77 Commits

Author SHA1 Message Date
Paul Miller
b8ddb603c1 Format 2024-01-25 12:41:48 +01:00
Paul Miller
943edbceba Implement ElligatorSwift 2024-01-14 11:00:03 +01:00
Paul Miller
0a663391bd Improve gitignore 2024-01-14 10:02:50 +01:00
Paul Miller
7be1dfc55d utils: copy concatBytes from hashes 2024-01-02 08:49:40 +01:00
Paul Miller
37eab5a28a Refactor tsconfig: use inheritance 2023-12-23 18:30:48 +01:00
Paul Miller
2706fe9f79 README: mention secp256r1. 2023-12-14 16:20:40 +03:00
Paul Miller
b39b0d1daf weierstrass: improve error wording for sqrt case 2023-12-13 15:58:51 +03:00
Paul Miller
4007ee975b Release 1.3.0. 2023-12-12 02:21:29 +03:00
Paul Miller
f8af434b9c Bump noble-hashes to 1.3.3 2023-12-12 02:18:35 +03:00
Paul Miller
be8033a2d8 readme 2023-12-11 01:43:32 +01:00
Paul Miller
b3c239981b readme 2023-12-11 01:42:57 +01:00
Paul Miller
18b0bc6317 readme: Mention zip215 2023-12-11 01:40:43 +01:00
Paul Miller
30f68c9e54 utils: improve isBytes 2023-12-11 00:04:11 +01:00
Paul Miller
ada1ea5a19 bls: fix types. Closes gh-101 2023-12-10 23:42:42 +01:00
Paul Miller
0a3a13b3dc Fix typescript esm config 2023-12-10 23:32:10 +01:00
Paul Miller
26a4fd4293 weierstrass, hash-to-curve: ensure to use utils.isBytes everywhere 2023-12-10 23:27:15 +01:00
Paul Miller
9db14fc6d0 utils: fix-up isBytes 2023-12-10 23:26:57 +01:00
Paul Miller
8e6c19de2b utils: make equalBytes constant-time 2023-12-10 23:04:01 +01:00
Paul Miller
4ffb68853d utils: make isBytes more resilient in weird envs, improve concatBytes type error resilience. 2023-12-10 23:00:49 +01:00
Paul Miller
008958364e weierstrass: reformat after new prettier 2023-12-10 22:58:13 +01:00
Paul Miller
1c535a3287 deps: Bump prettier and typescript, reduce their sizes 2023-12-10 22:58:01 +01:00
Paul Miller
b8b12671ac test: rename hash-to-curve vectors, remove colons. closes gh-102 2023-12-10 19:47:25 +01:00
Paul Miller
2f1460a4d7 BLS: Refactor mask-bit settings, improve encoding resiliency 2023-11-10 02:55:16 +01:00
Paul Miller
fb02e93ff6 ECDH tests: comment 2023-11-01 17:09:37 +01:00
Paul Miller
c525356916 ECDH tests: allow padded private keys 2023-11-01 17:06:40 +01:00
Paul Miller
a4abd8a202 ECDH tests: quick and dirty ASN1 parsing 2023-11-01 16:54:08 +01:00
Paul Miller
c19373a0b5 readme 2023-10-20 15:34:18 +02:00
Paul Miller
85006ed620 readme 2023-10-20 15:33:27 +02:00
Paul Miller
fae7f6612a README 2023-10-20 15:27:08 +02:00
Paul Miller
36894729c0 readme note on csprng 2023-10-20 15:16:43 +02:00
Paul Miller
eabab627c7 Merge pull request #93 from yhc125/patch-1 2023-10-16 17:36:43 +02:00
YoungHoon Cha
e1640eb74e Update README.md
Added libraries missing from the code examples.
2023-10-17 00:28:21 +09:00
Paul Miller
7f851873f9 Merge pull request #92 from secure12/main 2023-10-12 12:16:16 +02:00
Eric Ho
02099b9b4c Add weierstrassPoints return type 2023-10-11 19:16:56 +01:00
Eric Ho
3b14683806 Update weierstrass.ts 2023-10-11 18:41:11 +01:00
Paul Miller
47169740c6 readme 2023-10-07 15:19:21 +02:00
Paul Miller
45c7cb560d readme 2023-10-07 15:00:11 +02:00
Paul Miller
b36bf44f4b readme 2023-10-07 14:48:25 +02:00
Paul Miller
30763066ac readme 2023-10-07 14:43:51 +02:00
Paul Miller
911801ec0f readme 2023-10-07 14:39:45 +02:00
Paul Miller
8ba25a1c40 readme 2023-10-07 14:36:25 +02:00
Paul Miller
43a06b669a readme update 2023-10-07 14:35:00 +02:00
Paul Miller
e7720c1609 readme: Clarify ecdsa recovery 2023-10-05 06:46:31 +02:00
Paul Miller
2da6abb336 Fix x448 private keys: must be 56 bytes, not 57. Reported by @larabr 2023-10-03 01:15:43 +02:00
Paul Miller
4752ab1f1e utils: optimize hexToBytes by 4% 2023-09-25 20:22:57 +02:00
Paul Miller
f58002e6d4 utils: refactor hexToBytes a bit 2023-09-25 19:21:18 +02:00
Paul Miller
d0294bb2a6 Clarify build:release script. Closes gh-86 2023-09-21 23:07:52 +02:00
Paul Miller
2b41e387de Merge pull request #85 from sublimator/nd-impl-group-x-for-x-in-decafpoint-ristrettopoint-2023-09-20
feat: impl Group<X> for X in DecafPoint/RistrettoPoint
2023-09-20 21:06:36 +02:00
Nicholas Dudfield
08850c2d6a feat: impl Group<X> for X in DecafPoint/RistrettoPoint 2023-09-20 16:23:41 +07:00
Paul Miller
ce7a8fda55 bls, bn: clarify their security level in comments 2023-09-14 03:02:10 +02:00
Paul Miller
728b485cd8 Merge pull request #83 from arobsn/main
Improve `hexToBytes` performance
2023-09-14 00:17:42 +02:00
Paul Miller
eaefe9a272 benchmark add utils 2023-09-13 23:59:12 +02:00
Paul Miller
c935b398fe abstract/utils: reformat hexToBytes. 2023-09-13 23:57:34 +02:00
Paul Miller
ddad219e7a README 2023-09-13 23:29:05 +02:00
arobsn
1d83bab27d add char code based hexToBytes function 2023-09-13 18:14:13 -03:00
Paul Miller
4be208e4b2 README: add new project using curves 2023-09-10 21:55:06 +02:00
Paul Miller
77bee0d54e ed448: clarify why there are 56 or 57 byte keys 2023-09-10 03:00:51 +02:00
Paul Miller
6bcab6c24b readme: add example for chash 2023-09-07 23:44:46 +02:00
Paul Miller
7befd5f881 readme 2023-09-07 15:34:29 +02:00
Paul Miller
8f78471703 Merge pull request #82 from sublimator/patch-2
docs: audited by plural firms
2023-09-07 15:34:03 +02:00
Nicholas Dudfield
17294f4974 docs: audited by plural firms 2023-09-07 10:59:48 +07:00
Paul Miller
3890b79e7e readme 2023-09-06 20:22:45 +02:00
Paul Miller
2acebc8176 Add new audit of noble-curves by kudelski security. 2023-09-06 01:48:09 +02:00
Paul Miller
1e67754943 Merge pull request #81 from randombit/jack/check-short-sig-in-subgroup
Fix ShortSignature.fromHex to check the G1 point is valid
2023-08-31 20:26:17 +02:00
Jack Lloyd
156a1e909a Fix ShortSignature.fromHex to check the G1 point is valid 2023-08-31 13:28:53 -04:00
Paul Miller
ccea23a712 Fix README. Closes gh-80 2023-08-31 02:16:47 +02:00
Paul Miller
8661eef949 readme 2023-08-29 16:29:30 +02:00
Paul Miller
4743182bf7 README: update security section 2023-08-29 14:36:11 +02:00
Paul Miller
5c477a88fa README: update security section 2023-08-29 14:25:58 +02:00
Paul Miller
df9d461adf README: update security section 2023-08-29 14:00:53 +02:00
Paul Miller
5c21fa3855 Merge pull request #79 from randombit/jack/update-readme-for-bls-short-sigs
Update the README to describe BLS short signature support
2023-08-28 17:02:19 +02:00
Jack Lloyd
6661a7db7b Update the README to describe BLS short signature support 2023-08-28 09:22:25 -04:00
Paul Miller
cf5f2268fb ed448: add todo comment 2023-08-27 18:49:55 +02:00
Paul Miller
1d5286ffa7 single-file build: expose more methods 2023-08-27 18:49:55 +02:00
Paul Miller
e31efd91d8 Merge pull request #74 from randombit/jack/add-short-signatures
Add verification of BLS short signatures
2023-08-26 00:50:42 +02:00
Jack Lloyd
c5e0e070d1 Complete BLS short signature support 2023-08-24 16:38:12 -04:00
Jack Lloyd
b082d41c29 Add verification of BLS short signatures 2023-08-18 15:36:17 -04:00
52 changed files with 1222 additions and 464 deletions

18
.gitignore vendored
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@@ -1,13 +1,9 @@
build/
node_modules/
coverage/
node_modules
/*.js
/*.ts
/*.js.map
/*.d.ts.map
/esm/*.js
/esm/*.ts
/esm/*.js.map
/esm/*.d.ts.map
/esm/abstract
/abstract/
*.d.ts
*.d.ts.map
*.js.map
/build
/abstract
/esm/abstract

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@@ -1,4 +1,5 @@
{
"printWidth": 100,
"singleQuote": true
"singleQuote": true,
"trailingComma": "es5"
}

308
README.md
View File

@@ -2,13 +2,13 @@
Audited & minimal JS implementation of elliptic curve cryptography.
- 🔒 [**Audited**](#security) by an independent security firm
- 🔒 [**Audited**](#security) by independent security firms
- 🔻 Tree-shaking-friendly: use only what's necessary, other code won't be included
- 🏎 Ultra-fast, hand-optimized for caveats of JS engines
- 🔍 Unique tests ensure correctness: property-based, cross-library and Wycheproof vectors, fuzzing
- ➰ Short Weierstrass, Edwards, Montgomery curves
- ✍️ ECDSA, EdDSA, Schnorr, BLS signature schemes, ECDH key agreement
- 🔖 SUF-CMA and SBS (non-repudiation) for ed25519, ed448 and others
- 🔖 SUF-CMA, SBS (non-repudiation), ZIP215 (consensus friendliness) features for ed25519
- #⃣ hash-to-curve for encoding or hashing an arbitrary string to an elliptic curve point
- 🧜‍♂️ Poseidon ZK-friendly hash
@@ -16,16 +16,15 @@ Audited & minimal JS implementation of elliptic curve cryptography.
> **noble-crypto** — high-security, easily auditable set of contained cryptographic libraries and tools.
- No dependencies, protection against supply chain attacks
- Auditable TypeScript / JS code
- Supported on all major platforms
- Releases are signed with PGP keys and built transparently with NPM provenance
- Check out [homepage](https://paulmillr.com/noble/) & all libraries:
- 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),
4kb [secp256k1](https://github.com/paulmillr/noble-secp256k1) /
[ed25519](https://github.com/paulmillr/noble-ed25519)
[hashes](https://github.com/paulmillr/noble-hashes)
- [Check out homepage](https://paulmillr.com/noble/)
for reading resources, documentation and apps built with noble
## Usage
@@ -33,13 +32,19 @@ Audited & minimal JS implementation of elliptic curve cryptography.
We support all major platforms and runtimes.
For [Deno](https://deno.land), ensure to use [npm specifier](https://deno.land/manual@v1.28.0/node/npm_specifiers).
For React Native, you may need a [polyfill for crypto.getRandomValues](https://github.com/LinusU/react-native-get-random-values).
If you don't like NPM, a standalone [noble-curves.js](https://github.com/paulmillr/noble-curves/releases) is also available.
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.
```js
// import * from '@noble/curves'; // Error: use sub-imports, to ensure small app size
import { secp256k1 } from '@noble/curves/secp256k1'; // ESM and Common.js
// import { secp256k1 } from 'npm:@noble/curves@1.2.0/secp256k1'; // Deno
```
- [Implementations](#implementations)
- [ECDSA signature scheme](#ecdsa-signature-scheme)
- [ECDSA public key recovery & extra entropy](#ecdsa-public-key-recovery--extra-entropy)
- [ECDH (Elliptic Curve Diffie-Hellman)](#ecdh-elliptic-curve-diffie-hellman)
- [ECDH: Elliptic Curve Diffie-Hellman](#ecdh-elliptic-curve-diffie-hellman)
- [Schnorr signatures over secp256k1, BIP340](#schnorr-signatures-over-secp256k1-bip340)
- [ed25519, X25519, ristretto255](#ed25519-x25519-ristretto255)
- [ed448, X448, decaf448](#ed448-x448-decaf448)
@@ -47,37 +52,32 @@ If you don't like NPM, a standalone [noble-curves.js](https://github.com/paulmil
- [All available imports](#all-available-imports)
- [Accessing a curve's variables](#accessing-a-curves-variables)
- [Abstract API](#abstract-api)
- [abstract/weierstrass: Short Weierstrass curve](#abstractweierstrass-short-weierstrass-curve)
- [abstract/edwards: Twisted Edwards curve](#abstractedwards-twisted-edwards-curve)
- [abstract/montgomery: Montgomery curve](#abstractmontgomery-montgomery-curve)
- [abstract/bls: Barreto-Lynn-Scott curves](#abstractbls-barreto-lynn-scott-curves)
- [abstract/hash-to-curve: Hashing strings to curve points](#abstracthash-to-curve-hashing-strings-to-curve-points)
- [abstract/poseidon: Poseidon hash](#abstractposeidon-poseidon-hash)
- [abstract/modular: Modular arithmetics utilities](#abstractmodular-modular-arithmetics-utilities)
- [weierstrass: Short Weierstrass curve](#weierstrass-short-weierstrass-curve)
- [edwards: Twisted Edwards curve](#edwards-twisted-edwards-curve)
- [montgomery: Montgomery curve](#montgomery-montgomery-curve)
- [bls: Barreto-Lynn-Scott curves](#bls-barreto-lynn-scott-curves)
- [hash-to-curve: Hashing strings to curve points](#hash-to-curve-hashing-strings-to-curve-points)
- [poseidon: Poseidon hash](#poseidon-poseidon-hash)
- [modular: Modular arithmetics utilities](#modular-modular-arithmetics-utilities)
- [Creating private keys from hashes](#creating-private-keys-from-hashes)
- [abstract/utils: Useful utilities](#abstractutils-useful-utilities)
- [utils: Useful utilities](#utils-useful-utilities)
- [Security](#security)
- [Speed](#speed)
- [Contributing & testing](#contributing--testing)
- [Upgrading](#upgrading)
- [Contributing & testing](#contributing--testing)
- [Resources](#resources)
- [Demos](#demos)
- [Projects using curves](#projects-using-curves)
- [License](#license)
### Implementations
Implementations are utilizing [noble-hashes](https://github.com/paulmillr/noble-hashes).
