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https://github.com/paulmillr/noble-curves

Audited & minimal JS implementation of elliptic curve cryptography.
https://github.com/paulmillr/noble-curves

bitcoin bls12-381 bn128 bn254 crypto cryptography ecdsa ed25519 ed448 eddsa elliptic-curve-cryptography elliptic-curves ethereum jubjub noble p256 p384 p521 secp256k1 x448

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Audited & minimal JS implementation of elliptic curve cryptography.

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README 

        
# noble-curves



Audited & minimal JS implementation of elliptic curve cryptography.



- 🔒 [**Audited**](#security) by independent security firms

- 🔻 Tree-shakeable: unused code is excluded from your builds

- 🏎 Fast: hand-optimized for caveats of JS engines

- 🔍 Reliable: cross-library / wycheproof tests and fuzzing ensure correctness

- ➰ Short Weierstrass, Edwards, Montgomery curves

- ✍️ ECDSA, EdDSA, Schnorr, BLS, ECDH, hashing to curves, Poseidon ZK-friendly hash

- 🔖 SUF-CMA, SBS (non-repudiation), ZIP215 (consensus friendliness) features for ed25519 & ed448

- 🪶 93KB for everything with hashes, 26KB (11KB gzipped) for single-curve build



Curves have 4KB sister projects

[secp256k1](https://github.com/paulmillr/noble-secp256k1) & [ed25519](https://github.com/paulmillr/noble-ed25519).

They have smaller attack surface, but less features.



Take a glance at [GitHub Discussions](https://github.com/paulmillr/noble-curves/discussions) for questions and support.



### This library belongs to _noble_ cryptography



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



- Zero or minimal dependencies

- Highly readable TypeScript / JS code

- PGP-signed releases and transparent NPM builds

- All libraries:

  [ciphers](https://github.com/paulmillr/noble-ciphers),

  [curves](https://github.com/paulmillr/noble-curves),

  [hashes](https://github.com/paulmillr/noble-hashes),

  [post-quantum](https://github.com/paulmillr/noble-post-quantum),

  4kb [secp256k1](https://github.com/paulmillr/noble-secp256k1) /

  [ed25519](https://github.com/paulmillr/noble-ed25519)

- [Check out homepage](https://paulmillr.com/noble/)

  for reading resources, documentation and apps built with noble



## Usage



> `npm install @noble/curves`



> `deno add jsr:@noble/curves`



> `deno doc jsr:@noble/curves` # command-line documentation



We support all major platforms and runtimes.

For React Native, you may need a [polyfill for getRandomValues](https://github.com/LinusU/react-native-get-random-values).

A standalone file [noble-curves.js](https://github.com/paulmillr/noble-curves/releases) is also available.



```ts

// import * from '@noble/curves'; // Error: use sub-imports, to ensure small app size

import { secp256k1, schnorr } from '@noble/curves/secp256k1';

import { ed25519, ed25519ph, ed25519ctx, x25519 } 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 { bls12_381 } from '@noble/curves/bls12-381';

import { bn254 } from '@noble/curves/bn254'; // also known as alt_bn128

import { jubjub } from '@noble/curves/jubjub';

import { bytesToHex, hexToBytes, concatBytes, utf8ToBytes } from '@noble/curves/abstract/utils';

```



- [ECDSA signatures over secp256k1 and others](#ecdsa-signatures-over-secp256k1-and-others)

- [Hedged ECDSA with noise](#hedged-ecdsa-with-noise)

- [ECDH: Diffie-Hellman shared secrets](#ecdh-diffie-hellman-shared-secrets)

- [secp256k1 Schnorr signatures from BIP340](#secp256k1-schnorr-signatures-from-bip340)

- [ed25519](#ed25519) / [X25519](#x25519) / [ristretto255](#ristretto255)

- [ed448](#ed448) / [X448](#x448) / [decaf448](#decaf448)

- [bls12-381](#bls12-381)

- [bn254 aka alt_bn128](#bn254-aka-alt_bn128)

- [Low-level methods](#low-level-methods)

- [Abstract API](#abstract-api)

  - [weierstrass](#weierstrass-short-weierstrass-curve), [edwards](#edwards-twisted-edwards-curve), [montgomery](#montgomery-montgomery-curve), [bls](#bls-barreto-lynn-scott-curves)

  - [hash-to-curve](#hash-to-curve-hashing-strings-to-curve-points), [poseidon](#poseidon-poseidon-hash)

  - [modular](#modular-modular-arithmetics-utilities), [utils](#utils-useful-utilities)

- [Security](#security)

- [Speed](#speed)

- [Upgrading](#upgrading)

- [Contributing & testing](#contributing--testing)

- [License](#license)



### Implementations



#### ECDSA signatures over secp256k1 and others



```ts

import { secp256k1 } from '@noble/curves/secp256k1';

// import { p256 } from '@noble/curves/p256'; // or p384 / p521



const priv = secp256k1.utils.randomPrivateKey();

const pub = secp256k1.getPublicKey(priv);

const msg = new Uint8Array(32).fill(1); // message hash (not message) in ecdsa

const sig = secp256k1.sign(msg, priv); // `{prehash: true}` option is available

const isValid = secp256k1.verify(sig, msg, pub) === true;



// hex strings are also supported besides Uint8Array-s:

const privHex = '46c930bc7bb4db7f55da20798697421b98c4175a52c630294d75a84b9c126236';

const pub2 = secp256k1.getPublicKey(privHex);



// public key recovery

// let sig = secp256k1.Signature.fromCompact(sigHex); // or .fromDER(sigDERHex)

// sig = sig.addRecoveryBit(bit); // bit is not serialized into compact / der format

sig.recoverPublicKey(msg).toRawBytes(); // === pub; // public key recovery

```



The same code would work for NIST P256 (secp256r1), P384 (secp384r1) & P521 (secp521r1).



