noble-curves

Audited & minimal JS implementation of elliptic curve cryptography.

• 🔒 Audited 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 (36KB gzipped) for everything with hashes, 26KB (11KB gzipped) for single-curve build

Curves have 4KB sister projects secp256k1 & ed25519. They have smaller attack surface, but less features.

Take a glance at GitHub 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, curves, hashes, post-quantum, 4kb secp256k1 / ed25519
• Check out homepage 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. A standalone file noble-curves.js is also available.

1// import * from '@noble/curves'; // Error: use sub-imports, to ensure small app size
2import { secp256k1, schnorr } from '@noble/curves/secp256k1';
3import { ed25519, ed25519ph, ed25519ctx, x25519 } from '@noble/curves/ed25519';
4import { ed448, ed448ph, ed448ctx, x448 } from '@noble/curves/ed448';
5import { p256, p384, p521 } from '@noble/curves/nist';
6import { bls12_381 } from '@noble/curves/bls12-381';
7import { bn254 } from '@noble/curves/bn254';
8import { jubjub, babyjubjub } from '@noble/curves/misc';
9import { bytesToHex, hexToBytes, concatBytes, utf8ToBytes } from '@noble/curves/abstract/utils';

• ECDSA signatures over secp256k1 and others
• Hedged ECDSA with noise
• ECDH: Diffie-Hellman shared secrets
• secp256k1 Schnorr signatures from BIP340
• ed25519 / X25519 / ristretto255
• ed448 / X448 / decaf448
• bls12-381
• bn254 aka alt_bn128
• misc curves
• Low-level methods
• Abstract API
  • weierstrass, edwards, montgomery, bls
  • hash-to-curve, poseidon
  • modular
  • fft
  • Creating private keys from hashes
  • utils
• Security
• Speed
• Upgrading
• Contributing & testing
• License

Implementations

ECDSA signatures over secp256k1 and others

1import { secp256k1 } from '@noble/curves/secp256k1';
2// import { p256 } from '@noble/curves/nist'; // or p384 / p521
3
4const priv = secp256k1.utils.randomPrivateKey();
5const pub = secp256k1.getPublicKey(priv);
6const msg = new Uint8Array(32).fill(1); // message hash (not message) in ecdsa
7const sig = secp256k1.sign(msg, priv); // `{prehash: true}` option is available
8const isValid = secp256k1.verify(sig, msg, pub) === true;
9
10// hex strings are also supported besides Uint8Array-s:
11const privHex = '46c930bc7bb4db7f55da20798697421b98c4175a52c630294d75a84b9c126236';
12const pub2 = secp256k1.getPublicKey(privHex);
13
14// public key recovery
15// let sig = secp256k1.Signature.fromCompact(sigHex); // or .fromDER(sigDERHex)
16// sig = sig.addRecoveryBit(bit); // bit is not serialized into compact / der format
17sig.recoverPublicKey(msg).toRawBytes(); // === pub; // public key recovery

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

Hedged ECDSA with noise

1const noisySignature = secp256k1.sign(msg, priv, { extraEntropy: true });
2const ent = new Uint8Array(32).fill(3); // set custom entropy
3const 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 and spec draft.

ECDH: Diffie-Hellman shared secrets

1const someonesPub = secp256k1.getPublicKey(secp256k1.utils.randomPrivateKey());
2const shared = secp256k1.getSharedSecret(priv, someonesPub);
3// NOTE:
4// - `shared` includes parity byte: strip it using shared.slice(1)
5// - `shared` is not hashed: more secure way is sha256(shared) or hkdf(shared)

secp256k1 Schnorr signatures from BIP340

1import { schnorr } from '@noble/curves/secp256k1';
2const priv = schnorr.utils.randomPrivateKey();
3const pub = schnorr.getPublicKey(priv);
4const msg = new TextEncoder().encode('hello');
5const sig = schnorr.sign(msg, priv);
6const isValid = schnorr.verify(sig, msg, pub);

ed25519

1import { ed25519 } from '@noble/curves/ed25519';
2const priv = ed25519.utils.randomPrivateKey();
3const pub = ed25519.getPublicKey(priv);
4const msg = new TextEncoder().encode('hello');
5const sig = ed25519.sign(msg, priv);
6ed25519.verify(sig, msg, pub); // Default mode: follows ZIP215
7ed25519.verify(sig, msg, pub, { zip215: false }); // SBS / e-voting / RFC8032 / FIPS 186-5
8
9// Variants from RFC8032: with context, prehashed
10import { 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. Both options have SUF-CMA (strong unforgeability under chosen message attacks).