[Abstract API](#abstract-api) doesn't depend on them: you can use a different hashing library.
Implementations use [noble-hashes](https://github.com/paulmillr/noble-hashes).
If you want to use a different hashing library, [abstract API](#abstract-api) doesn't depend on them.
#### ECDSA signature scheme
Generic example that works for all curves, shown for secp256k1:
```ts
// import * from '@noble/curves'; // Error: use sub-imports, to ensure small app size
import { secp256k1 } from '@noble/curves/secp256k1'; // ESM and Common.js
// import { secp256k1 } from 'npm:@noble/curves@1.2.0/secp256k1'; // Deno
import { secp256k1 } from '@noble/curves/secp256k1';
const priv = secp256k1.utils.randomPrivateKey();
const pub = secp256k1.getPublicKey(priv);
const msg = new Uint8Array(32).fill(1); // message hash (not message) in ecdsa
@@ -89,16 +89,20 @@ const privHex = '46c930bc7bb4db7f55da20798697421b98c4175a52c630294d75a84b9c12623
const pub2 = secp256k1.getPublicKey(privHex);
```
We support P256 (secp256r1), P384 (secp384r1), P521 (secp521r1).
#### ECDSA public key recovery & extra entropy
```ts
// 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
// extraEntropy https://moderncrypto.org/mail-archive/curves/2017/000925.html
const sigImprovedSecurity = secp256k1.sign(msg, priv, { extraEntropy: true });
```
#### ECDH (Elliptic Curve Diffie-Hellman)
#### ECDH: Elliptic Curve Diffie-Hellman
```ts
// 1. The output includes parity byte. Strip it using shared.slice(1)
@@ -274,7 +278,7 @@ 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.
### abstract/weierstrass: Short Weierstrass curve
### weierstrass: Short Weierstrass curve
```ts
import { weierstrass } from '@noble/curves/abstract/weierstrass';
@@ -316,6 +320,10 @@ type CHash = {
outputLen: number;
create(): any;
};
// example
function sha256(message: Uint8Array) { return _internal_lowlvl(message) }
sha256.outputLen = 32; // 32 bytes of output for sha2-256
```
**Message hash** is expected instead of message itself:
@@ -441,7 +449,7 @@ const fast = secq256k1.utils.precompute(8, Point.fromHex(someonesPubKey));
fast.multiply(privKey); // much faster ECDH now
```
### abstract/edwards: Twisted Edwards curve
### edwards: Twisted Edwards curve
```ts
import { twistedEdwards } from '@noble/curves/abstract/edwards';
@@ -531,7 +539,7 @@ interface ExtPointConstructor extends GroupConstructor<ExtPointType> {
}
```
### abstract/montgomery: Montgomery curve
### montgomery: Montgomery curve
```typescript
import { montgomery } from '@noble/curves/abstract/montgomery';
@@ -558,7 +566,7 @@ Proper Elliptic Curve Points are not implemented yet.
You must specify curve params `Fp`, `a`, `Gu` coordinate of u, `montgomeryBits` and `nByteLength`.
### abstract/bls: Barreto-Lynn-Scott curves
### 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
@@ -567,6 +575,8 @@ use aggregated, batch-verifiable
using Boneh-Lynn-Shacham signature scheme.
The module doesn't expose `CURVE` property: use `G1.CURVE`, `G2.CURVE` instead.
Only BLS12-381 is implemented currently.
Defining BLS12-377 and BLS24 should be straightforward.
Main methods and properties are:
@@ -579,8 +589,13 @@ Main methods and properties are:
- `Signature` property with `fromHex`, `toHex` methods
- `fields` containing `Fp`, `Fp2`, `Fp6`, `Fp12`, `Fr`
Right now we only implement BLS12-381 (compatible with ETH and others),
but in theory defining BLS12-377, BLS24 should be straightforward. An example:
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) is also supported, using:
- `getPublicKeyForShortSignatures(privateKey)`
- `signShortSignature(message, privateKey)`
- `verifyShortSignature(signature, message, publicKey)`
- `aggregateShortSignatures(signatures)`
```ts
import { bls12_381 as bls } from '@noble/curves/bls12-381';
@@ -612,68 +627,19 @@ const isValid3 = bls.verifyBatch(aggSignature3, messages, publicKeys);
console.log({ publicKeys, signatures3, aggSignature3, isValid3 });
// Pairings, with and without final exponentiation
// bls.pairing(PointG1, PointG2);
// bls.pairing(PointG1, PointG2, false);
// bls.fields.Fp12.finalExponentiate(bls.fields.Fp12.mul(eGS, ePHm));
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
bls.G1.ProjectivePoint.BASE, bls.G2.ProjectivePoint.BASE
bls.fields.Fp, bls.fields.Fp2, bls.fields.Fp12, bls.fields.Fr
bls.params.x, bls.params.r, bls.params.G1b, bls.params.G2b
// hash-to-curve examples can be seen below
```
Full types:
```ts
getPublicKey: (privateKey: PrivKey) => Uint8Array;
sign: {
(message: Hex, privateKey: PrivKey): Uint8Array;
(message: ProjPointType<Fp2>, privateKey: PrivKey): ProjPointType<Fp2>;
};
verify: (
signature: Hex | ProjPointType<Fp2>,
message: Hex | ProjPointType<Fp2>,
publicKey: Hex | ProjPointType<Fp>
) => boolean;
verifyBatch: (
signature: Hex | ProjPointType<Fp2>,
messages: (Hex | ProjPointType<Fp2>)[],
publicKeys: (Hex | ProjPointType<Fp>)[]
) => boolean;
aggregatePublicKeys: {
(publicKeys: Hex[]): Uint8Array;
(publicKeys: ProjPointType<Fp>[]): ProjPointType<Fp>;
};
aggregateSignatures: {
(signatures: Hex[]): Uint8Array;
(signatures: ProjPointType<Fp2>[]): ProjPointType<Fp2>;
};
millerLoop: (ell: [Fp2, Fp2, Fp2][], g1: [Fp, Fp]) => Fp12;
pairing: (P: ProjPointType<Fp>, Q: ProjPointType<Fp2>, withFinalExponent?: boolean) => Fp12;
G1: CurvePointsRes<Fp> & ReturnType<typeof htf.createHasher<Fp>>;
G2: CurvePointsRes<Fp2> & ReturnType<typeof htf.createHasher<Fp2>>;
Signature: SignatureCoder<Fp2>;
params: {
x: bigint;
r: bigint;
G1b: bigint;
G2b: Fp2;
};
fields: {
Fp: IField<Fp>;
Fp2: IField<Fp2>;
Fp6: IField<Fp6>;
Fp12: IField<Fp12>;
Fr: IField<bigint>;
};
utils: {
randomPrivateKey: () => Uint8Array;
calcPairingPrecomputes: (p: AffinePoint<Fp2>) => [Fp2, Fp2, Fp2][];
};
```
### abstract/hash-to-curve: Hashing strings to curve points
### 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).
@@ -731,7 +697,7 @@ type Opts = {
};
```
### abstract/poseidon: Poseidon hash
### poseidon: Poseidon hash
Implements [Poseidon](https://www.poseidon-hash.info) ZK-friendly hash.
@@ -755,7 +721,7 @@ type PoseidonOpts = {
const instance = poseidon(opts: PoseidonOpts);
```
### abstract/modular: Modular arithmetics utilities
### modular: Modular arithmetics utilities
```ts
import * as mod from '@noble/curves/abstract/modular';
@@ -799,12 +765,13 @@ if you need to hash to **public key**.
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);
```
### abstract/utils: Useful utilities
### utils: Useful utilities
```ts
import * as utils from '@noble/curves/abstract/utils';
@@ -826,43 +793,61 @@ utils.equalBytes(Uint8Array.from([0xde]), Uint8Array.from([0xde]));
## Security
1. The library has been independently audited:
The library has been independently audited:
- in Feb 2023 by [Trail of Bits](https://www.trailofbits.com):
[PDF](https://github.com/trailofbits/publications/blob/master/reviews/2023-01-ryanshea-noblecurveslibrary-securityreview.pdf).
The audit has been funded by [Ryan Shea](https://www.shea.io).
Audit scope was abstract modules `curve`, `hash-to-curve`, `modular`, `poseidon`, `utils`, `weierstrass`,
and top-level modules `_shortw_utils` and `secp256k1`.
See [changes since v0.7.3 audit](https://github.com/paulmillr/noble-curves/compare/0.7.3..main).
- at version 1.2.0, in Sep 2023, by [Kudelski Security](https://kudelskisecurity.com)
- PDFs: [offline](./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: [online](https://github.com/trailofbits/publications/blob/master/reviews/2023-01-ryanshea-noblecurveslibrary-securityreview.pdf),
[offline](./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)
2. The library has been fuzzed by [Guido Vranken's cryptofuzz](https://github.com/guidovranken/cryptofuzz).
You can run the fuzzer by yourself to check it.
3. [Timing attack](https://en.wikipedia.org/wiki/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](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. Nonetheless we're targetting algorithmic constant time.
It is tested against property-based, cross-library and Wycheproof vectors,
and has fuzzing by [Guido Vranken's cryptofuzz](https://github.com/guidovranken/cryptofuzz).
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.
As for devDependencies used by the library:
If you see anything unusual: investigate and report.
- `@scure` base, bip32, bip39 (used in tests), micro-bmark (benchmark), micro-should (testing)
are developed by us and follow the same practices such as: minimal library size, auditability,
signed releases
- prettier (linter), fast-check (property-based testing), typescript versions
are locked and rarely updated. Every update is checked with `npm-diff`.
The packages are big, which makes it hard to audit their source code thoroughly and fully.
- They are only used if you clone the git repo and want to add some feature to it. End-users won't use them.
### Constant-timeness
As for key generation, we're deferring to built-in
_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. Nonetheless we're targetting algorithmic constant time.
### 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
* **Rare releasing** is followed to ensure less re-audit need for end-users
* **Dependencies** are minimized and locked-down:
- If your app has 500 dependencies, any dep could get hacked and you'll be downloading
malware with every install. We make sure to use as few dependencies as possible
- We prevent automatic dependency updates by locking-down version ranges. Every update is checked with `npm-diff`
- One dependency [noble-hashes](https://github.com/paulmillr/noble-hashes) is used, by the same author, to provide hashing functionality
* **Dev Dependencies** are only used if you want to contribute to the repo. They are disabled for end-users:
- scure-base, scure-bip32, scure-bip39, micro-bmark and micro-should are developed by the same author and follow identical security practices
- prettier (linter), fast-check (property-based testing) and typescript are used for code quality, vector generation and ts compilation. The packages are big, which makes it hard to audit their source code thoroughly and fully