#### Hedged ECDSA with noise



```ts

const noisySignature = secp256k1.sign(msg, priv, { extraEntropy: true });

const ent = new Uint8Array(32).fill(3); // set custom entropy

const noisySignature2 = secp256k1.sign(msg, priv, { extraEntropy: ent });

```



Hedged ECDSA is add-on, providing improved protection against fault attacks.

It adds noise to signatures. The technique is used by default in BIP340; we also implement them

optionally for ECDSA. Check out blog post

[Deterministic signatures are not your friends](https://paulmillr.com/posts/deterministic-signatures/)

and [spec draft](https://datatracker.ietf.org/doc/draft-irtf-cfrg-det-sigs-with-noise/).



#### ECDH: Diffie-Hellman shared secrets



```ts

const someonesPub = secp256k1.getPublicKey(secp256k1.utils.randomPrivateKey());

const shared = secp256k1.getSharedSecret(priv, someonesPub);

// NOTE:

// - `shared` includes parity byte: strip it using shared.slice(1)

// - `shared` is not hashed: more secure way is sha256(shared) or hkdf(shared)

```



#### secp256k1 Schnorr signatures from BIP340



```ts

import { schnorr } from '@noble/curves/secp256k1';

const priv = schnorr.utils.randomPrivateKey();

const pub = schnorr.getPublicKey(priv);

const msg = new TextEncoder().encode('hello');

const sig = schnorr.sign(msg, priv);

const isValid = schnorr.verify(sig, msg, pub);

```



#### ed25519



```ts

import { ed25519 } from '@noble/curves/ed25519';

const priv = ed25519.utils.randomPrivateKey();

const pub = ed25519.getPublicKey(priv);

const msg = new TextEncoder().encode('hello');

const sig = ed25519.sign(msg, priv);

ed25519.verify(sig, msg, pub); // Default mode: follows ZIP215

ed25519.verify(sig, msg, pub, { zip215: false }); // SBS / e-voting / RFC8032 / FIPS 186-5



// Variants from RFC8032: with context, prehashed

import { ed25519ctx, ed25519ph } from '@noble/curves/ed25519';

```



Default `verify` behavior follows ZIP215 and

can be used in consensus-critical applications.

If you need SBS (Strongly Binding Signatures) and FIPS 186-5 compliance,

use `zip215: false`. Check out [Edwards Signatures section for more info](#edwards-twisted-edwards-curve).

Both options have SUF-CMA (strong unforgeability under chosen message attacks).



#### X25519



```ts

// X25519 aka ECDH on Curve25519 from [RFC7748](https://www.rfc-editor.org/rfc/rfc7748)

import { x25519 } from '@noble/curves/ed25519';

const priv = 'a546e36bf0527c9d3b16154b82465edd62144c0ac1fc5a18506a2244ba449ac4';

const pub = 'e6db6867583030db3594c1a424b15f7c726624ec26b3353b10a903a6d0ab1c4c';

x25519.getSharedSecret(priv, pub) === x25519.scalarMult(priv, pub); // aliases

x25519.getPublicKey(priv) === x25519.scalarMultBase(priv);

x25519.getPublicKey(x25519.utils.randomPrivateKey());



// ed25519 => x25519 conversion

import { edwardsToMontgomeryPub, edwardsToMontgomeryPriv } from '@noble/curves/ed25519';

edwardsToMontgomeryPub(ed25519.getPublicKey(ed25519.utils.randomPrivateKey()));

edwardsToMontgomeryPriv(ed25519.utils.randomPrivateKey());

```



#### ristretto255



```ts

// ristretto255 from [RFC9496](https://www.rfc-editor.org/rfc/rfc9496)

import { utf8ToBytes } from '@noble/hashes/utils';

import { sha512 } from '@noble/hashes/sha512';

import {

  hashToCurve,

  encodeToCurve,

  RistrettoPoint,

  hashToRistretto255,

} from '@noble/curves/ed25519';



const msg = utf8ToBytes('Ristretto is traditionally a short shot of espresso coffee');

hashToCurve(msg);



const rp = RistrettoPoint.fromHex(

  '6a493210f7499cd17fecb510ae0cea23a110e8d5b901f8acadd3095c73a3b919'

);

RistrettoPoint.BASE.multiply(2n).add(rp).subtract(RistrettoPoint.BASE).toRawBytes();

RistrettoPoint.ZERO.equals(dp) === false;

// pre-hashed hash-to-curve

RistrettoPoint.hashToCurve(sha512(msg));

// full hash-to-curve including domain separation tag

hashToRistretto255(msg, { DST: 'ristretto255_XMD:SHA-512_R255MAP_RO_' });

```



#### ed448



```ts

import { ed448 } from '@noble/curves/ed448';

const priv = ed448.utils.randomPrivateKey();

const pub = ed448.getPublicKey(priv);

const msg = new TextEncoder().encode('whatsup');

const sig = ed448.sign(msg, priv);

ed448.verify(sig, msg, pub);



// Variants from RFC8032: prehashed

import { ed448ph } from '@noble/curves/ed448';