X25519

1// X25519 aka ECDH on Curve25519 from [RFC7748](https://www.rfc-editor.org/rfc/rfc7748)
2import { x25519 } from '@noble/curves/ed25519';
3const priv = 'a546e36bf0527c9d3b16154b82465edd62144c0ac1fc5a18506a2244ba449ac4';
4const pub = 'e6db6867583030db3594c1a424b15f7c726624ec26b3353b10a903a6d0ab1c4c';
5x25519.getSharedSecret(priv, pub) === x25519.scalarMult(priv, pub); // aliases
6x25519.getPublicKey(priv) === x25519.scalarMultBase(priv);
7x25519.getPublicKey(x25519.utils.randomPrivateKey());
8
9// ed25519 => x25519 conversion
10import { edwardsToMontgomeryPub, edwardsToMontgomeryPriv } from '@noble/curves/ed25519';
11edwardsToMontgomeryPub(ed25519.getPublicKey(ed25519.utils.randomPrivateKey()));
12edwardsToMontgomeryPriv(ed25519.utils.randomPrivateKey());

ristretto255

1// ristretto255 from [RFC9496](https://www.rfc-editor.org/rfc/rfc9496)
2import { utf8ToBytes } from '@noble/hashes/utils';
3import { sha512 } from '@noble/hashes/sha512';
4import {
5  hashToCurve,
6  encodeToCurve,
7  RistrettoPoint,
8  hashToRistretto255,
9} from '@noble/curves/ed25519';
10
11const msg = utf8ToBytes('Ristretto is traditionally a short shot of espresso coffee');
12hashToCurve(msg);
13
14const rp = RistrettoPoint.fromHex(
15  '6a493210f7499cd17fecb510ae0cea23a110e8d5b901f8acadd3095c73a3b919'
16);
17RistrettoPoint.BASE.multiply(2n).add(rp).subtract(RistrettoPoint.BASE).toRawBytes();
18RistrettoPoint.ZERO.equals(dp) === false;
19// pre-hashed hash-to-curve
20RistrettoPoint.hashToCurve(sha512(msg));
21// full hash-to-curve including domain separation tag
22hashToRistretto255(msg, { DST: 'ristretto255_XMD:SHA-512_R255MAP_RO_' });

ed448

1import { ed448 } from '@noble/curves/ed448';
2const priv = ed448.utils.randomPrivateKey();
3const pub = ed448.getPublicKey(priv);
4const msg = new TextEncoder().encode('whatsup');
5const sig = ed448.sign(msg, priv);
6ed448.verify(sig, msg, pub);
7
8// Variants from RFC8032: prehashed
9import { ed448ph } from '@noble/curves/ed448';

X448

1// X448 aka ECDH on Curve448 from [RFC7748](https://www.rfc-editor.org/rfc/rfc7748)
2import { x448 } from '@noble/curves/ed448';
3x448.getSharedSecret(priv, pub) === x448.scalarMult(priv, pub); // aliases
4x448.getPublicKey(priv) === x448.scalarMultBase(priv);
5
6// ed448 => x448 conversion
7import { edwardsToMontgomeryPub } from '@noble/curves/ed448';
8edwardsToMontgomeryPub(ed448.getPublicKey(ed448.utils.randomPrivateKey()));

decaf448

1// decaf448 from [RFC9496](https://www.rfc-editor.org/rfc/rfc9496)
2import { utf8ToBytes } from '@noble/hashes/utils';
3import { shake256 } from '@noble/hashes/sha3';
4import { hashToCurve, encodeToCurve, DecafPoint, hashToDecaf448 } from '@noble/curves/ed448';
5
6const msg = utf8ToBytes('Ristretto is traditionally a short shot of espresso coffee');
7hashToCurve(msg);
8
9const dp = DecafPoint.fromHex(
10  'c898eb4f87f97c564c6fd61fc7e49689314a1f818ec85eeb3bd5514ac816d38778f69ef347a89fca817e66defdedce178c7cc709b2116e75'
11);
12DecafPoint.BASE.multiply(2n).add(dp).subtract(DecafPoint.BASE).toRawBytes();
13DecafPoint.ZERO.equals(dp) === false;
14// pre-hashed hash-to-curve
15DecafPoint.hashToCurve(shake256(msg, { dkLen: 112 }));
16// full hash-to-curve including domain separation tag
17hashToDecaf448(msg, { DST: 'decaf448_XOF:SHAKE256_D448MAP_RO_' });