### 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.
## Speed
Benchmark results on Apple M2 with node v20:
@@ -944,13 +929,6 @@ ed25519 x 3,088 ops/sec @ 323μs/op
ed448 x 1,247 ops/sec @ 801μs/op
```
## 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
## Upgrading
Previously, the library was split into single-feature packages
@@ -971,7 +949,7 @@ Upgrading from noble-secp256k1 1.7:
- to use old behavior, which produced 65-byte uncompressed keys, set
argument `isCompressed` to `false`: `getPublicKey(priv, false)`
- `sign`
- is now sync; use `signAsync` for async version
- 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
@@ -980,6 +958,7 @@ Upgrading from noble-secp256k1 1.7:
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
@@ -1009,61 +988,18 @@ Upgrading from [@noble/bls12-381](https://github.com/paulmillr/noble-bls12-381):
- PointG2.fromSignature -> Signature.decode, PointG2.toSignature -> Signature.encode
- Fp2 ORDER was corrected
## 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
## Resources
- [Learning fast elliptic-curve cryptography](https://paulmillr.com/posts/noble-secp256k1-fast-ecc/)
- EdDSA
- [A Deep dive into Ed25519 Signatures](https://cendyne.dev/posts/2022-03-06-ed25519-signatures.html)
- [Ed25519 Deep Dive Addendum](https://cendyne.dev/posts/2022-09-11-ed25519-deep-dive-addendum.html)
- [Its 255:19AM. Do you know what your validation criteria are?](https://hdevalence.ca/blog/2020-10-04-its-25519am)
- [Taming the many EdDSAs](https://csrc.nist.gov/csrc/media/Presentations/2023/crclub-2023-03-08/images-media/20230308-crypto-club-slides--taming-the-many-EdDSAs.pdf)
that describes concepts of Strong UnForgeability under Chosen Message Attacks and Strongly Binding Signatures
- [Cofactor Explained: Clearing Elliptic Curves dirty little secret](https://loup-vaillant.fr/tutorials/cofactor)
- [Surrounded by Elligators](https://loup-vaillant.fr/articles/implementing-elligator)
- Pairings and BLS
- [BLS signatures for busy people](https://gist.github.com/paulmillr/18b802ad219b1aee34d773d08ec26ca2)
- [BLS12-381 for the rest of us](https://hackmd.io/@benjaminion/bls12-381)
- [Key concepts of pairings](https://medium.com/@alonmuroch_65570/bls-signatures-part-2-key-concepts-of-pairings-27a8a9533d0c)
- Pairing over bls12-381:
[fields](https://research.nccgroup.com/2020/07/06/pairing-over-bls12-381-part-1-fields/),
[curves](https://research.nccgroup.com/2020/07/13/pairing-over-bls12-381-part-2-curves/),
[pairings](https://research.nccgroup.com/2020/08/13/pairing-over-bls12-381-part-3-pairing/)
- [Estimating the bit security of pairing-friendly curves](https://research.nccgroup.com/2022/02/03/estimating-the-bit-security-of-pairing-friendly-curves/)
### Demos
- [Elliptic Curve Calculator](https://paulmillr.com/noble): add / multiply points, sign messages
- [BLS threshold signatures](https://genthresh.com)
### Projects using curves
- HDkey libraries: [scure-bip32](https://github.com/paulmillr/scure-bip32), [bip32](https://github.com/bitcoinjs/bip32)
- Social networks: [nostr](https://github.com/nbd-wtf/nostr-tools), [bluesky](https://github.com/bluesky-social/atproto)
- Ethereum libraries:
- [ethereum-cryptography](https://github.com/ethereum/js-ethereum-cryptography)
- [micro-eth-signer](https://github.com/paulmillr/micro-eth-signer),
[ethers](https://github.com/ethers-io/ethers.js) (old noble),
[viem.sh](https://viem.sh),
[@ethereumjs](https://github.com/ethereumjs/ethereumjs-monorepo)
- [metamask's eth-sig-util](https://github.com/MetaMask/eth-sig-util)
- [gridplus lattice sdk](https://github.com/GridPlus/lattice-eth2-utils)
- Bitcoin libraries:
- [scure-btc-signer](https://github.com/paulmillr/scure-btc-signer)
- [tapscript](https://github.com/cmdruid/tapscript)
- Solana libraries: [micro-sol-signer](https://github.com/paulmillr/micro-sol-signer), [solana-web3.js](https://github.com/solana-labs/solana-web3.js)
- Other web3 stuff:
- [scure-starknet](https://github.com/paulmillr/scure-starknet)
- [aztec](https://github.com/AztecProtocol/aztec-packages)
- [polkadot.js](https://github.com/polkadot-js/common), [drand-client](https://github.com/drand/drand-client), [moneroj](https://github.com/beritani/moneroj), [tronlib](https://github.com/CoinSpace/tronlib)
- [protonmail](https://github.com/ProtonMail/WebClients) (old noble for now)
- [did-jwt](https://github.com/decentralized-identity/did-jwt), [hpke-js](https://github.com/dajiaji/hpke-js),
[js-libp2p-noise](https://github.com/ChainSafe/js-libp2p-noise)
- [ed25519-keygen](https://github.com/paulmillr/ed25519-keygen) SSH, PGP, TOR key generation
- [secp256k1 compatibility layer](https://github.com/ethereum/js-ethereum-cryptography/blob/2.0.0/src/secp256k1-compat.ts)
for users who want to switch from secp256k1-node or tiny-secp256k1. Allows to see which methods map to corresponding noble code.
- [BLS BBS signatures](https://github.com/Wind4Greg/BBS-Draft-Checks) following [draft-irtf-cfrg-bbs-signatures-latest](https://identity.foundation/bbs-signature/draft-irtf-cfrg-bbs-signatures.html)
- [KZG trusted setup ceremony](https://github.com/dsrvlabs/czg-keremony)
- See [full list of projects on GitHub](https://github.com/paulmillr/noble-curves/network/dependents).
Check out [paulmillr.com/noble](https://paulmillr.com/noble/)
for useful resources, articles, documentation and demos
related to the library.
## License

View File

@@ -1,5 +1,7 @@
# Security Policy
See [README's Security section](./README.md#security) for detailed description of internal security practices.
## Supported Versions
| Version | Supported |

Binary file not shown.

View File

@@ -1,11 +1,7 @@
# Audit
The library has been audited during Jan-Feb 2023 by an independent security firm [Trail of Bits](https://www.trailofbits.com):
[PDF](https://github.com/trailofbits/publications/blob/master/reviews/2023-01-ryanshea-noblecurveslibrary-securityreview.pdf).
The audit has been funded by Ryan Shea. Audit scope was abstract modules `curve`, `hash-to-curve`, `modular`, `poseidon`, `utils`, `weierstrass`, and top-level modules `_shortw_utils` and `secp256k1`. See [changes since audit](https://github.com/paulmillr/noble-curves/compare/0.7.3..main).
All audits of the library are described in [README's Security section](../README.md#security)
File in the directory was saved from
`2023-01-trailofbits-audit-curves.pdf` file in the directory was saved from
[github.com/trailofbits/publications](https://github.com/trailofbits/publications).
Check out their repo and verify checksums to ensure the PDF in this directory has not been altered.
See information about fuzzing in root [README](../README.md).

9
benchmark/utils.js Normal file
View File

@@ -0,0 +1,9 @@
import { hexToBytes } from '../abstract/utils.js';
import { run, mark } from 'micro-bmark';
run(async () => {
const hex32 = '0123456789abcdef'.repeat(4);
const hex256 = hex32.repeat(8);
await mark('hexToBytes 32b', 5000000, () => hexToBytes(hex32));
await mark('hexToBytes 256b', 500000, () => hexToBytes(hex256));
});

View File

@@ -1,8 +1,17 @@
import { bytesToHex, concatBytes, hexToBytes } from '@noble/curves/abstract/utils';
export { secp256k1 } from '@noble/curves/secp256k1';
export { ed25519, x25519 } from '@noble/curves/ed25519';
export { ed448, x448 } from '@noble/curves/ed448';
export { secp256k1, schnorr as secp256k1_schnorr } from '@noble/curves/secp256k1';
export {
ed25519,
x25519,
edwardsToMontgomeryPub as ed25519_edwardsToMontgomeryPub,
edwardsToMontgomeryPriv as ed25519_edwardsToMontgomeryPriv,
} from '@noble/curves/ed25519';
export {
ed448,
x448,
edwardsToMontgomeryPub as ed448_edwardsToMontgomeryPub,
} from '@noble/curves/ed448';
export { p256 } from '@noble/curves/p256';
export { p384 } from '@noble/curves/p384';
export { p521 } from '@noble/curves/p521';

34
package-lock.json generated
View File

@@ -1,31 +1,31 @@
{
"name": "@noble/curves",
"version": "1.2.0",
"version": "1.3.0",
"lockfileVersion": 3,
"requires": true,
"packages": {
"": {
"name": "@noble/curves",
"version": "1.2.0",
"version": "1.3.0",
"license": "MIT",
"dependencies": {
"@noble/hashes": "1.3.2"
"@noble/hashes": "1.3.3"
},
"devDependencies": {
"fast-check": "3.0.0",
"micro-bmark": "0.3.1",
"micro-should": "0.4.0",
"prettier": "2.8.4",
"typescript": "5.0.2"
"prettier": "3.1.1",
"typescript": "5.3.2"
},
"funding": {
"url": "https://paulmillr.com/funding/"
}
},
"node_modules/@noble/hashes": {
"version": "1.3.2",
"resolved": "https://registry.npmjs.org/@noble/hashes/-/hashes-1.3.2.tgz",
"integrity": "sha512-MVC8EAQp7MvEcm30KWENFjgR+Mkmf+D189XJTkFIlwohU5hcBbn1ZkKq7KVTi2Hme3PMGF390DaL52beVrIihQ==",
"version": "1.3.3",
"resolved": "https://registry.npmjs.org/@noble/hashes/-/hashes-1.3.3.tgz",
"integrity": "sha512-V7/fPHgl+jsVPXqqeOzT8egNj2iBIVt+ECeMMG8TdcnTikP3oaBtUVqpT/gYCR68aEBJSF+XbYUxStjbFMqIIA==",
"engines": {
"node": ">= 16"
},
@@ -62,15 +62,15 @@
"dev": true
},
"node_modules/prettier": {
"version": "2.8.4",
"resolved": "https://registry.npmjs.org/prettier/-/prettier-2.8.4.tgz",
"integrity": "sha512-vIS4Rlc2FNh0BySk3Wkd6xmwxB0FpOndW5fisM5H8hsZSxU2VWVB5CWIkIjWvrHjIhxk2g3bfMKM87zNTrZddw==",
"version": "3.1.1",
"resolved": "https://registry.npmjs.org/prettier/-/prettier-3.1.1.tgz",
"integrity": "sha512-22UbSzg8luF4UuZtzgiUOfcGM8s4tjBv6dJRT7j275NXsy2jb4aJa4NNveul5x4eqlF1wuhuR2RElK71RvmVaw==",
"dev": true,
"bin": {
"prettier": "bin-prettier.js"
"prettier": "bin/prettier.cjs"
},
"engines": {
"node": ">=10.13.0"
"node": ">=14"
},
"funding": {
"url": "https://github.com/prettier/prettier?sponsor=1"
@@ -93,16 +93,16 @@
]
},
"node_modules/typescript": {
"version": "5.0.2",
"resolved": "https://registry.npmjs.org/typescript/-/typescript-5.0.2.tgz",
"integrity": "sha512-wVORMBGO/FAs/++blGNeAVdbNKtIh1rbBL2EyQ1+J9lClJ93KiiKe8PmFIVdXhHcyv44SL9oglmfeSsndo0jRw==",
"version": "5.3.2",
"resolved": "https://registry.npmjs.org/typescript/-/typescript-5.3.2.tgz",
"integrity": "sha512-6l+RyNy7oAHDfxC4FzSJcz9vnjTKxrLpDG5M2Vu4SHRVNg6xzqZp6LYSR9zjqQTu8DU/f5xwxUdADOkbrIX2gQ==",
"dev": true,
"bin": {
"tsc": "bin/tsc",
"tsserver": "bin/tsserver"
},
"engines": {
"node": ">=12.20"
"node": ">=14.17"
}
}
}

View File

@@ -1,6 +1,6 @@
{
"name": "@noble/curves",
"version": "1.2.0",
"version": "1.3.0",
"description": "Audited & minimal JS implementation of elliptic curve cryptography",
"files": [
"abstract",
@@ -14,7 +14,7 @@
"scripts": {
"bench": "cd benchmark; node secp256k1.js; node curves.js; node ecdh.js; node hash-to-curve.js; node modular.js; node bls.js; node ristretto255.js; node decaf448.js",
"build": "tsc && tsc -p tsconfig.esm.json",
"build:release": "rollup -c rollup.config.js",
"build:release": "cd build; npm install && npm run build",
"build:clean": "rm *.{js,d.ts,d.ts.map,js.map} esm/*.{js,d.ts,d.ts.map,js.map} 2> /dev/null",
"lint": "prettier --check 'src/**/*.{js,ts}' 'test/*.js'",
"format": "prettier --write 'src/**/*.{js,ts}' 'test/*.js'",
@@ -24,18 +24,18 @@
"homepage": "https://paulmillr.com/noble/",
"repository": {
"type": "git",
"url": "https://github.com/paulmillr/noble-curves.git"
"url": "git+https://github.com/paulmillr/noble-curves.git"
},
"license": "MIT",
"dependencies": {
"@noble/hashes": "1.3.2"
"@noble/hashes": "1.3.3"
},
"devDependencies": {
"fast-check": "3.0.0",
"micro-bmark": "0.3.1",
"micro-should": "0.4.0",
"prettier": "2.8.4",
"typescript": "5.0.2"
"prettier": "3.1.1",
"typescript": "5.3.2"
},
"sideEffects": false,
"main": "index.js",
@@ -178,4 +178,4 @@
"schnorr"
],
"funding": "https://paulmillr.com/funding/"
}
}

View File

@@ -14,7 +14,11 @@
import { AffinePoint } from './curve.js';
import { IField, getMinHashLength, mapHashToField } from './modular.js';
import { Hex, PrivKey, CHash, bitLen, bitGet, ensureBytes } from './utils.js';
import * as htf from './hash-to-curve.js';
// prettier-ignore
import {
MapToCurve, Opts as HTFOpts, H2CPointConstructor, htfBasicOpts,
createHasher
} from './hash-to-curve.js';
import {
CurvePointsType,
ProjPointType as ProjPointType,
@@ -27,6 +31,12 @@ type Fp = bigint; // Can be different field?