```



#### X448



```ts

// X448 aka ECDH on Curve448 from [RFC7748](https://www.rfc-editor.org/rfc/rfc7748)

import { x448 } from '@noble/curves/ed448';

x448.getSharedSecret(priv, pub) === x448.scalarMult(priv, pub); // aliases

x448.getPublicKey(priv) === x448.scalarMultBase(priv);



// ed448 => x448 conversion

import { edwardsToMontgomeryPub } from '@noble/curves/ed448';

edwardsToMontgomeryPub(ed448.getPublicKey(ed448.utils.randomPrivateKey()));

```



#### decaf448



```ts

// decaf448 from [RFC9496](https://www.rfc-editor.org/rfc/rfc9496)

import { utf8ToBytes } from '@noble/hashes/utils';

import { shake256 } from '@noble/hashes/sha3';

import { hashToCurve, encodeToCurve, DecafPoint, hashToDecaf448 } from '@noble/curves/ed448';



const msg = utf8ToBytes('Ristretto is traditionally a short shot of espresso coffee');

hashToCurve(msg);



const dp = DecafPoint.fromHex(

  'c898eb4f87f97c564c6fd61fc7e49689314a1f818ec85eeb3bd5514ac816d38778f69ef347a89fca817e66defdedce178c7cc709b2116e75'

);

DecafPoint.BASE.multiply(2n).add(dp).subtract(DecafPoint.BASE).toRawBytes();

DecafPoint.ZERO.equals(dp) === false;

// pre-hashed hash-to-curve

DecafPoint.hashToCurve(shake256(msg, { dkLen: 112 }));

// full hash-to-curve including domain separation tag

hashToDecaf448(msg, { DST: 'decaf448_XOF:SHAKE256_D448MAP_RO_' });

```



#### bls12-381



```ts

import { bls12_381 as bls } from '@noble/curves/bls12-381';



// G1 keys, G2 signatures

const privateKey = '67d53f170b908cabb9eb326c3c337762d59289a8fec79f7bc9254b584b73265c';

const message = '64726e3da8';

const publicKey = bls.getPublicKey(privateKey);

const signature = bls.sign(message, privateKey);

const isValid = bls.verify(signature, message, publicKey);

console.log({ publicKey, signature, isValid });



// G2 keys, G1 signatures

// getPublicKeyForShortSignatures(privateKey)

// signShortSignature(message, privateKey)

// verifyShortSignature(signature, message, publicKey)

// aggregateShortSignatures(signatures)



// Custom DST

const htfEthereum = { DST: 'BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_POP_' };

const signatureEth = bls.sign(message, privateKey, htfEthereum);

const isValidEth = bls.verify(signature, message, publicKey, htfEthereum);



// Aggregation

const aggregatedKey = bls.aggregatePublicKeys([

  bls.utils.randomPrivateKey(),

  bls.utils.randomPrivateKey(),

]);

// const aggregatedSig = bls.aggregateSignatures(sigs)



// Pairings, with and without final exponentiation

// bls.pairing(PointG1, PointG2);

// bls.pairing(PointG1, PointG2, false);

// bls.fields.Fp12.finalExponentiate(bls.fields.Fp12.mul(PointG1, PointG2));



// Others

// bls.G1.ProjectivePoint.BASE, bls.G2.ProjectivePoint.BASE;

// bls.fields.Fp, bls.fields.Fp2, bls.fields.Fp12, bls.fields.Fr;

```



See [abstract/bls](#bls-barreto-lynn-scott-curves).

For example usage, check out [the implementation of BLS EVM precompiles](https://github.com/ethereumjs/ethereumjs-monorepo/blob/361f4edbc239e795a411ac2da7e5567298b9e7e5/packages/evm/src/precompiles/bls12_381/noble.ts).



#### bn254 aka alt_bn128



```ts

import { bn254 } from '@noble/curves/bn254';



console.log(bn254.G1, bn254.G2, bn254.pairing);

```



The API mirrors [BLS](#bls12-381). The curve was previously called alt_bn128.

The implementation is compatible with [EIP-196](https://eips.ethereum.org/EIPS/eip-196) and

[EIP-197](https://eips.ethereum.org/EIPS/eip-197).



Keep in mind that we don't implement Point methods toHex / toRawBytes. It's because

different implementations of bn254 do it differently - there is no standard. Points of divergence:



- Endianness: LE vs BE (byte-swapped)

- Flags as first hex bits (similar to BLS) vs no-flags

- Imaginary part last in G2 vs first (c0, c1 vs c1, c0)



For example usage, check out [the implementation of bn254 EVM precompiles](https://github.com/paulmillr/noble-curves/blob/3ed792f8ad9932765b84d1064afea8663a255457/test/bn254.test.js#L697).



#### Low-level methods



```ts

import { secp256k1 } from '@noble/curves/secp256k1';



// Curve's variables

// Every curve has `CURVE` object that contains its parameters, field, and others

console.log(secp256k1.CURVE.p); // field modulus

console.log(secp256k1.CURVE.n); // curve order

console.log(secp256k1.CURVE.a, secp256k1.CURVE.b); // equation params

console.log(secp256k1.CURVE.Gx, secp256k1.CURVE.Gy); // base point coordinates



// MSM

const p = secp256k1.ProjectivePoint;

const points = [p.BASE, p.BASE.multiply(2n), p.BASE.multiply(4n), p.BASE.multiply(8n)];

p.msm(points, [3n, 5n, 7n, 11n]).equals(p.BASE.multiply(129n)); // 129*G

```



Multi-scalar-multiplication (MSM) is basically `(Pa + Qb + Rc + ...)`.