bls12-381

1import { bls12_381 } from '@noble/curves/bls12-381';
2import { hexToBytes, utf8ToBytes } from '@noble/curves/abstract/utils';
3
4// private keys are 32 bytes
5const privKey = hexToBytes('67d53f170b908cabb9eb326c3c337762d59289a8fec79f7bc9254b584b73265c');
6// const privKey = bls12_381.utils.randomPrivateKey();
7
8// Long signatures (G2), short public keys (G1)
9const blsl = bls12_381.longSignatures;
10const publicKey = blsl.getPublicKey(privateKey);
11// Sign msg with custom (Ethereum) DST
12const msg = utf8ToBytes('hello');
13const DST = 'BLS_SIG_BLS12381G2_XMD:SHA-256_SSWU_RO_POP_';
14const msgp = blsl.hash(msg, DST);
15const signature = blsl.sign(msgp, privateKey);
16const isValid = blsl.verify(signature, msgp, publicKey);
17console.log({ publicKey, signature, isValid });
18
19// Short signatures (G1), long public keys (G2)
20const blss = bls12_381.shortSignatures;
21const publicKey2 = blss.getPublicKey(privateKey);
22const msgp2 = blss.hash(utf8ToBytes('hello'), 'BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_NUL_')
23const signature2 = blss.sign(msgp2, privateKey);
24const isValid2 = blss.verify(signature2, msgp2, publicKey);
25console.log({ publicKey2, signature2, isValid2 });
26
27// Aggregation
28const aggregatedKey = bls12_381.longSignatures.aggregatePublicKeys([
29  bls12_381.utils.randomPrivateKey(),
30  bls12_381.utils.randomPrivateKey(),
31]);
32// const aggregatedSig = bls.aggregateSignatures(sigs)
33
34// Pairings, with and without final exponentiation
35// bls.pairing(PointG1, PointG2);
36// bls.pairing(PointG1, PointG2, false);
37// bls.fields.Fp12.finalExponentiate(bls.fields.Fp12.mul(PointG1, PointG2));
38
39// Others
40// bls.G1.ProjectivePoint.BASE, bls.G2.ProjectivePoint.BASE;
41// bls.fields.Fp, bls.fields.Fp2, bls.fields.Fp12, bls.fields.Fr;

See abstract/bls. For example usage, check out the implementation of BLS EVM precompiles.

bn254 aka alt_bn128

1import { bn254 } from '@noble/curves/bn254';
2
3console.log(bn254.G1, bn254.G2, bn254.pairing);

The API mirrors BLS. The curve was previously called alt_bn128. The implementation is compatible with EIP-196 and EIP-197.

We don't implement Point methods toHex / toRawBytes. To work around this limitation, has to initialize points on their own from BigInts. Reason it's not implemented is because 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.

misc curves

1import { jubjub, babyjubjub } from '@noble/curves/misc';

Miscellaneous, rarely used curves are contained in the module. Jubjub curves have Fp over scalar fields of other curves. They are friendly to ZK proofs. jubjub Fp = bls n. babyjubjub Fp = bn254 n.

Low-level methods

1import { secp256k1 } from '@noble/curves/secp256k1';
2
3// Curve's variables
4// Every curve has `CURVE` object that contains its parameters, field, and others
5console.log(secp256k1.CURVE.p); // field modulus
6console.log(secp256k1.CURVE.n); // curve order
7console.log(secp256k1.CURVE.a, secp256k1.CURVE.b); // equation params
8console.log(secp256k1.CURVE.Gx, secp256k1.CURVE.Gy); // base point coordinates
9
10// MSM
11const p = secp256k1.ProjectivePoint;
12const points = [p.BASE, p.BASE.multiply(2n), p.BASE.multiply(4n), p.BASE.multiply(8n)];
13p.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. 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). 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