// prettier-ignore
const _2n = BigInt(2), _3n = BigInt(3);
export type ShortSignatureCoder<Fp> = {
fromHex(hex: Hex): ProjPointType<Fp>;
toRawBytes(point: ProjPointType<Fp>): Uint8Array;
toHex(point: ProjPointType<Fp>): string;
};
export type SignatureCoder<Fp2> = {
fromHex(hex: Hex): ProjPointType<Fp2>;
toRawBytes(point: ProjPointType<Fp2>): Uint8Array;
@@ -35,13 +45,14 @@ export type SignatureCoder<Fp2> = {
export type CurveType<Fp, Fp2, Fp6, Fp12> = {
G1: Omit<CurvePointsType<Fp>, 'n'> & {
mapToCurve: htf.MapToCurve<Fp>;
htfDefaults: htf.Opts;
ShortSignature: SignatureCoder<Fp>;
mapToCurve: MapToCurve<Fp>;
htfDefaults: HTFOpts;
};
G2: Omit<CurvePointsType<Fp2>, 'n'> & {
Signature: SignatureCoder<Fp2>;
mapToCurve: htf.MapToCurve<Fp2>;
htfDefaults: htf.Opts;
mapToCurve: MapToCurve<Fp2>;
htfDefaults: HTFOpts;
};
fields: {
Fp: IField<Fp>;
@@ -63,26 +74,39 @@ export type CurveType<Fp, Fp2, Fp6, Fp12> = {
x: bigint;
r: bigint;
};
htfDefaults: htf.Opts;
htfDefaults: HTFOpts;
hash: CHash; // Because we need outputLen for DRBG
randomBytes: (bytesLength?: number) => Uint8Array;
};
export type CurveFn<Fp, Fp2, Fp6, Fp12> = {
getPublicKey: (privateKey: PrivKey) => Uint8Array;
getPublicKeyForShortSignatures: (privateKey: PrivKey) => Uint8Array;
sign: {
(message: Hex, privateKey: PrivKey): Uint8Array;
(message: ProjPointType<Fp2>, privateKey: PrivKey): ProjPointType<Fp2>;
};
signShortSignature: {
(message: Hex, privateKey: PrivKey): Uint8Array;
(message: ProjPointType<Fp>, privateKey: PrivKey): ProjPointType<Fp>;
};
verify: (
signature: Hex | ProjPointType<Fp2>,
message: Hex | ProjPointType<Fp2>,
publicKey: Hex | ProjPointType<Fp>
publicKey: Hex | ProjPointType<Fp>,
htfOpts?: htfBasicOpts
) => boolean;
verifyShortSignature: (
signature: Hex | ProjPointType<Fp>,
message: Hex | ProjPointType<Fp>,
publicKey: Hex | ProjPointType<Fp2>,
htfOpts?: htfBasicOpts
) => boolean;
verifyBatch: (
signature: Hex | ProjPointType<Fp2>,
messages: (Hex | ProjPointType<Fp2>)[],
publicKeys: (Hex | ProjPointType<Fp>)[]
publicKeys: (Hex | ProjPointType<Fp>)[],
htfOpts?: htfBasicOpts
) => boolean;
aggregatePublicKeys: {
(publicKeys: Hex[]): Uint8Array;
@@ -92,11 +116,16 @@ export type CurveFn<Fp, Fp2, Fp6, Fp12> = {
(signatures: Hex[]): Uint8Array;
(signatures: ProjPointType<Fp2>[]): ProjPointType<Fp2>;
};
aggregateShortSignatures: {
(signatures: Hex[]): Uint8Array;
(signatures: ProjPointType<Fp>[]): ProjPointType<Fp>;
};
millerLoop: (ell: [Fp2, Fp2, Fp2][], g1: [Fp, Fp]) => Fp12;
pairing: (P: ProjPointType<Fp>, Q: ProjPointType<Fp2>, withFinalExponent?: boolean) => Fp12;
G1: CurvePointsRes<Fp> & ReturnType<typeof htf.createHasher<Fp>>;
G2: CurvePointsRes<Fp2> & ReturnType<typeof htf.createHasher<Fp2>>;
G1: CurvePointsRes<Fp> & ReturnType<typeof createHasher<Fp>>;
G2: CurvePointsRes<Fp2> & ReturnType<typeof createHasher<Fp2>>;
Signature: SignatureCoder<Fp2>;
ShortSignature: ShortSignatureCoder<Fp>;
params: {
x: bigint;
r: bigint;
@@ -198,7 +227,7 @@ export function bls<Fp2, Fp6, Fp12>(
const G1_ = weierstrassPoints({ n: Fr.ORDER, ...CURVE.G1 });
const G1 = Object.assign(
G1_,
htf.createHasher(G1_.ProjectivePoint, CURVE.G1.mapToCurve, {
createHasher(G1_.ProjectivePoint, CURVE.G1.mapToCurve, {
...CURVE.htfDefaults,
...CURVE.G1.htfDefaults,
})
@@ -224,12 +253,13 @@ export function bls<Fp2, Fp6, Fp12>(
const G2_ = weierstrassPoints({ n: Fr.ORDER, ...CURVE.G2 });
const G2 = Object.assign(
G2_,
htf.createHasher(G2_.ProjectivePoint as htf.H2CPointConstructor<Fp2>, CURVE.G2.mapToCurve, {
createHasher(G2_.ProjectivePoint as H2CPointConstructor<Fp2>, CURVE.G2.mapToCurve, {
...CURVE.htfDefaults,
...CURVE.G2.htfDefaults,
})
);
const { ShortSignature } = CURVE.G1;
const { Signature } = CURVE.G2;
// Calculates bilinear pairing
@@ -251,26 +281,37 @@ export function bls<Fp2, Fp6, Fp12>(
function normP1(point: G1Hex): G1 {
return point instanceof G1.ProjectivePoint ? (point as G1) : G1.ProjectivePoint.fromHex(point);
}
function normP1Hash(point: G1Hex, htfOpts?: htfBasicOpts): G1 {
return point instanceof G1.ProjectivePoint
? point
: (G1.hashToCurve(ensureBytes('point', point), htfOpts) as G1);
}
function normP2(point: G2Hex): G2 {
return point instanceof G2.ProjectivePoint ? point : Signature.fromHex(point);
}
function normP2Hash(point: G2Hex, htfOpts?: htf.htfBasicOpts): G2 {
function normP2Hash(point: G2Hex, htfOpts?: htfBasicOpts): G2 {
return point instanceof G2.ProjectivePoint
? point
: (G2.hashToCurve(ensureBytes('point', point), htfOpts) as G2);
}
// Multiplies generator by private key.
// Multiplies generator (G1) by private key.
// P = pk x G
function getPublicKey(privateKey: PrivKey): Uint8Array {
return G1.ProjectivePoint.fromPrivateKey(privateKey).toRawBytes(true);
}
// Multiplies generator (G2) by private key.
// P = pk x G
function getPublicKeyForShortSignatures(privateKey: PrivKey): Uint8Array {
return G2.ProjectivePoint.fromPrivateKey(privateKey).toRawBytes(true);
}
// Executes `hashToCurve` on the message and then multiplies the result by private key.
// S = pk x H(m)
function sign(message: Hex, privateKey: PrivKey, htfOpts?: htf.htfBasicOpts): Uint8Array;
function sign(message: G2, privateKey: PrivKey, htfOpts?: htf.htfBasicOpts): G2;
function sign(message: G2Hex, privateKey: PrivKey, htfOpts?: htf.htfBasicOpts): Uint8Array | G2 {
function sign(message: Hex, privateKey: PrivKey, htfOpts?: htfBasicOpts): Uint8Array;
function sign(message: G2, privateKey: PrivKey, htfOpts?: htfBasicOpts): G2;
function sign(message: G2Hex, privateKey: PrivKey, htfOpts?: htfBasicOpts): Uint8Array | G2 {
const msgPoint = normP2Hash(message, htfOpts);
msgPoint.assertValidity();
const sigPoint = msgPoint.multiply(G1.normPrivateKeyToScalar(privateKey));
@@ -278,13 +319,31 @@ export function bls<Fp2, Fp6, Fp12>(
return Signature.toRawBytes(sigPoint);
}
function signShortSignature(
message: Hex,
privateKey: PrivKey,
htfOpts?: htfBasicOpts
): Uint8Array;
function signShortSignature(message: G1, privateKey: PrivKey, htfOpts?: htfBasicOpts): G1;
function signShortSignature(
message: G1Hex,
privateKey: PrivKey,
htfOpts?: htfBasicOpts
): Uint8Array | G1 {
const msgPoint = normP1Hash(message, htfOpts);
msgPoint.assertValidity();
const sigPoint = msgPoint.multiply(G1.normPrivateKeyToScalar(privateKey));
if (message instanceof G1.ProjectivePoint) return sigPoint;
return ShortSignature.toRawBytes(sigPoint);
}
// Checks if pairing of public key & hash is equal to pairing of generator & signature.
// e(P, H(m)) == e(G, S)
function verify(
signature: G2Hex,
message: G2Hex,
publicKey: G1Hex,
htfOpts?: htf.htfBasicOpts
htfOpts?: htfBasicOpts
): boolean {
const P = normP1(publicKey);
const Hm = normP2Hash(message, htfOpts);
@@ -298,6 +357,26 @@ export function bls<Fp2, Fp6, Fp12>(
return Fp12.eql(exp, Fp12.ONE);
}
// Checks if pairing of public key & hash is equal to pairing of generator & signature.
// e(S, G) == e(H(m), P)
function verifyShortSignature(
signature: G1Hex,
message: G1Hex,
publicKey: G2Hex,
htfOpts?: htfBasicOpts
): boolean {
const P = normP2(publicKey);
const Hm = normP1Hash(message, htfOpts);
const G = G2.ProjectivePoint.BASE;
const S = normP1(signature);
// Instead of doing 2 exponentiations, we use property of billinear maps
// and do one exp after multiplying 2 points.
const eHmP = pairing(Hm, P, false);
const eSG = pairing(S, G.negate(), false);
const exp = Fp12.finalExponentiate(Fp12.mul(eSG, eHmP));
return Fp12.eql(exp, Fp12.ONE);
}
// Adds a bunch of public key points together.
// pk1 + pk2 + pk3 = pkA
function aggregatePublicKeys(publicKeys: Hex[]): Uint8Array;
@@ -328,13 +407,27 @@ export function bls<Fp2, Fp6, Fp12>(
return Signature.toRawBytes(aggAffine);
}
// Adds a bunch of signature points together.
function aggregateShortSignatures(signatures: Hex[]): Uint8Array;
function aggregateShortSignatures(signatures: G1[]): G1;
function aggregateShortSignatures(signatures: G1Hex[]): Uint8Array | G1 {
if (!signatures.length) throw new Error('Expected non-empty array');
const agg = signatures.map(normP1).reduce((sum, s) => sum.add(s), G1.ProjectivePoint.ZERO);
const aggAffine = agg; //.toAffine();
if (signatures[0] instanceof G1.ProjectivePoint) {
aggAffine.assertValidity();
return aggAffine;
}
return ShortSignature.toRawBytes(aggAffine);
}
// https://ethresear.ch/t/fast-verification-of-multiple-bls-signatures/5407
// e(G, S) = e(G, SUM(n)(Si)) = MUL(n)(e(G, Si))
function verifyBatch(
signature: G2Hex,
messages: G2Hex[],
publicKeys: G1Hex[],
htfOpts?: htf.htfBasicOpts
htfOpts?: htfBasicOpts
): boolean {
// @ts-ignore
// console.log('verifyBatch', bytesToHex(signature as any), messages, publicKeys.map(bytesToHex));
@@ -370,16 +463,21 @@ export function bls<Fp2, Fp6, Fp12>(
return {
getPublicKey,
getPublicKeyForShortSignatures,
sign,
signShortSignature,
verify,
verifyBatch,
verifyShortSignature,
aggregatePublicKeys,
aggregateSignatures,
aggregateShortSignatures,
millerLoop,
pairing,
G1,
G2,
Signature,
ShortSignature,
fields: {
Fr,
Fp,

View File

@@ -1,7 +1,8 @@
/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import type { Group, GroupConstructor, AffinePoint } from './curve.js';
import { mod, IField } from './modular.js';
import { bytesToNumberBE, CHash, concatBytes, utf8ToBytes, validateObject } from './utils.js';
import type { CHash } from './utils.js';
import { bytesToNumberBE, isBytes, concatBytes, utf8ToBytes, validateObject } from './utils.js';
/**
* * `DST` is a domain separation tag, defined in section 2.2.5
@@ -22,7 +23,7 @@ export type Opts = {
};
function validateDST(dst: UnicodeOrBytes): Uint8Array {
if (dst instanceof Uint8Array) return dst;
if (isBytes(dst)) return dst;
if (typeof dst === 'string') return utf8ToBytes(dst);
throw new Error('DST must be Uint8Array or string');
}
@@ -51,8 +52,8 @@ function strxor(a: Uint8Array, b: Uint8Array): Uint8Array {
return arr;
}
function isBytes(item: unknown): void {
if (!(item instanceof Uint8Array)) throw new Error('Uint8Array expected');
function abytes(item: unknown): void {
if (!isBytes(item)) throw new Error('Uint8Array expected');
}
function isNum(item: unknown): void {
if (!Number.isSafeInteger(item)) throw new Error('number expected');
@@ -66,8 +67,8 @@ export function expand_message_xmd(
lenInBytes: number,
H: CHash
): Uint8Array {
isBytes(msg);
isBytes(DST);
abytes(msg);
abytes(DST);
isNum(lenInBytes);
// https://www.rfc-editor.org/rfc/rfc9380#section-5.3.3
if (DST.length > 255) DST = H(concatBytes(utf8ToBytes('H2C-OVERSIZE-DST-'), DST));
@@ -100,8 +101,8 @@ export function expand_message_xof(
k: number,
H: CHash
): Uint8Array {
isBytes(msg);
isBytes(DST);
abytes(msg);
abytes(DST);
isNum(lenInBytes);
// https://www.rfc-editor.org/rfc/rfc9380#section-5.3.3
// DST = H('H2C-OVERSIZE-DST-' || a_very_long_DST, Math.ceil((lenInBytes * k) / 8));
@@ -139,7 +140,7 @@ export function hash_to_field(msg: Uint8Array, count: number, options: Opts): bi
hash: 'hash',
});
const { p, k, m, hash, expand, DST: _DST } = options;
isBytes(msg);
abytes(msg);
isNum(count);
const DST = validateDST(_DST);
const log2p = p.toString(2).length;

View File

@@ -150,17 +150,15 @@ export function montgomery(curveDef: CurveType): CurveFn {
function decodeUCoordinate(uEnc: Hex): bigint {
// Section 5: When receiving such an array, implementations of X25519
// MUST mask the most significant bit in the final byte.
