
utils.bytesToNumberLE(Uint8Array.from([0xde, 0xad, 0xbe, 0xef]));
utils.numberToBytesBE(123n);
utils.numberToBytesLE(123n);
utils.numberToHexUnpadded(123n);
utils.concatBytes(Uint8Array.from([0xde, 0xad]), Uint8Array.from([0xbe, 0xef]));
utils.nLength(255n);
utils.hashToPrivateScalar(sha512_of_something, secp256r1.n);
utils.equalBytes(Uint8Array.from([0xde]), Uint8Array.from([0xde]));

Security

The library had no prior security audit.

Timing attack considerations: JIT-compiler and Garbage Collector make "constant time" extremely hard to achieve in a scripting language. Which means any other JS library can't have constant-timeness. Even statically typed Rust, a language without GC, makes it harder to achieve constant-time for some cases. If your goal is absolute security, don't use any JS lib — including bindings to native ones. Use low-level libraries & languages. Nonetheless we're targetting algorithmic constant time.

We consider infrastructure attacks like rogue NPM modules very important; that's why it's crucial to minimize the amount of 3rd-party dependencies & native bindings. If your app uses 500 dependencies, any dep could get hacked and you'll be downloading malware with every npm install. Our goal is to minimize this attack vector.

Speed

Benchmark results on Apple M2 with node v18.10:

secp256k1
init x 57 ops/sec @ 17ms/op
getPublicKey x 4,946 ops/sec @ 202μs/op
sign x 3,914 ops/sec @ 255μs/op
verify x 682 ops/sec @ 1ms/op
getSharedSecret x 427 ops/sec @ 2ms/op
recoverPublicKey x 683 ops/sec @ 1ms/op
schnorr.sign x 539 ops/sec @ 1ms/op
schnorr.verify x 716 ops/sec @ 1ms/op

P256
init x 30 ops/sec @ 32ms/op
getPublicKey x 5,008 ops/sec @ 199μs/op
sign x 3,970 ops/sec @ 251μs/op
verify x 515 ops/sec @ 1ms/op

P384
init x 14 ops/sec @ 66ms/op
getPublicKey x 2,434 ops/sec @ 410μs/op
sign x 1,942 ops/sec @ 514μs/op
verify x 206 ops/sec @ 4ms/op

P521
init x 7 ops/sec @ 126ms/op
getPublicKey x 1,282 ops/sec @ 779μs/op
sign x 1,077 ops/sec @ 928μs/op
verify x 110 ops/sec @ 9ms/op

ed25519
init x 37 ops/sec @ 26ms/op
getPublicKey x 8,147 ops/sec @ 122μs/op
sign x 3,979 ops/sec @ 251μs/op
verify x 848 ops/sec @ 1ms/op

ed448
init x 17 ops/sec @ 58ms/op
getPublicKey x 3,083 ops/sec @ 324μs/op
sign x 1,473 ops/sec @ 678μs/op
verify x 323 ops/sec @ 3ms/op

bls12-381
init x 30 ops/sec @ 33ms/op
getPublicKey x 788 ops/sec @ 1ms/op
sign x 45 ops/sec @ 21ms/op
verify x 32 ops/sec @ 30ms/op
pairing x 88 ops/sec @ 11ms/op

stark
init x 31 ops/sec @ 31ms/op
pedersen
├─old x 84 ops/sec @ 11ms/op
└─noble x 802 ops/sec @ 1ms/op
poseidon x 7,466 ops/sec @ 133μs/op
verify
├─old x 300 ops/sec @ 3ms/op
└─noble x 474 ops/sec @ 2ms/op

Upgrading

If you're coming from single-curve noble packages, the following changes need to be kept in mind:

• 2d affine (x, y) points have been removed to reduce complexity and improve speed
• Removed number support as a type for private keys. bigint is still supported
• mod, invert are no longer present in utils. Use @noble/curves/abstract/modular.js now.

Upgrading from @noble/secp256k1 1.7:

• Compressed (33-byte) public keys are now returned by default, instead of uncompressed
• Methods are now synchronous. Setting secp.utils.hmacSha256 is no longer required
• sign()
  • der, recovered options were removed
  • canonical was renamed to lowS
  • Return type is now { r: bigint, s: bigint, recovery: number } instance of Signature
• verify()
  • strict was renamed to lowS
• recoverPublicKey(): moved to sig instance Signature#recoverPublicKey(msgHash)
• Point was removed: use ProjectivePoint in xyz coordinates
• utils: Many methods were removed, others were moved to schnorr namespace

Upgrading from @noble/ed25519 1.7:

• Methods are now synchronous. Setting secp.utils.hmacSha256 is no longer required
• ed25519ph, ed25519ctx
• Point was removed: use ExtendedPoint in xyzt coordinates
• Signature was removed
• getSharedSecret was removed: use separate x25519 sub-module
• bigint is no longer allowed in getPublicKey, sign, verify. Reason: ed25519 is LE, can lead to bugs

Contributing & testing

1. Clone the repository
2. npm install to install build dependencies like TypeScript
3. npm run build to compile TypeScript code
4. npm run test will execute all main tests

License

The MIT License (MIT)

Copyright (c) 2022 Paul Miller (https://paulmillr.com)

See LICENSE file.