1import { weierstrass } from '@noble/curves/abstract/weierstrass';
2import { Field } from '@noble/curves/abstract/modular';
3import { sha256 } from '@noble/hashes/sha256';
4import { hmac } from '@noble/hashes/hmac';
5import { concatBytes, randomBytes } from '@noble/hashes/utils';
6
7const hmacSha256 = (key: Uint8Array, ...msgs: Uint8Array[]) =>
8  hmac(sha256, key, concatBytes(...msgs));
9
10// secQ (not secP) - secq256k1 is a cycle of secp256k1 with Fp/N flipped.
11// https://personaelabs.org/posts/spartan-ecdsa
12// https://zcash.github.io/halo2/background/curves.html#cycles-of-curves
13const secq256k1 = weierstrass({
14  a: 0n,
15  b: 7n,
16  Fp: Field(2n ** 256n - 432420386565659656852420866394968145599n),
17  n: 2n ** 256n - 2n ** 32n - 2n ** 9n - 2n ** 8n - 2n ** 7n - 2n ** 6n - 2n ** 4n - 1n,
18  Gx: 55066263022277343669578718895168534326250603453777594175500187360389116729240n,
19  Gy: 32670510020758816978083085130507043184471273380659243275938904335757337482424n,
20  hash: sha256,
21  hmac: hmacSha256,
22  randomBytes,
23});
24
25// NIST secp192r1 aka p192
26// https://www.secg.org/sec2-v2.pdf, https://neuromancer.sk/std/secg/secp192r1
27const secp192r1 = weierstrass({
28  a: 0xfffffffffffffffffffffffffffffffefffffffffffffffcn,
29  b: 0x64210519e59c80e70fa7e9ab72243049feb8deecc146b9b1n,
30  Fp: Field(0xfffffffffffffffffffffffffffffffeffffffffffffffffn),
31  n: 0xffffffffffffffffffffffff99def836146bc9b1b4d22831n,
32  Gx: 0x188da80eb03090f67cbf20eb43a18800f4ff0afd82ff1012n,
33  Gy: 0x07192b95ffc8da78631011ed6b24cdd573f977a11e794811n,
34  hash: sha256,
35  hmac: hmacSha256,
36  randomBytes,
37});

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.
  • Consider hedged ECDSA with noise for adding randomness into for signatures, to get improved security against fault attacks.

More examples:

1// All curves expose same generic interface.
2const priv = secq256k1.utils.randomPrivateKey();
3secq256k1.getPublicKey(priv); // Convert private key to public.
4const sig = secq256k1.sign(msg, priv); // Sign msg with private key.
5const sig2 = secq256k1.sign(msg, priv, { prehash: true }); // hash(msg)
6secq256k1.verify(sig, msg, priv); // Verify if sig is correct.
7
8// Default behavior is "try DER, then try compact if fails". Can be explicit:
9secq256k1.verify(sig.toCompactHex(), msg, priv, { format: 'compact' });
10
11const Point = secq256k1.ProjectivePoint;
12const point = Point.BASE; // Elliptic curve Point class and BASE point static var.
13point.add(point).equals(point.double()); // add(), equals(), double() methods
14point.subtract(point).equals(Point.ZERO); // subtract() method, ZERO static var
15point.negate(); // Flips point over x/y coordinate.
16point.multiply(31415n); // Multiplication of Point by scalar.
17
18point.assertValidity(); // Checks for being on-curve
19point.toAffine(); // Converts to 2d affine xy coordinates
20
21secq256k1.CURVE.n;
22secq256k1.CURVE.p;
23secq256k1.CURVE.Fp.mod();
24secq256k1.CURVE.hash();
25
26// precomputes
27const fast = secq256k1.utils.precompute(8, Point.fromHex(someonesPubKey));
28fast.multiply(privKey); // much faster ECDH now

edwards: Twisted Edwards curve

1import { twistedEdwards } from '@noble/curves/abstract/edwards';
2import { Field } from '@noble/curves/abstract/modular';
3import { sha512 } from '@noble/hashes/sha512';
4import { randomBytes } from '@noble/hashes/utils';
5
6const Fp = Field(2n ** 255n - 19n);
7const ed25519 = twistedEdwards({
8  a: Fp.create(-1n),
9  d: Fp.div(-121665n, 121666n), // -121665n/121666n mod p
10  Fp: Fp,
11  n: 2n ** 252n + 27742317777372353535851937790883648493n,
12  h: 8n,
13  Gx: 15112221349535400772501151409588531511454012693041857206046113283949847762202n,
14  Gy: 46316835694926478169428394003475163141307993866256225615783033603165251855960n,
15  hash: sha512,
16  randomBytes,
17  adjustScalarBytes(bytes) {
18    // optional; but mandatory in ed25519
19    bytes[0] &= 248;
20    bytes[31] &= 127;
21    bytes[31] |= 64;
22    return bytes;
23  },
24} 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, following ZIP215 rules
• zip215: false switches verification criteria to strict RFC8032 / FIPS 186-5 and additionally provides non-repudiation with SBS, 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 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 and use aggregated, batch-verifiable threshold signatures, 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.