// This is very ugly way, but it works because fieldLen-1 is outside of bounds for X448, so this becomes NOOP
// fieldLen - scalaryBytes = 1 for X448 and = 0 for X25519
const u = ensureBytes('u coordinate', uEnc, montgomeryBytes);
// u[fieldLen-1] crashes QuickJS (TypeError: out-of-bound numeric index)
if (fieldLen === montgomeryBytes) u[fieldLen - 1] &= 127; // 0b0111_1111
if (fieldLen === 32) u[31] &= 127; // 0b0111_1111
return bytesToNumberLE(u);
}
function decodeScalar(n: Hex): bigint {
const bytes = ensureBytes('scalar', n);
if (bytes.length !== montgomeryBytes && bytes.length !== fieldLen)
throw new Error(`Expected ${montgomeryBytes} or ${fieldLen} bytes, got ${bytes.length}`);
const len = bytes.length;
if (len !== montgomeryBytes && len !== fieldLen)
throw new Error(`Expected ${montgomeryBytes} or ${fieldLen} bytes, got ${len}`);
return bytesToNumberLE(adjustScalarBytes(bytes));
}
function scalarMult(scalar: Hex, u: Hex): Uint8Array {

View File

@@ -6,7 +6,6 @@
const _0n = BigInt(0);
const _1n = BigInt(1);
const _2n = BigInt(2);
const u8a = (a: any): a is Uint8Array => a instanceof Uint8Array;
export type Hex = Uint8Array | string; // hex strings are accepted for simplicity
export type PrivKey = Hex | bigint; // bigints are accepted to ease learning curve
export type CHash = {
@@ -17,6 +16,14 @@ export type CHash = {
};
export type FHash = (message: Uint8Array | string) => Uint8Array;
export function isBytes(a: unknown): a is Uint8Array {
return (
a instanceof Uint8Array ||
(a != null && typeof a === 'object' && a.constructor.name === 'Uint8Array')
);
}
// Array where index 0xf0 (240) is mapped to string 'f0'
const hexes = /* @__PURE__ */ Array.from({ length: 256 }, (_, i) =>
i.toString(16).padStart(2, '0')
);
@@ -24,7 +31,7 @@ const hexes = /* @__PURE__ */ Array.from({ length: 256 }, (_, i) =>
* @example bytesToHex(Uint8Array.from([0xca, 0xfe, 0x01, 0x23])) // 'cafe0123'
*/
export function bytesToHex(bytes: Uint8Array): string {
if (!u8a(bytes)) throw new Error('Uint8Array expected');
if (!isBytes(bytes)) throw new Error('Uint8Array expected');
// pre-caching improves the speed 6x
let hex = '';
for (let i = 0; i < bytes.length; i++) {
@@ -44,20 +51,32 @@ export function hexToNumber(hex: string): bigint {
return BigInt(hex === '' ? '0' : `0x${hex}`);
}
// We use optimized technique to convert hex string to byte array
const asciis = { _0: 48, _9: 57, _A: 65, _F: 70, _a: 97, _f: 102 } as const;
function asciiToBase16(char: number): number | undefined {
if (char >= asciis._0 && char <= asciis._9) return char - asciis._0;
if (char >= asciis._A && char <= asciis._F) return char - (asciis._A - 10);
if (char >= asciis._a && char <= asciis._f) return char - (asciis._a - 10);
return;
}
/**
* @example hexToBytes('cafe0123') // Uint8Array.from([0xca, 0xfe, 0x01, 0x23])
*/
export function hexToBytes(hex: string): Uint8Array {
if (typeof hex !== 'string') throw new Error('hex string expected, got ' + typeof hex);
const len = hex.length;
if (len % 2) throw new Error('padded hex string expected, got unpadded hex of length ' + len);
const array = new Uint8Array(len / 2);
for (let i = 0; i < array.length; i++) {
const j = i * 2;
const hexByte = hex.slice(j, j + 2);
const byte = Number.parseInt(hexByte, 16);
if (Number.isNaN(byte) || byte < 0) throw new Error('Invalid byte sequence');
array[i] = byte;
const hl = hex.length;
const al = hl / 2;
if (hl % 2) throw new Error('padded hex string expected, got unpadded hex of length ' + hl);
const array = new Uint8Array(al);
for (let ai = 0, hi = 0; ai < al; ai++, hi += 2) {
const n1 = asciiToBase16(hex.charCodeAt(hi));
const n2 = asciiToBase16(hex.charCodeAt(hi + 1));
if (n1 === undefined || n2 === undefined) {
const char = hex[hi] + hex[hi + 1];
throw new Error('hex string expected, got non-hex character "' + char + '" at index ' + hi);
}
array[ai] = n1 * 16 + n2;
}
return array;
}
@@ -67,7 +86,7 @@ export function bytesToNumberBE(bytes: Uint8Array): bigint {
return hexToNumber(bytesToHex(bytes));
}
export function bytesToNumberLE(bytes: Uint8Array): bigint {
if (!u8a(bytes)) throw new Error('Uint8Array expected');
if (!isBytes(bytes)) throw new Error('Uint8Array expected');
return hexToNumber(bytesToHex(Uint8Array.from(bytes).reverse()));
}
@@ -99,7 +118,7 @@ export function ensureBytes(title: string, hex: Hex, expectedLength?: number): U
} catch (e) {
throw new Error(`${title} must be valid hex string, got "${hex}". Cause: ${e}`);
}
} else if (u8a(hex)) {
} else if (isBytes(hex)) {
// Uint8Array.from() instead of hash.slice() because node.js Buffer
// is instance of Uint8Array, and its slice() creates **mutable** copy
res = Uint8Array.from(hex);
@@ -116,21 +135,27 @@ export function ensureBytes(title: string, hex: Hex, expectedLength?: number): U
* Copies several Uint8Arrays into one.
*/
export function concatBytes(...arrays: Uint8Array[]): Uint8Array {
const r = new Uint8Array(arrays.reduce((sum, a) => sum + a.length, 0));
let pad = 0; // walk through each item, ensure they have proper type
arrays.forEach((a) => {
if (!u8a(a)) throw new Error('Uint8Array expected');
r.set(a, pad);
let sum = 0;
for (let i = 0; i < arrays.length; i++) {
const a = arrays[i];
if (!isBytes(a)) throw new Error('Uint8Array expected');
sum += a.length;
}
const res = new Uint8Array(sum);
for (let i = 0, pad = 0; i < arrays.length; i++) {
const a = arrays[i];
res.set(a, pad);
pad += a.length;
});
return r;
}
return res;
}
export function equalBytes(b1: Uint8Array, b2: Uint8Array) {
// We don't care about timing attacks here
if (b1.length !== b2.length) return false;
for (let i = 0; i < b1.length; i++) if (b1[i] !== b2[i]) return false;
return true;
// Compares 2 u8a-s in kinda constant time
export function equalBytes(a: Uint8Array, b: Uint8Array) {
if (a.length !== b.length) return false;
let diff = 0;
for (let i = 0; i < a.length; i++) diff |= a[i] ^ b[i];
return diff === 0;
}
// Global symbols in both browsers and Node.js since v11
@@ -248,7 +273,7 @@ const validatorFns = {
function: (val: any) => typeof val === 'function',
boolean: (val: any) => typeof val === 'boolean',
string: (val: any) => typeof val === 'string',
stringOrUint8Array: (val: any) => typeof val === 'string' || val instanceof Uint8Array,
stringOrUint8Array: (val: any) => typeof val === 'string' || isBytes(val),
isSafeInteger: (val: any) => Number.isSafeInteger(val),
array: (val: any) => Array.isArray(val),
field: (val: any, object: any) => (object as any).Fp.isValid(val),

View File

@@ -123,6 +123,7 @@ function validatePointOpts<T>(curve: CurvePointsType<T>) {
}
export type CurvePointsRes<T> = {
CURVE: ReturnType<typeof validatePointOpts<T>>;
ProjectivePoint: ProjConstructor<T>;
normPrivateKeyToScalar: (key: PrivKey) => bigint;
weierstrassEquation: (x: T) => T;
@@ -157,7 +158,7 @@ export const DER = {
// parse DER signature
const { Err: E } = DER;
const data = typeof hex === 'string' ? h2b(hex) : hex;
if (!(data instanceof Uint8Array)) throw new Error('ui8a expected');
if (!ut.isBytes(data)) throw new Error('ui8a expected');
let l = data.length;
if (l < 2 || data[0] != 0x30) throw new E('Invalid signature tag');
if (data[1] !== l - 2) throw new E('Invalid signature: incorrect length');
@@ -187,7 +188,7 @@ export const DER = {
// prettier-ignore
const _0n = BigInt(0), _1n = BigInt(1), _2n = BigInt(2), _3n = BigInt(3), _4n = BigInt(4);
export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
export function weierstrassPoints<T>(opts: CurvePointsType<T>): CurvePointsRes<T> {
const CURVE = validatePointOpts(opts);
const { Fp } = CURVE; // All curves has same field / group length as for now, but they can differ
@@ -237,7 +238,7 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
function normPrivateKeyToScalar(key: PrivKey): bigint {
const { allowedPrivateKeyLengths: lengths, nByteLength, wrapPrivateKey, n } = CURVE;
if (lengths && typeof key !== 'bigint') {
if (key instanceof Uint8Array) key = ut.bytesToHex(key);
if (ut.isBytes(key)) key = ut.bytesToHex(key);
// Normalize to hex string, pad. E.g. P521 would norm 130-132 char hex to 132-char bytes
if (typeof key !== 'string' || !lengths.includes(key.length)) throw new Error('Invalid key');
key = key.padStart(nByteLength * 2, '0');
@@ -269,7 +270,11 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
static readonly BASE = new Point(CURVE.Gx, CURVE.Gy, Fp.ONE);
static readonly ZERO = new Point(Fp.ZERO, Fp.ONE, Fp.ZERO);
constructor(readonly px: T, readonly py: T, readonly pz: T) {
constructor(
readonly px: T,
readonly py: T,
readonly pz: T
) {
if (px == null || !Fp.isValid(px)) throw new Error('x required');
if (py == null || !Fp.isValid(py)) throw new Error('y required');
if (pz == null || !Fp.isValid(pz)) throw new Error('z required');
@@ -728,7 +733,13 @@ export function weierstrass(curveDef: CurveType): CurveFn {
const x = ut.bytesToNumberBE(tail);
if (!isValidFieldElement(x)) throw new Error('Point is not on curve');
const y2 = weierstrassEquation(x); // y² = x³ + ax + b
let y = Fp.sqrt(y2); // y = y² ^ (p+1)/4
let y: bigint;
try {
y = Fp.sqrt(y2); // y = y² ^ (p+1)/4
} catch (sqrtError) {
const suffix = sqrtError instanceof Error ? ': ' + sqrtError.message : '';
throw new Error('Point is not on curve' + suffix);
}
const isYOdd = (y & _1n) === _1n;
// ECDSA
const isHeadOdd = (head & 1) === 1;
@@ -763,7 +774,11 @@ export function weierstrass(curveDef: CurveType): CurveFn {
* ECDSA signature with its (r, s) properties. Supports DER & compact representations.
*/
class Signature implements SignatureType {
constructor(readonly r: bigint, readonly s: bigint, readonly recovery?: number) {
constructor(
readonly r: bigint,
readonly s: bigint,
readonly recovery?: number
) {
this.assertValidity();
}
@@ -884,7 +899,7 @@ export function weierstrass(curveDef: CurveType): CurveFn {
* Quick and dirty check for item being public key. Does not validate hex, or being on-curve.
*/
function isProbPub(item: PrivKey | PubKey): boolean {
const arr = item instanceof Uint8Array;
const arr = ut.isBytes(item);
const str = typeof item === 'string';
const len = (arr || str) && (item as Hex).length;
if (arr) return len === compressedLen || len === uncompressedLen;
@@ -1048,7 +1063,7 @@ export function weierstrass(curveDef: CurveType): CurveFn {
let _sig: Signature | undefined = undefined;
let P: ProjPointType<bigint>;
try {
if (typeof sg === 'string' || sg instanceof Uint8Array) {
if (typeof sg === 'string' || ut.isBytes(sg)) {
// Signature can be represented in 2 ways: compact (2*nByteLength) & DER (variable-length).
// Since DER can also be 2*nByteLength bytes, we check for it first.
try {

View File

@@ -1,15 +1,9 @@
/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
// bls12-381 pairing-friendly Barreto-Lynn-Scott elliptic curve construction allows to:
// - Construct zk-SNARKs at the 128-bit security
// - Use threshold signatures, which allows a user to sign lots of messages with one signature and
// verify them swiftly in a batch, using Boneh-Lynn-Shacham signature scheme.
//
// The library uses G1 for public keys and G2 for signatures. Support for G1 signatures is planned.
// Compatible with Algorand, Chia, Dfinity, Ethereum, FIL, Zcash. Matches specs
// [pairing-curves-11](https://tools.ietf.org/html/draft-irtf-cfrg-pairing-friendly-curves-11),
// [bls-sigs-04](https:/cfrg-hash-to/tools.ietf.org/html/draft-irtf-cfrg-bls-signature-04),
// [hash-to-curve-12](https://tools.ietf.org/html/draft-irtf--curve-12).
// bls12-381 is pairing-friendly Barreto-Lynn-Scott elliptic curve construction allowing to:
// - Construct zk-SNARKs at the 120-bit security
// - Efficiently verify N aggregate signatures with 1 pairing and N ec additions:
// the Boneh-Lynn-Shacham signature scheme is orders of magnitude more efficient than Schnorr
//
// ### Summary
// 1. BLS Relies on Bilinear Pairing (expensive)
@@ -25,8 +19,17 @@
// - `S = pk x H(m)` - signing
// - `e(P, H(m)) == e(G, S)` - verification using pairings
// - `e(G, S) = e(G, SUM(n)(Si)) = MUL(n)(e(G, Si))` - signature aggregation
// Filecoin uses little endian byte arrays for private keys -
// so ensure to reverse byte order if you'll use it with FIL.
//
// ### Compatibility and notes
// 1. It is compatible with Algorand, Chia, Dfinity, Ethereum, Filecoin, ZEC
// Filecoin uses little endian byte arrays for private keys - make sure to reverse byte order.