It's 10-30x faster vs naive addition for large amount of points.

Pippenger algorithm is used underneath.



## Abstract API



Implementations use [noble-hashes](https://github.com/paulmillr/noble-hashes).

If you want to use a different hashing library, abstract API doesn't depend on them.



Abstract API allows to define custom curves. All arithmetics is done with JS

bigints over finite fields, which is defined from `modular` sub-module. For

scalar multiplication, we use

[precomputed tables with w-ary non-adjacent form (wNAF)](https://paulmillr.com/posts/noble-secp256k1-fast-ecc/).

Precomputes are enabled for weierstrass and edwards BASE points of a curve. You

could precompute any other point (e.g. for ECDH) using `utils.precompute()`

method: check out examples.



### weierstrass: Short Weierstrass curve



```ts

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

import { Field } from '@noble/curves/abstract/modular';

import { sha256 } from '@noble/hashes/sha256';

import { hmac } from '@noble/hashes/hmac';

import { concatBytes, randomBytes } from '@noble/hashes/utils';



const hmacSha256 = (key: Uint8Array, ...msgs: Uint8Array[]) =>

  hmac(sha256, key, concatBytes(...msgs));



// secQ (not secP) - secq256k1 is a cycle of secp256k1 with Fp/N flipped.

// https://personaelabs.org/posts/spartan-ecdsa

// https://zcash.github.io/halo2/background/curves.html#cycles-of-curves

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: hmacSha256,

  randomBytes,

});



// NIST secp192r1 aka p192

// https://www.secg.org/sec2-v2.pdf, https://neuromancer.sk/std/secg/secp192r1

const secp192r1 = weierstrass({

  a: 0xfffffffffffffffffffffffffffffffefffffffffffffffcn,

  b: 0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1n,

  Fp: Field(0xfffffffffffffffffffffffffffffffeffffffffffffffffn),

  n: 0xffffffffffffffffffffffff99def836146bc9b1b4d22831n,

  Gx: 0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012n,

  Gy: 0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811n,

  hash: sha256,

  hmac: hmacSha256,

  randomBytes,

});

```



Short Weierstrass curve's formula is `y² = x³ + ax + b`. `weierstrass`

expects arguments `a`, `b`, field `Fp`, curve order `n`, cofactor `h`

and coordinates `Gx`, `Gy` of generator point.

`hmac` and `hash` must be specified for deterministic `k` generation.



**Weierstrass points:**



- Are exported as `ProjectivePoint`

- Are represented in projective (homogeneous) coordinates: (x, y, z) ∋ (x=x/z, y=y/z)

- Use complete exception-free formulas for addition and doubling

- Can be decoded/encoded from/to Uint8Array / hex strings using

  `ProjectivePoint.fromHex` and `ProjectivePoint#toRawBytes()`

- Have `assertValidity()` which checks for being on-curve

- Have `toAffine()` and `x` / `y` getters which convert to 2d xy affine coordinates



**ECDSA signatures:**



- Are represented by `Signature` instances with `r, s` and optional `recovery` properties

- Have `recoverPublicKey()`, `toCompactRawBytes()` and `toDERRawBytes()` methods

- Can be prehashed, or non-prehashed:

  - `sign(msgHash, privKey)` (default, prehash: false) - you did hashing before

  - `sign(msg, privKey, {prehash: true})` - curves will do hashing for you

- Are generated deterministically, following [RFC6979](https://www.rfc-editor.org/rfc/rfc6979).

  - Consider [hedged ECDSA with noise](#hedged-ecdsa-with-noise) for adding randomness into

    for signatures, to get improved security against fault attacks.



More examples:



```typescript

// All curves expose same generic interface.

const priv = secq256k1.utils.randomPrivateKey();

secq256k1.getPublicKey(priv); // Convert private key to public.

const sig = secq256k1.sign(msg, priv); // Sign msg with private key.

const sig2 = secq256k1.sign(msg, priv, { prehash: true }); // hash(msg)

secq256k1.verify(sig, msg, priv); // Verify if sig is correct.



// Default behavior is "try DER, then try compact if fails". Can be explicit:

secq256k1.verify(sig.toCompactHex(), msg, priv, { format: 'compact' });



const Point = secq256k1.ProjectivePoint;

const point = Point.BASE; // Elliptic curve Point class and BASE point static var.

point.add(point).equals(point.double()); // add(), equals(), double() methods

point.subtract(point).equals(Point.ZERO); // subtract() method, ZERO static var

point.negate(); // Flips point over x/y coordinate.

point.multiply(31415n); // Multiplication of Point by scalar.



point.assertValidity(); // Checks for being on-curve

point.toAffine(); // Converts to 2d affine xy coordinates



secq256k1.CURVE.n;

secq256k1.CURVE.p;

secq256k1.CURVE.Fp.mod();

secq256k1.CURVE.hash();



// precomputes

const fast = secq256k1.utils.precompute(8, Point.fromHex(someonesPubKey));

fast.multiply(privKey); // much faster ECDH now