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

1import { hashToCurve, encodeToCurve } from '@noble/curves/secp256k1';
2import { randomBytes } from '@noble/hashes/utils';
3hashToCurve('0102abcd');
4console.log(hashToCurve(randomBytes()));
5console.log(encodeToCurve(randomBytes()));
6
7import { bls12_381 } from '@noble/curves/bls12-381';
8bls12_381.G1.hashToCurve(randomBytes(), { DST: 'another' });
9bls12_381.G2.hashToCurve(randomBytes(), { DST: 'custom' });

Low-level methods from the spec:

1// produces a uniformly random byte string using a cryptographic hash function H that outputs b bits.
2function expand_message_xmd(
3  msg: Uint8Array,
4  DST: Uint8Array,
5  lenInBytes: number,
6  H: CHash // For CHash see abstract/weierstrass docs section
7): Uint8Array;
8// produces a uniformly random byte string using an extendable-output function (XOF) H.
9function expand_message_xof(
10  msg: Uint8Array,
11  DST: Uint8Array,
12  lenInBytes: number,
13  k: number,
14  H: CHash
15): Uint8Array;
16// Hashes arbitrary-length byte strings to a list of one or more elements of a finite field F
17function hash_to_field(msg: Uint8Array, count: number, options: Opts): bigint[][];
18
19/**
20 * * `DST` is a domain separation tag, defined in section 2.2.5
21 * * `p` characteristic of F, where F is a finite field of characteristic p and order q = p^m
22 * * `m` is extension degree (1 for prime fields)
23 * * `k` is the target security target in bits (e.g. 128), from section 5.1
24 * * `expand` is `xmd` (SHA2, SHA3, BLAKE) or `xof` (SHAKE, BLAKE-XOF)
25 * * `hash` conforming to `utils.CHash` interface, with `outputLen` / `blockLen` props
26 */
27type UnicodeOrBytes = string | Uint8Array;
28type Opts = {
29  DST: UnicodeOrBytes;
30  p: bigint;
31  m: number;
32  k: number;
33  expand?: 'xmd' | 'xof';
34  hash: CHash;
35};

poseidon: Poseidon hash

Implements Poseidon ZK-friendly hash: permutation and sponge.

There are many poseidon variants with different constants. We don't provide them: you should construct them manually. Check out micro-starknet package for a proper example.

1import { poseidon, poseidonSponge } from '@noble/curves/abstract/poseidon';
2
3const rate = 2;
4const capacity = 1;
5const { mds, roundConstants } = poseidon.grainGenConstants({
6  Fp,
7  t: rate + capacity,
8  roundsFull: 8,
9  roundsPartial: 31,
10});
11const opts = {
12  Fp,
13  rate,
14  capacity,
15  sboxPower: 17,
16  mds,
17  roundConstants,
18  roundsFull: 8,
19  roundsPartial: 31,
20};
21const permutation = poseidon.poseidon(opts);
22const sponge = poseidon.poseidonSponge(opts); // use carefully, not specced

modular: Modular arithmetics utilities

1import * as mod from '@noble/curves/abstract/modular';
2
3// Finite Field utils
4const fp = mod.Field(2n ** 255n - 19n); // Finite field over 2^255-19
5fp.mul(591n, 932n); // multiplication
6fp.pow(481n, 11024858120n); // exponentiation
7fp.div(5n, 17n); // division: 5/17 mod 2^255-19 == 5 * invert(17)
8fp.inv(5n); // modular inverse
9fp.sqrt(21n); // square root
10
11// Non-Field generic utils are also available
12mod.mod(21n, 10n); // 21 mod 10 == 1n; fixed version of 21 % 10
13mod.invert(17n, 10n); // invert(17) mod 10; modular multiplicative inverse
14mod.invertBatch([1n, 2n, 4n], 21n); // => [1n, 11n, 16n] in one inversion

Field operations are not constant-time: they are using JS bigints, see 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.

fft: Fast Fourier Transform

Experimental implementation of NTT / FFT (Fast Fourier Transform) over finite fields. API may change at any time. The code has not been audited. Feature requests are welcome.