// 2. Some projects use G2 for public keys and G1 for signatures. It's called "short signature"
// 3. Curve security level is about 120 bits as per Barbulescu-Duquesne 2017
// https://hal.science/hal-01534101/file/main.pdf
// 4. Compatible with specs:
// [cfrg-pairing-friendly-curves-11](https://tools.ietf.org/html/draft-irtf-cfrg-pairing-friendly-curves-11),
// [cfrg-bls-signature-05](https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-bls-signature-05),
// [RFC 9380](https://www.rfc-editor.org/rfc/rfc9380).
import { sha256 } from '@noble/hashes/sha256';
import { randomBytes } from '@noble/hashes/utils';
import { bls, CurveFn } from './abstract/bls.js';
@@ -37,7 +40,6 @@ import {
numberToBytesBE,
bytesToNumberBE,
bitLen,
bitSet,
bitGet,
Hex,
bitMask,
@@ -1016,11 +1018,41 @@ const htfDefaults = Object.freeze({
// Encoding utils
// Point on G1 curve: (x, y)
const C_BIT_POS = Fp.BITS; // C_bit, compression bit for serialization flag
const I_BIT_POS = Fp.BITS + 1; // I_bit, point-at-infinity bit for serialization flag
const S_BIT_POS = Fp.BITS + 2; // S_bit, sign bit for serialization flag
// Compressed point of infinity
const COMPRESSED_ZERO = Fp.toBytes(bitSet(bitSet(_0n, I_BIT_POS, true), S_BIT_POS, true)); // set compressed & point-at-infinity bits
const COMPRESSED_ZERO = setMask(Fp.toBytes(_0n), { infinity: true, compressed: true }); // set compressed & point-at-infinity bits
function parseMask(bytes: Uint8Array) {
// Copy, so we can remove mask data. It will be removed also later, when Fp.create will call modulo.
bytes = bytes.slice();
const mask = bytes[0] & 0b1110_0000;
const compressed = !!((mask >> 7) & 1); // compression bit (0b1000_0000)
const infinity = !!((mask >> 6) & 1); // point at infinity bit (0b0100_0000)
const sort = !!((mask >> 5) & 1); // sort bit (0b0010_0000)
bytes[0] &= 0b0001_1111; // clear mask (zero first 3 bits)
return { compressed, infinity, sort, value: bytes };
}
function setMask(
bytes: Uint8Array,
mask: { compressed?: boolean; infinity?: boolean; sort?: boolean }
) {
if (bytes[0] & 0b1110_0000) throw new Error('setMask: non-empty mask');
if (mask.compressed) bytes[0] |= 0b1000_0000;
if (mask.infinity) bytes[0] |= 0b0100_0000;
if (mask.sort) bytes[0] |= 0b0010_0000;
return bytes;
}
function signatureG1ToRawBytes(point: ProjPointType<Fp>) {
point.assertValidity();
const isZero = point.equals(bls12_381.G1.ProjectivePoint.ZERO);
const { x, y } = point.toAffine();
if (isZero) return COMPRESSED_ZERO.slice();
const P = Fp.ORDER;
const sort = Boolean((y * _2n) / P);
return setMask(numberToBytesBE(x, Fp.BYTES), { compressed: true, sort });
}
function signatureG2ToRawBytes(point: ProjPointType<Fp2>) {
// NOTE: by some reasons it was missed in bls12-381, looks like bug
@@ -1032,10 +1064,12 @@ function signatureG2ToRawBytes(point: ProjPointType<Fp2>) {
const { re: x0, im: x1 } = Fp2.reim(x);
const { re: y0, im: y1 } = Fp2.reim(y);
const tmp = y1 > _0n ? y1 * _2n : y0 * _2n;
const aflag1 = Boolean((tmp / Fp.ORDER) & _1n);
const z1 = bitSet(bitSet(x1, 381, aflag1), S_BIT_POS, true);
const sort = Boolean((tmp / Fp.ORDER) & _1n);
const z2 = x0;
return concatB(numberToBytesBE(z1, len), numberToBytesBE(z2, len));
return concatB(
setMask(numberToBytesBE(x1, len), { sort, compressed: true }),
numberToBytesBE(z2, len)
);
}
// To verify curve parameters, see pairing-friendly-curves spec:
@@ -1074,7 +1108,7 @@ export const bls12_381: CurveFn<Fp, Fp2, Fp6, Fp12> = bls({
),
a: Fp.ZERO,
b: _4n,
htfDefaults: { ...htfDefaults, m: 1 },
htfDefaults: { ...htfDefaults, m: 1, DST: 'BLS_SIG_BLS12381G1_XMD:SHA-256_SSWU_RO_NUL_' },
wrapPrivateKey: true,
allowInfinityPoint: true,
// Checks is the point resides in prime-order subgroup.
@@ -1116,26 +1150,30 @@ export const bls12_381: CurveFn<Fp, Fp2, Fp6, Fp12> = bls({
return isogenyMapG1(x, y);
},
fromBytes: (bytes: Uint8Array): AffinePoint<Fp> => {
bytes = bytes.slice();
if (bytes.length === 48) {
const { compressed, infinity, sort, value } = parseMask(bytes);
if (value.length === 48 && compressed) {
// TODO: Fp.bytes
const P = Fp.ORDER;
const compressedValue = bytesToNumberBE(bytes);
const bflag = bitGet(compressedValue, I_BIT_POS);
const compressedValue = bytesToNumberBE(value);
// Zero
if (bflag === _1n) return { x: _0n, y: _0n };
const x = Fp.create(compressedValue & Fp.MASK);
if (infinity) {
if (x !== _0n) throw new Error('G1: non-empty compressed point at infinity');
return { x: _0n, y: _0n };
}
const right = Fp.add(Fp.pow(x, _3n), Fp.create(bls12_381.params.G1b)); // y² = x³ + b
let y = Fp.sqrt(right);
if (!y) throw new Error('Invalid compressed G1 point');
const aflag = bitGet(compressedValue, C_BIT_POS);
if ((y * _2n) / P !== aflag) y = Fp.neg(y);
if ((y * _2n) / P !== BigInt(sort)) y = Fp.neg(y);
return { x: Fp.create(x), y: Fp.create(y) };
} else if (bytes.length === 96) {
} else if (value.length === 96 && !compressed) {
// Check if the infinity flag is set
if ((bytes[0] & (1 << 6)) !== 0) return bls12_381.G1.ProjectivePoint.ZERO.toAffine();
const x = bytesToNumberBE(bytes.subarray(0, Fp.BYTES));
const y = bytesToNumberBE(bytes.subarray(Fp.BYTES));
const x = bytesToNumberBE(value.subarray(0, Fp.BYTES));
const y = bytesToNumberBE(value.subarray(Fp.BYTES));
if (infinity) {
if (x !== _0n || y !== _0n) throw new Error('G1: non-empty point at infinity');
return bls12_381.G1.ProjectivePoint.ZERO.toAffine();
}
return { x: Fp.create(x), y: Fp.create(y) };
} else {
throw new Error('Invalid point G1, expected 48/96 bytes');
@@ -1147,10 +1185,8 @@ export const bls12_381: CurveFn<Fp, Fp2, Fp6, Fp12> = bls({
if (isCompressed) {
if (isZero) return COMPRESSED_ZERO.slice();
const P = Fp.ORDER;
let num;
num = bitSet(x, C_BIT_POS, Boolean((y * _2n) / P)); // set aflag
num = bitSet(num, S_BIT_POS, true);
return numberToBytesBE(num, Fp.BYTES);
const sort = Boolean((y * _2n) / P);
return setMask(numberToBytesBE(x, Fp.BYTES), { compressed: true, sort });
} else {
if (isZero) {
// 2x PUBLIC_KEY_LENGTH
@@ -1161,6 +1197,30 @@ export const bls12_381: CurveFn<Fp, Fp2, Fp6, Fp12> = bls({
}
}
},
ShortSignature: {
fromHex(hex: Hex): ProjPointType<Fp> {
const { infinity, sort, value } = parseMask(ensureBytes('signatureHex', hex, 48));
const P = Fp.ORDER;
const compressedValue = bytesToNumberBE(value);
// Zero
if (infinity) return bls12_381.G1.ProjectivePoint.ZERO;
const x = Fp.create(compressedValue & Fp.MASK);
const right = Fp.add(Fp.pow(x, _3n), Fp.create(bls12_381.params.G1b)); // y² = x³ + b
let y = Fp.sqrt(right);
if (!y) throw new Error('Invalid compressed G1 point');
const aflag = BigInt(sort);
if ((y * _2n) / P !== aflag) y = Fp.neg(y);
const point = bls12_381.G1.ProjectivePoint.fromAffine({ x, y });
point.assertValidity();
return point;
},
toRawBytes(point: ProjPointType<Fp>) {
return signatureG1ToRawBytes(point);
},
toHex(point: ProjPointType<Fp>) {
return bytesToHex(signatureG1ToRawBytes(point));
},
},
},
// G2 is the order-q subgroup of E2(Fp²) : y² = x³+4(1+√1),
// where Fp2 is Fp[√1]/(x2+1). #E2(Fp2 ) = h2q, where
@@ -1232,45 +1292,45 @@ export const bls12_381: CurveFn<Fp, Fp2, Fp6, Fp12> = bls({
return Q; // [x²-x-1]P + [x-1]Ψ(P) + Ψ²(2P)
},
fromBytes: (bytes: Uint8Array): AffinePoint<Fp2> => {
bytes = bytes.slice();
const m_byte = bytes[0] & 0xe0;
if (m_byte === 0x20 || m_byte === 0x60 || m_byte === 0xe0) {
throw new Error('Invalid encoding flag: ' + m_byte);
const { compressed, infinity, sort, value } = parseMask(bytes);
if (
(!compressed && !infinity && sort) || // 00100000
(!compressed && infinity && sort) || // 01100000
(sort && infinity && compressed) // 11100000
) {
throw new Error('Invalid encoding flag: ' + (bytes[0] & 0b1110_0000));
}
const bitC = m_byte & 0x80; // compression bit
const bitI = m_byte & 0x40; // point at infinity bit
const bitS = m_byte & 0x20; // sign bit
const L = Fp.BYTES;
const slc = (b: Uint8Array, from: number, to?: number) => bytesToNumberBE(b.slice(from, to));
if (bytes.length === 96 && bitC) {
if (value.length === 96 && compressed) {
const b = bls12_381.params.G2b;
const P = Fp.ORDER;
bytes[0] = bytes[0] & 0x1f; // clear flags
if (bitI) {
if (infinity) {
// check that all bytes are 0
if (bytes.reduce((p, c) => (p !== 0 ? c + 1 : c), 0) > 0) {
if (value.reduce((p, c) => (p !== 0 ? c + 1 : c), 0) > 0) {
throw new Error('Invalid compressed G2 point');
}
return { x: Fp2.ZERO, y: Fp2.ZERO };
}
const x_1 = slc(bytes, 0, L);
const x_0 = slc(bytes, L, 2 * L);
const x_1 = slc(value, 0, L);
const x_0 = slc(value, L, 2 * L);
const x = Fp2.create({ c0: Fp.create(x_0), c1: Fp.create(x_1) });
const right = Fp2.add(Fp2.pow(x, _3n), b); // y² = x³ + 4 * (u+1) = x³ + b
let y = Fp2.sqrt(right);
const Y_bit = y.c1 === _0n ? (y.c0 * _2n) / P : (y.c1 * _2n) / P ? _1n : _0n;
y = bitS > 0 && Y_bit > 0 ? y : Fp2.neg(y);
y = sort && Y_bit > 0 ? y : Fp2.neg(y);
return { x, y };
} else if (bytes.length === 192 && !bitC) {
// Check if the infinity flag is set
if ((bytes[0] & (1 << 6)) !== 0) {
} else if (value.length === 192 && !compressed) {
if (infinity) {
if (value.reduce((p, c) => (p !== 0 ? c + 1 : c), 0) > 0) {
throw new Error('Invalid uncompressed G2 point');
}
return { x: Fp2.ZERO, y: Fp2.ZERO };
}
const x1 = slc(bytes, 0, L);
const x0 = slc(bytes, L, 2 * L);
const y1 = slc(bytes, 2 * L, 3 * L);
const y0 = slc(bytes, 3 * L, 4 * L);
const x1 = slc(value, 0, L);
const x0 = slc(value, L, 2 * L);
const y1 = slc(value, 2 * L, 3 * L);
const y0 = slc(value, 3 * L, 4 * L);
return { x: Fp2.fromBigTuple([x0, x1]), y: Fp2.fromBigTuple([y0, y1]) };
} else {
throw new Error('Invalid point G2, expected 96/192 bytes');
@@ -1283,10 +1343,10 @@ export const bls12_381: CurveFn<Fp, Fp2, Fp6, Fp12> = bls({
if (isCompressed) {
if (isZero) return concatB(COMPRESSED_ZERO, numberToBytesBE(_0n, len));
const flag = Boolean(y.c1 === _0n ? (y.c0 * _2n) / P : (y.c1 * _2n) / P);
// set compressed & sign bits (looks like different offsets than for G1/Fp?)
let x_1 = bitSet(x.c1, C_BIT_POS, flag);
x_1 = bitSet(x_1, S_BIT_POS, true);
return concatB(numberToBytesBE(x_1, len), numberToBytesBE(x.c0, len));
return concatB(
setMask(numberToBytesBE(x.c1, len), { compressed: true, sort: flag }),
numberToBytesBE(x.c0, len)
);
} else {
if (isZero) return concatB(new Uint8Array([0x40]), new Uint8Array(4 * len - 1)); // bytes[0] |= 1 << 6;
const { re: x0, im: x1 } = Fp2.reim(x);
@@ -1302,17 +1362,15 @@ export const bls12_381: CurveFn<Fp, Fp2, Fp6, Fp12> = bls({
Signature: {
// TODO: Optimize, it's very slow because of sqrt.
fromHex(hex: Hex): ProjPointType<Fp2> {
hex = ensureBytes('signatureHex', hex);
const { infinity, sort, value } = parseMask(ensureBytes('signatureHex', hex));
const P = Fp.ORDER;
const half = hex.length / 2;
if (half !== 48 && half !== 96)
throw new Error('Invalid compressed signature length, must be 96 or 192');
const z1 = bytesToNumberBE(hex.slice(0, half));
const z2 = bytesToNumberBE(hex.slice(half));
const z1 = bytesToNumberBE(value.slice(0, half));
const z2 = bytesToNumberBE(value.slice(half));
// Indicates the infinity point
const bflag1 = bitGet(z1, I_BIT_POS);
if (bflag1 === _1n) return bls12_381.G2.ProjectivePoint.ZERO;
if (infinity) return bls12_381.G2.ProjectivePoint.ZERO;
const x1 = Fp.create(z1 & Fp.MASK);
const x2 = Fp.create(z2);
const x = Fp2.create({ c0: x2, c1: x1 });
@@ -1324,7 +1382,7 @@ export const bls12_381: CurveFn<Fp, Fp2, Fp6, Fp12> = bls({
// Choose the y whose leftmost bit of the imaginary part is equal to the a_flag1
// If y1 happens to be zero, then use the bit of y0
const { re: y0, im: y1 } = Fp2.reim(y);
const aflag1 = bitGet(z1, 381);
const aflag1 = BigInt(sort);
const isGreater = y1 > _0n && (y1 * _2n) / P !== aflag1;
const isZero = y1 === _0n && (y0 * _2n) / P !== aflag1;
if (isGreater || isZero) y = Fp2.neg(y);

View File

@@ -6,8 +6,9 @@ import { Field } from './abstract/modular.js';
/**
* bn254 pairing-friendly curve.