```



### edwards: Twisted Edwards curve



```ts

import { twistedEdwards } from '@noble/curves/abstract/edwards';

import { Field } from '@noble/curves/abstract/modular';

import { sha512 } from '@noble/hashes/sha512';

import { randomBytes } from '@noble/hashes/utils';



const Fp = Field(2n ** 255n - 19n);

const ed25519 = twistedEdwards({

  a: Fp.create(-1n),

  d: Fp.div(-121665n, 121666n), // -121665n/121666n mod p

  Fp: Fp,

  n: 2n ** 252n + 27742317777372353535851937790883648493n,

  h: 8n,

  Gx: 15112221349535400772501151409588531511454012693041857206046113283949847762202n,

  Gy: 46316835694926478169428394003475163141307993866256225615783033603165251855960n,

  hash: sha512,

  randomBytes,

  adjustScalarBytes(bytes) {

    // optional; but mandatory in ed25519

    bytes[0] &= 248;

    bytes[31] &= 127;

    bytes[31] |= 64;

    return bytes;

  },

} as const);

```



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:**



- Are exported as `ExtendedPoint`

- Are represented in extended coordinates: (x, y, z, t) ∋ (x=x/z, y=y/z)

- Use complete exception-free formulas for addition and doubling

- Can be decoded/encoded from/to Uint8Array / hex strings using `ExtendedPoint.fromHex` and `ExtendedPoint#toRawBytes()`

- Have `assertValidity()` which checks for being on-curve

- Have `toAffine()` and `x` / `y` getters which convert to 2d xy affine coordinates

- Have `isTorsionFree()`, `clearCofactor()` and `isSmallOrder()` utilities to handle torsions



**EdDSA signatures:**



- `zip215: true` is default behavior. It has slightly looser verification logic

  to be [consensus-friendly](https://hdevalence.ca/blog/2020-10-04-its-25519am), following [ZIP215](https://zips.z.cash/zip-0215) rules

- `zip215: false` switches verification criteria to strict

  [RFC8032](https://www.rfc-editor.org/rfc/rfc8032) / [FIPS 186-5](https://csrc.nist.gov/publications/detail/fips/186/5/final)

  and additionally provides [non-repudiation with SBS](https://eprint.iacr.org/2020/1244),

  which is useful for:

  - Contract Signing: if A signed an agreement with B using key that allows repudiation, it can later claim that it signed a different contract

  - E-voting: malicious voters may pick keys that allow repudiation in order to deny results

  - Blockchains: transaction of amount X might also be valid for a different amount Y

- Both modes have SUF-CMA (strong unforgeability under chosen message attacks).



Check out [RFC9496](https://datatracker.ietf.org/doc/html/rfc9496) for description of

ristretto and decaf groups which we implement.



### montgomery: Montgomery curve



The module contains methods for x-only ECDH on Curve25519 / Curve448 from RFC7748.

Proper Elliptic Curve Points are not implemented yet.



### bls: Barreto-Lynn-Scott curves



The module abstracts BLS (Barreto-Lynn-Scott) pairing-friendly elliptic curve construction.

They allow to construct [zk-SNARKs](https://z.cash/technology/zksnarks/) and

use aggregated, batch-verifiable

[threshold signatures](https://medium.com/snigirev.stepan/bls-signatures-better-than-schnorr-5a7fe30ea716),

using Boneh-Lynn-Shacham signature scheme.



The module doesn't expose `CURVE` property: use `G1.CURVE`, `G2.CURVE` instead.

Only BLS12-381 is currently implemented.

Defining BLS12-377 and BLS24 should be straightforward.



The default BLS uses short public keys (with public keys in G1 and signatures in G2).

Short signatures (public keys in G2 and signatures in G1) are also supported.



### hash-to-curve: Hashing strings to curve points



The module allows to hash arbitrary strings to elliptic curve points. Implements [RFC 9380](https://www.rfc-editor.org/rfc/rfc9380).



Every curve has exported `hashToCurve` and `encodeToCurve` methods. You should always prefer `hashToCurve` for security:



```ts

import { hashToCurve, encodeToCurve } from '@noble/curves/secp256k1';

import { randomBytes } from '@noble/hashes/utils';

hashToCurve('0102abcd');

console.log(hashToCurve(randomBytes()));

console.log(encodeToCurve(randomBytes()));



import { bls12_381 } from '@noble/curves/bls12-381';

bls12_381.G1.hashToCurve(randomBytes(), { DST: 'another' });

bls12_381.G2.hashToCurve(randomBytes(), { DST: 'custom' });

```



Low-level methods from the spec:



```ts

// 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 // For CHash see abstract/weierstrass docs section

): Uint8Array;

// produces a uniformly random byte string using an extendable-output function (XOF) H.

function expand_message_xof(

  msg: Uint8Array,

  DST: Uint8Array,

  lenInBytes: number,

  k: number,

  H: CHash

): Uint8Array;

// Hashes arbitrary-length byte strings to a list of one or more elements of a finite field F

function hash_to_field(msg: Uint8Array, count: number, options: Opts): bigint[][];



/**

 * * `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;

};

```



### poseidon: Poseidon hash



Implements [Poseidon](https://www.poseidon-hash.info) ZK-friendly hash.



There are many poseidon variants with different constants.

We don't provide them: you should construct them manually.