1import * as fft from '@noble/curves/abstract/fft.js';

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.

To make the bias negligible, we follow FIPS 186-5 A.2 and RFC 9380. 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 if you need to hash to public key.

1import { p256 } from '@noble/curves/nist';
2import { sha256 } from '@noble/hashes/sha256';
3import { hkdf } from '@noble/hashes/hkdf';
4import * as mod from '@noble/curves/abstract/modular';
5const someKey = new Uint8Array(32).fill(2); // Needs to actually be random, not .fill(2)
6const derived = hkdf(sha256, someKey, undefined, 'application', 48); // 48 bytes for 32-byte priv
7const validPrivateKey = mod.hashToPrivateScalar(derived, p256.CURVE.n);

utils: Useful utilities

1import * as utils from '@noble/curves/abstract/utils';
2
3utils.bytesToHex(Uint8Array.from([0xde, 0xad, 0xbe, 0xef]));
4utils.hexToBytes('deadbeef');
5utils.numberToHexUnpadded(123n);
6utils.hexToNumber();
7
8utils.bytesToNumberBE(Uint8Array.from([0xde, 0xad, 0xbe, 0xef]));
9utils.bytesToNumberLE(Uint8Array.from([0xde, 0xad, 0xbe, 0xef]));
10utils.numberToBytesBE(123n, 32);
11utils.numberToBytesLE(123n, 64);
12
13utils.concatBytes(Uint8Array.from([0xde, 0xad]), Uint8Array.from([0xbe, 0xef]));
14utils.nLength(255n);
15utils.equalBytes(Uint8Array.from([0xde]), Uint8Array.from([0xde]));

Security

The library has been independently audited:

• at version 1.6.0, in Sep 2024, by Cure53
  • PDFs: website, in-repo
  • Changes since audit
  • 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
• at version 1.2.0, in Sep 2023, by Kudelski Security
  • PDFs: in-repo
  • Changes since audit
  • Scope: scure-starknet and its related abstract modules of noble-curves: curve, modular, poseidon, weierstrass
  • The audit has been funded by Starkware
• at version 0.7.3, in Feb 2023, by Trail of Bits
  • PDFs: website, in-repo
  • Changes since audit
  • 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

It is tested against property-based, cross-library and Wycheproof vectors, and is being fuzzed in the separate repo.

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 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 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.

Memory dumping

Use low-level languages instead of JS / WASM if your goal is absolute security.

The library mostly uses Uint8Arrays and bigints.

• Uint8Arrays have .fill(0) which instructs to fill content with zeroes but there are no guarantees in JS
• bigints are immutable and don't have a method to zeroize their content: a user needs to wait until the next garbage collection cycle
• hex strings are also immutable: there is no way to zeroize them
• await fn() will write all internal variables to memory. With async functions there are no guarantees when the code chunk would be executed. Which means attacker can have plenty of time to read data from memory.

This means some secrets could stay in memory longer than anticipated. However, if an attacker can read application memory, it's doomed anyway: there is no way to guarantee anything about zeroizing sensitive data without complex tests-suite which will dump process memory and verify that there is no sensitive data left. For JS it means testing all browsers (including mobile). And, of course, it will be useless without using the same test-suite in the actual application that consumes the library.

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 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 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 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.

NIST prohibits classical cryptography (RSA, DSA, ECDSA, ECDH) after 2035. Australian ASD prohibits it after 2030.

Speed

1npm run bench:install && npm run bench

noble-curves spends 10+ ms to generate 20MB+ of base point precomputes. This is done one-time per curve.

The generation is deferred until any method (pubkey, sign, verify) is called. User can force precompute generation by manually calling Point.BASE.precompute(windowSize, false). 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, noble-ed25519 and 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 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:

• 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 for general coding practices and rules.

See paulmillr.com/noble for useful resources, articles, documentation and demos related to the library.

MuSig2 signature scheme and BIP324 ElligatorSwift mapping for secp256k1 are available in a separate package.

License

The MIT License (MIT)

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

See LICENSE file.