* Previously known as alt_bn_128, when it had 128-bit security.
* Recent research shown it's weaker, the naming has been adjusted to its prime bit count.
* https://github.com/zcash/zcash/issues/2502
* Barbulescu-Duquesne 2017 shown it's weaker: just about 100 bits,
* so the naming has been adjusted to its prime bit count
* https://hal.science/hal-01534101/file/main.pdf
*/
export const bn254 = weierstrass({
a: BigInt(0),

View File

@@ -13,7 +13,7 @@ import {
numberToBytesLE,
} from './abstract/utils.js';
import { createHasher, htfBasicOpts, expand_message_xmd } from './abstract/hash-to-curve.js';
import { AffinePoint } from './abstract/curve.js';
import { AffinePoint, Group } from './abstract/curve.js';
/**
* ed25519 Twisted Edwards curve with following addons:
@@ -343,7 +343,7 @@ function calcElligatorRistrettoMap(r0: bigint): ExtendedPoint {
* but it should work in its own namespace: do not combine those two.
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448
*/
class RistPoint {
class RistPoint implements Group<RistPoint> {
static BASE: RistPoint;
static ZERO: RistPoint;
// Private property to discourage combining ExtendedPoint + RistrettoPoint
@@ -471,6 +471,14 @@ class RistPoint {
multiplyUnsafe(scalar: bigint): RistPoint {
return new RistPoint(this.ep.multiplyUnsafe(scalar));
}
double(): RistPoint {
return new RistPoint(this.ep.double());
}
negate(): RistPoint {
return new RistPoint(this.ep.negate());
}
}
export const RistrettoPoint = /* @__PURE__ */ (() => {
if (!RistPoint.BASE) RistPoint.BASE = new RistPoint(ed25519.ExtendedPoint.BASE);

View File

@@ -13,7 +13,7 @@ import {
Hex,
numberToBytesLE,
} from './abstract/utils.js';
import { AffinePoint } from './abstract/curve.js';
import { AffinePoint, Group } from './abstract/curve.js';
/**
* Edwards448 (not Ed448-Goldilocks) curve with following addons:
@@ -103,6 +103,7 @@ const ED448_DEF = {
n: BigInt(
'181709681073901722637330951972001133588410340171829515070372549795146003961539585716195755291692375963310293709091662304773755859649779'
),
// RFC 7748 has 56-byte keys, RFC 8032 has 57-byte keys
nBitLength: 456,
// Cofactor
h: BigInt(4),
@@ -137,8 +138,9 @@ export const ed448ph = /* @__PURE__ */ twistedEdwards({ ...ED448_DEF, prehash: s
export const x448 = /* @__PURE__ */ (() =>
montgomery({
a: BigInt(156326),
// RFC 7748 has 56-byte keys, RFC 8032 has 57-byte keys
montgomeryBits: 448,
nByteLength: 57,
nByteLength: 56,
P: ed448P,
Gu: BigInt(5),
powPminus2: (x: bigint): bigint => {
@@ -164,11 +166,14 @@ export function edwardsToMontgomeryPub(edwardsPub: string | Uint8Array): Uint8Ar
const _1n = BigInt(1);
return Fp.toBytes(Fp.create((y - _1n) * Fp.inv(y + _1n)));
}
export const edwardsToMontgomery = edwardsToMontgomeryPub; // deprecated
// TODO: add edwardsToMontgomeryPriv, similar to ed25519 version
// Hash To Curve Elligator2 Map
const ELL2_C1 = (Fp.ORDER - BigInt(3)) / BigInt(4); // 1. c1 = (q - 3) / 4 # Integer arithmetic
const ELL2_J = BigInt(156326);
function map_to_curve_elligator2_curve448(u: bigint) {
let tv1 = Fp.sqr(u); // 1. tv1 = u^2
let e1 = Fp.eql(tv1, Fp.ONE); // 2. e1 = tv1 == 1
@@ -198,6 +203,7 @@ function map_to_curve_elligator2_curve448(u: bigint) {
y = Fp.cmov(y, Fp.neg(y), e2 !== e3); // 26. y = CMOV(y, -y, e2 XOR e3)
return { xn, xd, yn: y, yd: Fp.ONE }; // 27. return (xn, xd, y, 1)
}
function map_to_curve_elligator2_edwards448(u: bigint) {
let { xn, xd, yn, yd } = map_to_curve_elligator2_curve448(u); // 1. (xn, xd, yn, yd) = map_to_curve_elligator2_curve448(u)
let xn2 = Fp.sqr(xn); // 2. xn2 = xn^2
@@ -323,7 +329,7 @@ function calcElligatorDecafMap(r0: bigint): ExtendedPoint {
* but it should work in its own namespace: do not combine those two.
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448
*/
class DcfPoint {
class DcfPoint implements Group<DcfPoint> {
static BASE: DcfPoint;
static ZERO: DcfPoint;
// Private property to discourage combining ExtendedPoint + DecafPoint
@@ -445,7 +451,16 @@ class DcfPoint {
multiplyUnsafe(scalar: bigint): DcfPoint {
return new DcfPoint(this.ep.multiplyUnsafe(scalar));
}
double(): DcfPoint {
return new DcfPoint(this.ep.double());
}
negate(): DcfPoint {
return new DcfPoint(this.ep.negate());
}
}
export const DecafPoint = /* @__PURE__ */ (() => {
// decaf448 base point is ed448 base x 2
// https://github.com/dalek-cryptography/curve25519-dalek/blob/59837c6ecff02b77b9d5ff84dbc239d0cf33ef90/vendor/ristretto.sage#L699

View File

@@ -1,7 +1,7 @@
/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { sha256 } from '@noble/hashes/sha256';
import { randomBytes } from '@noble/hashes/utils';
import { Field, mod, pow2 } from './abstract/modular.js';
import { Field, mod, pow2, FpIsSquare } from './abstract/modular.js';
import { ProjPointType as PointType, mapToCurveSimpleSWU } from './abstract/weierstrass.js';
import type { Hex, PrivKey } from './abstract/utils.js';
import { bytesToNumberBE, concatBytes, ensureBytes, numberToBytesBE } from './abstract/utils.js';
@@ -272,3 +272,129 @@ const htf = /* @__PURE__ */ (() =>
))();
export const hashToCurve = /* @__PURE__ */ (() => htf.hashToCurve)();
export const encodeToCurve = /* @__PURE__ */ (() => htf.encodeToCurve)();
// ElligatorSwift: Schnorr-like x-only ECDH with public keys indistinguishable
// from uniformly random bytes.
// https://github.com/bitcoin/bips/blob/master/bip-0324.mediawiki,
// https://github.com/bitcoin/bitcoin/blob/master/src/secp256k1/doc/ellswift.md
// SwiftEC: Shallue-van de Woestijne Indifferentiable Function to Elliptic Curves.
// https://eprint.iacr.org/2022/759.pdf
//
// Curve25519 & P-521 are incompatible with SwiftEC. Differences from SwiftEC:
// - undefined inputs are remapped
// - y-parity is encoded in u/t values
const MINUS_3_SQRT = Fp.sqrt(Fp.create(BigInt(-3)));
const _3n = BigInt(3);
const _4n = BigInt(4);
const _7n = BigInt(7);
const isSquare = FpIsSquare(Fp);
const isValidX = (x: bigint) => isSquare(Fp.add(Fp.mul(Fp.mul(x, x), x), _7n));
const trySqrt = (x: bigint): bigint | void => {
try {
return Fp.sqrt(x);
} catch (_e) {}
};
export const elligatorSwift = /* @__PURE__ */ {
// (internal stuff, exported for tests only): decode(u, _inv(x, u)) = x
_inv: (x: bigint, u: bigint, ellCase: number): bigint | void => {
if (!Number.isSafeInteger(ellCase) || ellCase < 0 || ellCase > 7)
throw new Error(`elligatorSwift._inv: wrong case=${ellCase}`);
let v: bigint, s: bigint;
// Most rejections happens in 3 condition (in comments, ~33% each)
const u2 = Fp.mul(u, u); // u**2
const u3 = Fp.mul(u2, u); // u**3
if ((ellCase & 2) === 0) {
if (isValidX(Fp.sub(Fp.neg(x), u))) return; // [1 condition]
v = x;
s = Fp.div(Fp.neg(Fp.add(u3, _7n)), Fp.add(Fp.add(u2, Fp.mul(u, v)), Fp.mul(v, v))); // = -(u**3 + 7) / (u**2 + u*v + v**2)
} else {
s = Fp.sub(x, u); // x - u
if (Fp.is0(s)) return;
const t0 = Fp.add(u3, _7n); // (u**3 + 7)
const t1 = Fp.mul(Fp.mul(_3n, s), u2); // 3 * s * u**2
// r = (-s * (4 * (u**3 + 7) + 3 * s * u**2)).sqrt()
const r = trySqrt(Fp.mul(Fp.neg(s), Fp.add(Fp.mul(_4n, t0), t1)));
if (r === undefined) return; // [2 condition]
if (ellCase & 1 && Fp.is0(r)) return;
v = Fp.div(Fp.add(Fp.neg(u), Fp.div(r, s)), _2n); // v = (-u + r / s) / 2
}
const w = trySqrt(s);
if (w === undefined) return; // [3 condition]
const last = ellCase & 5; // ellCase = 0..8, last = 0,1,4,5
const t0 = last & 1 ? Fp.add(_1n, MINUS_3_SQRT) : Fp.sub(_1n, MINUS_3_SQRT);
const w0 = last === 0 || last === 5 ? Fp.neg(w) : w; // -w | w
// w0 * (u * t0 / 2 + v)
return Fp.mul(w0, Fp.add(Fp.div(Fp.mul(u, t0), _2n), v));
},
// Encode public key (point or x coordinate bigint) into 64-byte pseudorandom encoding
encode: (x: bigint | PointType<bigint>): Uint8Array => {
if (x instanceof secp256k1.ProjectivePoint) x = x.x;
if (typeof x !== 'bigint') {
throw new Error(
'elligatorSwift.encode: wrong public key. Should be Projective point or x coordinate (bigint)'
);
}
// 200k test cycles per keygen: avg=4 max=48
// seems too much, but same as for reference implementation
while (true) {
// random scalar 1..Fp.ORDER
const u = Fp.create(Fp.fromBytes(secp256k1.utils.randomPrivateKey()));
const ellCase = randomBytes(1)[0] & 7; // [0..8)
const t = elligatorSwift._inv(x, u, ellCase);
if (!t) continue;
return concatBytes(numberToBytesBE(u, 32), numberToBytesBE(t, 32));
}
},
// Decode elligatorSwift point to xonly
decode: (data: Hex): Uint8Array => {
const _data = ensureBytes('data', data, 64);
let u = Fp.create(Fp.fromBytes(_data.subarray(0, 32)));
let t = Fp.create(Fp.fromBytes(_data.subarray(32, 64)));
if (Fp.is0(u)) u = Fp.create(_1n);
if (Fp.is0(t)) t = Fp.create(_1n);
const u3 = Fp.mul(Fp.mul(u, u), u); // u**3
const u3plus7 = Fp.add(u3, _7n);
// u**3 + t**2 + 7 == 0 -> t = 2 * t
if (Fp.is0(Fp.add(u3plus7, Fp.mul(t, t)))) t = Fp.add(t, t);
// X = (u**3 + 7 - t**2) / (2 * t)
const x = Fp.div(Fp.sub(u3plus7, Fp.mul(t, t)), Fp.add(t, t));
// Y = (X + t) / (MINUS_3_SQRT * u);
const y = Fp.div(Fp.add(x, t), Fp.mul(MINUS_3_SQRT, u));
// try different cases
let res = Fp.add(u, Fp.mul(Fp.mul(y, y), _4n)); // u + 4 * Y ** 2,
if (isValidX(res)) return numberToBytesBE(res, 32);
res = Fp.div(Fp.sub(Fp.div(Fp.neg(x), y), u), _2n); // (-X / Y - u) / 2
if (isValidX(res)) return numberToBytesBE(res, 32);
res = Fp.div(Fp.sub(Fp.div(x, y), u), _2n); // (X / Y - u) / 2
if (isValidX(res)) return numberToBytesBE(res, 32);
throw new Error('elligatorSwift: cannot decode public key');
},
// Generate pair (public key, secret key)
keygen: () => {
const privateKey = secp256k1.utils.randomPrivateKey();
const publicKey = elligatorSwift.encode(Point.fromPrivateKey(privateKey));
return { privateKey, publicKey };
},
// Generates shared secret between a pub key and a priv key
getSharedSecret: (privateKeyA: Hex, publicKeyB: Hex) => {
const pub = elligatorSwift.decode(publicKeyB);
const priv = ensureBytes('privKey', privateKeyA, 32);
const point = lift_x(Fp.fromBytes(pub));
const d = bytesToNumberBE(priv);