Check out [micro-starknet](https://github.com/paulmillr/micro-starknet) package for a proper example.



```ts

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



type PoseidonOpts = {

  Fp: Field;

  t: number;

  roundsFull: number;

  roundsPartial: number;

  sboxPower?: number;

  reversePartialPowIdx?: boolean;

  mds: bigint[][];

  roundConstants: bigint[][];

};

const instance = poseidon(opts: PoseidonOpts);

```



### modular: Modular arithmetics utilities



```ts

import * as mod from '@noble/curves/abstract/modular';



// Finite Field utils

const fp = mod.Field(2n ** 255n - 19n); // Finite field over 2^255-19

fp.mul(591n, 932n); // multiplication

fp.pow(481n, 11024858120n); // exponentiation

fp.div(5n, 17n); // division: 5/17 mod 2^255-19 == 5 * invert(17)

fp.inv(5n); // modular inverse

fp.sqrt(21n); // square root



// Non-Field generic utils are also available

mod.mod(21n, 10n); // 21 mod 10 == 1n; fixed version of 21 % 10

mod.invert(17n, 10n); // invert(17) mod 10; modular multiplicative inverse

mod.invertBatch([1n, 2n, 4n], 21n); // => [1n, 11n, 16n] in one inversion

```



Field operations are not constant-time: they are using JS bigints, see [security](#security).

The fact is mostly irrelevant, but the important method to keep in mind is `pow`,

which may leak exponent bits, when used naïvely.



`mod.Field` is always **field over prime number**. Non-prime fields aren't supported for now.

We don't test for prime-ness for speed and because algorithms are probabilistic anyway.

Initializing a non-prime field could make your app suspectible to

DoS (infilite loop) on Tonelli-Shanks square root calculation.



Unlike `mod.inv`, `mod.invertBatch` won't throw on `0`: make sure to throw an error yourself.



#### Creating private keys from hashes



You can't simply make a 32-byte private key from a 32-byte hash.

Doing so will make the key [biased](https://research.kudelskisecurity.com/2020/07/28/the-definitive-guide-to-modulo-bias-and-how-to-avoid-it/).



To make the bias negligible, we follow [FIPS 186-5 A.2](https://csrc.nist.gov/publications/detail/fips/186/5/final)

and [RFC 9380](https://www.rfc-editor.org/rfc/rfc9380#section-5.2).

This means, for 32-byte key, we would need 48-byte hash to get 2^-128 bias, which matches curve security level.



`hashToPrivateScalar()` that hashes to **private key** was created for this purpose.

Use [abstract/hash-to-curve](#hash-to-curve-hashing-strings-to-curve-points)

if you need to hash to **public key**.



```ts

import { p256 } from '@noble/curves/p256';

import { sha256 } from '@noble/hashes/sha256';

import { hkdf } from '@noble/hashes/hkdf';

import * as mod from '@noble/curves/abstract/modular';

const someKey = new Uint8Array(32).fill(2); // Needs to actually be random, not .fill(2)

const derived = hkdf(sha256, someKey, undefined, 'application', 48); // 48 bytes for 32-byte priv

const validPrivateKey = mod.hashToPrivateScalar(derived, p256.CURVE.n);

```



### utils: Useful utilities



```ts

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



utils.bytesToHex(Uint8Array.from([0xde, 0xad, 0xbe, 0xef]));

utils.hexToBytes('deadbeef');

utils.numberToHexUnpadded(123n);

utils.hexToNumber();



utils.bytesToNumberBE(Uint8Array.from([0xde, 0xad, 0xbe, 0xef]));

utils.bytesToNumberLE(Uint8Array.from([0xde, 0xad, 0xbe, 0xef]));

utils.numberToBytesBE(123n, 32);

utils.numberToBytesLE(123n, 64);



utils.concatBytes(Uint8Array.from([0xde, 0xad]), Uint8Array.from([0xbe, 0xef]));

utils.nLength(255n);

utils.equalBytes(Uint8Array.from([0xde]), Uint8Array.from([0xde]));

```



## Security



The library has been independently audited:



- at version 1.6.0, in Sep 2024, by [Cure53](https://cure53.de)

  - PDFs: [website](https://cure53.de/audit-report_noble-crypto-libs.pdf), [in-repo](./audit/2024-09-cure53-audit-nbl4.pdf)

  - [Changes since audit](https://github.com/paulmillr/noble-curves/compare/1.6.0..main)

  - Scope: ed25519, ed448, their add-ons, bls12-381, bn254,

    hash-to-curve, low-level primitives bls, tower, edwards, montgomery.

  - The audit has been funded by [OpenSats](https://opensats.org)

- at version 1.2.0, in Sep 2023, by [Kudelski Security](https://kudelskisecurity.com)

  - PDFs: [in-repo](./audit/2023-09-kudelski-audit-starknet.pdf)

  - [Changes since audit](https://github.com/paulmillr/noble-curves/compare/1.2.0..main)

  - Scope: [scure-starknet](https://github.com/paulmillr/scure-starknet) and its related

    abstract modules of noble-curves: `curve`, `modular`, `poseidon`, `weierstrass`

  - The audit has been funded by [Starkware](https://starkware.co)

- at version 0.7.3, in Feb 2023, by [Trail of Bits](https://www.trailofbits.com)

  - PDFs: [website](https://github.com/trailofbits/publications/blob/master/reviews/2023-01-ryanshea-noblecurveslibrary-securityreview.pdf),

    [in-repo](./audit/2023-01-trailofbits-audit-curves.pdf)

  - [Changes since audit](https://github.com/paulmillr/noble-curves/compare/0.7.3..main)

  - Scope: abstract modules `curve`, `hash-to-curve`, `modular`, `poseidon`, `utils`, `weierstrass` and

    top-level modules `_shortw_utils` and `secp256k1`

  - The audit has been funded by [Ryan Shea](https://www.shea.io)



It is tested against property-based, cross-library and Wycheproof vectors,

and is being fuzzed in [the separate repo](https://github.com/paulmillr/fuzzing).



If you see anything unusual: investigate and report.



### Constant-timeness



We're targetting algorithmic constant time. _JIT-compiler_ and _Garbage Collector_ make "constant time"

extremely hard to achieve [timing attack](https://en.wikipedia.org/wiki/Timing_attack) resistance

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](https://www.chosenplaintext.ca/open-source/rust-timing-shield/security)

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.



### Supply chain security



- **Commits** are signed with PGP keys, to prevent forgery. Make sure to verify commit signatures

- **Releases** are transparent and built on GitHub CI. Make sure to verify [provenance](https://docs.npmjs.com/generating-provenance-statements) logs

  - Use GitHub CLI to verify single-file builds:

    `gh attestation verify --owner paulmillr noble-curves.js`

- **Rare releasing** is followed to ensure less re-audit need for end-users

- **Dependencies** are minimized and locked-down: any dependency could get hacked and users will be downloading malware with every install.

  - We make sure to use as few dependencies as possible

  - Automatic dep updates are prevented by locking-down version ranges; diffs are checked with `npm-diff`

- **Dev Dependencies** are disabled for end-users; they are only used to develop / build the source code



For this package, there is 1 dependency; and a few dev dependencies:



- [noble-hashes](https://github.com/paulmillr/noble-hashes) provides cryptographic hashing functionality

- micro-bmark, micro-should and jsbt are used for benchmarking / testing / build tooling and developed by the same author

- prettier, fast-check and typescript are used for code quality / test generation / ts compilation. It's hard to audit their source code thoroughly and fully because of their size



### Randomness



We're deferring to built-in

[crypto.getRandomValues](https://developer.mozilla.org/en-US/docs/Web/API/Crypto/getRandomValues)

which is considered cryptographically secure (CSPRNG).



In the past, browsers had bugs that made it weak: it may happen again.

Implementing a userspace CSPRNG to get resilient to the weakness

is even worse: there is no reliable userspace source of quality entropy.



### Quantum computers



Cryptographically relevant quantum computer, if built, will allow to

break elliptic curve cryptography (both ECDSA / EdDSA & ECDH) using Shor's algorithm.



Consider switching to newer / hybrid algorithms, such as SPHINCS+. They are available in

[noble-post-quantum](https://github.com/paulmillr/noble-post-quantum).



NIST prohibits classical cryptography (RSA, DSA, ECDSA, ECDH) [after 2035](https://nvlpubs.nist.gov/nistpubs/ir/2024/NIST.IR.8547.ipd.pdf). Australian ASD prohibits it [after 2030](https://www.cyber.gov.au/resources-business-and-government/essential-cyber-security/ism/cyber-security-guidelines/guidelines-cryptography).



## Speed



```sh

npm run bench:install && npm run bench

```



During first call of most methods, `init` is done, which calculates base point precomputes.

The method consumes 20MB+ of memory and takes some time.

You can adjust how many precomputes are generated,

by using `_setWindowSize`. Check out the source code.



Benchmark results on Apple M4:



```

# secp256k1

init 10ms

getPublicKey x 9,099 ops/sec @ 109μs/op

sign x 7,182 ops/sec @ 139μs/op

verify x 1,188 ops/sec @ 841μs/op

getSharedSecret x 735 ops/sec @ 1ms/op

recoverPublicKey x 1,265 ops/sec @ 790μs/op

schnorr.sign x 957 ops/sec @ 1ms/op

schnorr.verify x 1,210 ops/sec @ 825μs/op



# ed25519

init 14ms

getPublicKey x 14,216 ops/sec @ 70μs/op

sign x 6,849 ops/sec @ 145μs/op

verify x 1,400 ops/sec @ 713μs/op



# ed448

init 37ms

getPublicKey x 5,273 ops/sec @ 189μs/op

sign x 2,494 ops/sec @ 400μs/op

verify x 476 ops/sec @ 2ms/op



# p256

init 17ms

getPublicKey x 8,977 ops/sec @ 111μs/op

sign x 7,236 ops/sec @ 138μs/op

verify x 877 ops/sec @ 1ms/op



# p384

init 42ms

getPublicKey x 4,084 ops/sec @ 244μs/op

sign x 3,247 ops/sec @ 307μs/op

verify x 331 ops/sec @ 3ms/op



# p521

init 83ms

getPublicKey x 2,049 ops/sec @ 487μs/op

sign x 1,748 ops/sec @ 571μs/op

verify x 170 ops/sec @ 5ms/op



# ristretto255

add x 931,966 ops/sec @ 1μs/op

multiply x 15,444 ops/sec @ 64μs/op

encode x 21,367 ops/sec @ 46μs/op

decode x 21,715 ops/sec @ 46μs/op



# decaf448

add x 478,011 ops/sec @ 2μs/op

multiply x 416 ops/sec @ 2ms/op

encode x 8,562 ops/sec @ 116μs/op

decode x 8,636 ops/sec @ 115μs/op



# ECDH

x25519 x 1,981 ops/sec @ 504μs/op

x448 x 743 ops/sec @ 1ms/op

secp256k1 x 728 ops/sec @ 1ms/op

p256 x 705 ops/sec @ 1ms/op

p384 x 268 ops/sec @ 3ms/op

p521 x 137 ops/sec @ 7ms/op



# hash-to-curve

hashToPrivateScalar x 1,754,385 ops/sec @ 570ns/op

hash_to_field x 135,703 ops/sec @ 7μs/op

hashToCurve secp256k1 x 3,194 ops/sec @ 313μs/op

hashToCurve p256 x 5,962 ops/sec @ 167μs/op

hashToCurve p384 x 2,230 ops/sec @ 448μs/op

hashToCurve p521 x 1,063 ops/sec @ 940μs/op

hashToCurve ed25519 x 4,047 ops/sec @ 247μs/op

hashToCurve ed448 x 1,691 ops/sec @ 591μs/op

hash_to_ristretto255 x 8,733 ops/sec @ 114μs/op

hash_to_decaf448 x 3,882 ops/sec @ 257μs/op



# modular over secp256k1 P field

invert a x 866,551 ops/sec @ 1μs/op

invert b x 693,962 ops/sec @ 1μs/op

sqrt p = 3 mod 4 x 25,738 ops/sec @ 38μs/op

sqrt tonneli-shanks x 847 ops/sec @ 1ms/op



# bls12-381

init 22ms

getPublicKey x 1,325 ops/sec @ 754μs/op

sign x 80 ops/sec @ 12ms/op

verify x 62 ops/sec @ 15ms/op

pairing x 166 ops/sec @ 6ms/op

pairing10 x 54 ops/sec @ 18ms/op ± 23.48% (15ms..36ms)

MSM 4096 scalars x points 3286ms

aggregatePublicKeys/8 x 173 ops/sec @ 5ms/op

aggregatePublicKeys/32 x 46 ops/sec @ 21ms/op

aggregatePublicKeys/128 x 11 ops/sec @ 84ms/op

aggregatePublicKeys/512 x 2 ops/sec @ 335ms/op

aggregatePublicKeys/2048 x 0 ops/sec @ 1346ms/op

aggregateSignatures/8 x 82 ops/sec @ 12ms/op

aggregateSignatures/32 x 21 ops/sec @ 45ms/op

aggregateSignatures/128 x 5 ops/sec @ 178ms/op

aggregateSignatures/512 x 1 ops/sec @ 705ms/op

aggregateSignatures/2048 x 0 ops/sec @ 2823ms/op

```



## Upgrading



Previously, the library was split into single-feature packages

[noble-secp256k1](https://github.com/paulmillr/noble-secp256k1),

[noble-ed25519](https://github.com/paulmillr/noble-ed25519) and

[noble-bls12-381](https://github.com/paulmillr/noble-bls12-381).



Curves continue their original work. The single-feature packages changed their

direction towards providing minimal 4kb implementations of cryptography,

which means they have less features.



Upgrading from noble-secp256k1 2.0 or noble-ed25519 2.0: no changes, libraries are compatible.



Upgrading from noble-secp256k1 1.7:



- `getPublicKey`

  - now produce 33-byte compressed signatures by default

  - to use old behavior, which produced 65-byte uncompressed keys, set

    argument `isCompressed` to `false`: `getPublicKey(priv, false)`

- `sign`

  - is now sync

  - now returns `Signature` instance with `{ r, s, recovery }` properties

  - `canonical` option was renamed to `lowS`

  - `recovered` option has been removed because recovery bit is always returned now

  - `der` option has been removed. There are 2 options:

    1. Use compact encoding: `fromCompact`, `toCompactRawBytes`, `toCompactHex`.

       Compact encoding is simply a concatenation of 32-byte r and 32-byte s.

    2. If you must use DER encoding, switch to noble-curves (see above).

- `verify`

  - is now sync

  - `strict` option was renamed to `lowS`

- `getSharedSecret`

  - now produce 33-byte compressed signatures by default

  - to use old behavior, which produced 65-byte uncompressed keys, set

    argument `isCompressed` to `false`: `getSharedSecret(a, b, false)`

- `recoverPublicKey(msg, sig, rec)` was changed to `sig.recoverPublicKey(msg)`

- `number` type for private keys have been removed: use `bigint` instead

- `Point` (2d xy) has been changed to `ProjectivePoint` (3d xyz)

- `utils` were split into `utils` (same api as in noble-curves) and

  `etc` (`hmacSha256Sync` and others)



Upgrading from [@noble/ed25519](https://github.com/paulmillr/noble-ed25519) 1.7:



- Methods are now sync by default

- `bigint` is no longer allowed in `getPublicKey`, `sign`, `verify`. Reason: ed25519 is LE, can lead to bugs

- `Point` (2d xy) has been changed to `ExtendedPoint` (xyzt)

- `Signature` was removed: just use raw bytes or hex now

- `utils` were split into `utils` (same api as in noble-curves) and

  `etc` (`sha512Sync` and others)

- `getSharedSecret` was moved to `x25519` module

- `toX25519` has been moved to `edwardsToMontgomeryPub` and `edwardsToMontgomeryPriv` methods



Upgrading from [@noble/bls12-381](https://github.com/paulmillr/noble-bls12-381):



- Methods and classes were renamed:

  - PointG1 -> G1.Point, PointG2 -> G2.Point

  - PointG2.fromSignature -> Signature.decode, PointG2.toSignature -> Signature.encode

- Fp2 ORDER was corrected



## Contributing & testing



- `npm install && npm run build && npm test` will build the code and run tests.

- `npm run lint` / `npm run format` will run linter / fix linter issues.

- `npm run bench` will run benchmarks, which may need their deps first (`npm run bench:install`)

- `npm run build:release` will build single file



Check out [github.com/paulmillr/guidelines](https://github.com/paulmillr/guidelines)

for general coding practices and rules.



See [paulmillr.com/noble](https://paulmillr.com/noble/)

for useful resources, articles, documentation and demos

related to the library.



## License



The MIT License (MIT)



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



See LICENSE file.