return numberToBytesBE(point.multiply(d).x, 32);
},
// BIP324 shared secret
getSharedSecretBip324: (
privateKeyOurs: Hex,
publicKeyTheirs: Hex,
publicKeyOurs: Hex,
initiating: boolean
) => {
const ours = ensureBytes('publicKeyOurs', publicKeyOurs);
const theirs = ensureBytes('publicKeyTheirs', publicKeyTheirs);
const ecdhPoint = elligatorSwift.getSharedSecret(privateKeyOurs, theirs);
const pubs = initiating ? [ours, theirs] : [theirs, ours];
return taggedHash('bip324_ellswift_xonly_ecdh', ...pubs, ecdhPoint);
},
};

View File

@@ -5,10 +5,16 @@ import { describe, should } from 'micro-should';
import { wNAF } from '../esm/abstract/curve.js';
import { bytesToHex, utf8ToBytes } from '../esm/abstract/utils.js';
import { hash_to_field } from '../esm/abstract/hash-to-curve.js';
import { bls12_381 as bls } from '../esm/bls12-381.js';
import { bls12_381 as bls, bls12_381 } from '../esm/bls12-381.js';
import * as utils from '../esm/abstract/utils.js';
import zkVectors from './bls12-381/zkcrypto/converted.json' assert { type: 'json' };
import pairingVectors from './bls12-381/go_pairing_vectors/pairing.json' assert { type: 'json' };
const G1_VECTORS = readFileSync('./test/bls12-381/bls12-381-g1-test-vectors.txt', 'utf-8')
.trim()
.split('\n')
.map((l) => l.split(':'));
const G2_VECTORS = readFileSync('./test/bls12-381/bls12-381-g2-test-vectors.txt', 'utf-8')
.trim()
.split('\n')
@@ -852,6 +858,13 @@ describe('bls12-381/basic', () => {
});
// should aggregate signatures
should(`produce correct short signatures (${G1_VECTORS.length} vectors)`, () => {
for (let vector of G1_VECTORS) {
const [priv, msg, expected] = vector;
const sig = bls.signShortSignature(msg, priv);
deepStrictEqual(bytesToHex(sig), expected);
}
});
should(`produce correct signatures (${G2_VECTORS.length} vectors)`, () => {
for (let vector of G2_VECTORS) {
const [priv, msg, expected] = vector;
@@ -1182,6 +1195,35 @@ describe('verify()', () => {
deepStrictEqual(res, false);
}
});
should('verify signed message (short signatures)', () => {
for (let i = 0; i < NUM_RUNS; i++) {
const [priv, msg] = G1_VECTORS[i];
const sig = bls.signShortSignature(msg, priv);
const pub = bls.getPublicKeyForShortSignatures(priv);
const res = bls.verifyShortSignature(sig, msg, pub);
deepStrictEqual(res, true, `${priv}-${msg}`);
}
});
should('not verify signature with wrong message (short signatures)', () => {
for (let i = 0; i < NUM_RUNS; i++) {
const [priv, msg] = G1_VECTORS[i];
const invMsg = G1_VECTORS[i + 1][1];
const sig = bls.signShortSignature(msg, priv);
const pub = bls.getPublicKeyForShortSignatures(priv);
const res = bls.verifyShortSignature(sig, invMsg, pub);
deepStrictEqual(res, false);
}
});
should('not verify signature with wrong key', () => {
for (let i = 0; i < NUM_RUNS; i++) {
const [priv, msg] = G1_VECTORS[i];
const sig = bls.signShortSignature(msg, priv);
const invPriv = G1_VECTORS[i + 1][1].padStart(64, '0');
const invPub = bls.getPublicKeyForShortSignatures(invPriv);
const res = bls.verifyShortSignature(sig, msg, invPub);
deepStrictEqual(res, false);
}
});
describe('batch', () => {
should('verify multi-signature', () => {
fc.assert(
@@ -1375,6 +1417,37 @@ describe('bls12-381 deterministic', () => {
}
}
});
should(`zkcrypt/G1 & G2 encoding edge cases`, () => {
const Fp = bls12_381.fields.Fp;
const S_BIT_POS = Fp.BITS; // C_bit, compression bit for serialization flag
const I_BIT_POS = Fp.BITS + 1; // I_bit, point-at-infinity bit for serialization flag
const C_BIT_POS = Fp.BITS + 2; // S_bit, sort bit for serialization flag
const VECTORS = [
{ pos: C_BIT_POS, shift: 7 }, // compression_flag_set = Choice::from((bytes[0] >> 7) & 1);
{ pos: I_BIT_POS, shift: 6 }, // infinity_flag_set = Choice::from((bytes[0] >> 6) & 1)
{ pos: S_BIT_POS, shift: 5 }, // sort_flag_set = Choice::from((bytes[0] >> 5) & 1)
];
for (const { pos, shift } of VECTORS) {
const d = utils.numberToBytesBE(utils.bitSet(0n, pos, Boolean(true)), Fp.BYTES);
deepStrictEqual((d[0] >> shift) & 1, 1, `${pos}`);
}
const baseC = G1Point.BASE.toRawBytes();
deepStrictEqual(baseC.length, 48);
const baseU = G1Point.BASE.toRawBytes(false);
deepStrictEqual(baseU.length, 96);
const compressedBit = baseU.slice();
compressedBit[0] |= 0b1000_0000; // add compression bit
throws(() => G1Point.fromHex(compressedBit), 'compressed bit'); // uncompressed point with compressed length
const uncompressedBit = baseC.slice();
uncompressedBit[0] &= 0b0111_1111; // remove compression bit
throws(() => G1Point.fromHex(uncompressedBit), 'uncompressed bit');
const infinityUncompressed = baseU.slice();
infinityUncompressed[0] |= 0b0100_0000;
throws(() => G1Point.fromHex(compressedBit), 'infinity uncompressed');
const infinityCompressed = baseC.slice();
infinityCompressed[0] |= 0b0100_0000;
throws(() => G1Point.fromHex(compressedBit), 'infinity compressed');
});
});
// ESM is broken.

View File

@@ -0,0 +1,128 @@
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5ae1bb72838eb38c1f4340c6e22686961e3e34b4d2d518d76ab4f881fd479c94:332c3bbd162c8cb774480ef81cede2aa52a0967a869663cab7c8d105e33038562599881f6ef77aa4e3f4225131fc864a27db9c58f5198e6190d5eec728ec4e698b96d1f982edc5d22231b3d76be10a7deeab8f32af36a2cf5d8aed42102ae20679cbb5c8befc8a65f13837710bf86f344590bc74f96e052eae:80e602213b6c748b3e68430ccd1c4fa288aa5c0a411466dd8994ff19ae8be60ad281da36e8a35e2f2b427ac2edc255cc
4113f2add0abb7f05cf56a1281457e9e4f7cfb39d093854b3b4e3bbde237ea23:a83d1cd6999e4940b0d71be96e000e46246a6e451c646756d893b3edb330f2f445858ed12d1d35412fc0eee90204aba0d4fa88583bbc37ee389f88fe0c8dd4c34ffaa8d8c6affeb7feb5f3064af0c06ad3f7a51e4b58f8edc95bd22a02ec4bbe2b2c660762e345d00548995ddcfb10299dd192344d45791ea8bb:a37b03f854b9cb12dc7fc109bea2ccc3423418b89d283715fb8c18db78a68d60b19bdd7e6acf869090b97f538a35df66
17c979dc43d5da70f57027cffc420c6df4245db222a2c3060c7499b2663f6093:c20c4351abcf4ecda9b5478cff8af859ec743eb0d67f6ce2b347349460a2854475d0a6e8165b5174434ca77c088e641c109ac8b8d5879586c034eadfa6bc3a1e66faa02a3dc0fed87af670cc8626df5c71a8360e1911159632558e7a99d4830ce21bcbf3f551465bd1061888b83db174343692b6d796460791ab8d:8ff5917a45cdda175f9d6f4a4a99308617ceec290794fda33d7585f4a3964febf0274479f77210373f3ca6350ae0639b
50adb11ff487000da2714e104241c0039dc6c2eca5e976a6bca8cc80fd2f22d5:3ca6b6f64aab22ea7dcf85eeb05e5eff6102731bb47c94bb737e77b389334742be939b79219fff38452b411052bba588b7872654d0b40f7be1516b8f243047a9aefb40fb70f93e53def38147fe564214605ef2d7508bd7ef44fdfee7882d754e735ba282e044bb953f18f131212d879def4d4d6923d0dcc0d3bb6a14:989e4389faf99c5cbf6798a147fd85cf1331e1e7faa966d6fbed2cd43c673d6a382e0e304f560e5ba816eb1a0b306cf9
55bdf02943f5a8130bcb537972f870043f5d7e3561df8717740e1a7a39bf73df:e288bfeeead0c3050ec6834694323af7bd77dcb52fee6cb54167a73181e487583ac75e63c95e71760cd9a1584b711842f602237f72afb268ef039a044d293d4091abc1807cbec9041ece11905b32ace59db1114047f60ae679c53b465afe03a8ba02ee89e85efebbb93226eaf1cd5c6ce1ef913fbf549934dfbda69dff:94c74729eaecf336b25d82a10798adc97e70cb345313413cf6550d622b0b92b9d4dcf606afb1b7e88db6b3ce106aba04
49a477ecf8786a9f5a44a9da0a83da977b844198068f1cadce28c599d9ebddbd:9944eeed83dde7b21b72ff1491cad3d7a1ceae3f9c5c880be49024d9ec055c55189de80b521df30fd17d558f7bc6ff6d5c9dcacae3ec1242929fae1fd8bd7fedea51acd344a1fa0120f60b1a4679e5177588fc27d713173f4fd47cccc16feb8b44a2d670d7b04c8c14bf37527230f3daa6d7c3a6e958c78376c5940f1063:8045a0358069c43170b238e7dc98952bf84e1caa0925905922b5822ab28b498297e901614909376fa04ced4c852c39da
6bbc2807b27b635285670a68a42d8fafb461fab581e4e2773c199b29088bc554:0c149764b95e6461e820206e5e2b7fa4acf62a3a132db955d5c4ff1cafbfef3b3816aab1bc9bbcf14af47a4e7753f0243842d9b53b3c3c26b9a2e244f2eb06461e2128949e9b437c96cddbddb52d9d6062d4692d05dc624f94fac39401ce51389576f0fd52e670841602645b4ed6a76cc845ec8454538782b3b179e44cd5ec:aaaa82c3fd810924f41b74bdf484460810f5f3274f8102520099f9ca384a5965e50202a36edfa5b0999c7b3c2548ae74

View File

@@ -23,29 +23,29 @@ import { default as xof_shake128_36 } from './hash-to-curve/expand_message_xof_S
import { default as xof_shake128_256 } from './hash-to-curve/expand_message_xof_SHAKE128_256.json' assert { type: 'json' };
import { default as xof_shake256_36 } from './hash-to-curve/expand_message_xof_SHAKE256_36.json' assert { type: 'json' };
// P256
import { default as p256_ro } from './hash-to-curve/P256_XMD:SHA-256_SSWU_RO_.json' assert { type: 'json' };
import { default as p256_nu } from './hash-to-curve/P256_XMD:SHA-256_SSWU_NU_.json' assert { type: 'json' };
import { default as p256_ro } from './hash-to-curve/P256_XMD_SHA-256_SSWU_RO_.json' assert { type: 'json' };
import { default as p256_nu } from './hash-to-curve/P256_XMD_SHA-256_SSWU_NU_.json' assert { type: 'json' };
// P384
import { default as p384_ro } from './hash-to-curve/P384_XMD:SHA-384_SSWU_RO_.json' assert { type: 'json' };
import { default as p384_nu } from './hash-to-curve/P384_XMD:SHA-384_SSWU_NU_.json' assert { type: 'json' };
import { default as p384_ro } from './hash-to-curve/P384_XMD_SHA-384_SSWU_RO_.json' assert { type: 'json' };
import { default as p384_nu } from './hash-to-curve/P384_XMD_SHA-384_SSWU_NU_.json' assert { type: 'json' };
// P521
import { default as p521_ro } from './hash-to-curve/P521_XMD:SHA-512_SSWU_RO_.json' assert { type: 'json' };
import { default as p521_nu } from './hash-to-curve/P521_XMD:SHA-512_SSWU_NU_.json' assert { type: 'json' };
import { default as p521_ro } from './hash-to-curve/P521_XMD_SHA-512_SSWU_RO_.json' assert { type: 'json' };
import { default as p521_nu } from './hash-to-curve/P521_XMD_SHA-512_SSWU_NU_.json' assert { type: 'json' };
// secp256k1
import { default as secp256k1_ro } from './hash-to-curve/secp256k1_XMD:SHA-256_SSWU_RO_.json' assert { type: 'json' };
import { default as secp256k1_nu } from './hash-to-curve/secp256k1_XMD:SHA-256_SSWU_NU_.json' assert { type: 'json' };
import { default as secp256k1_ro } from './hash-to-curve/secp256k1_XMD_SHA-256_SSWU_RO_.json' assert { type: 'json' };
import { default as secp256k1_nu } from './hash-to-curve/secp256k1_XMD_SHA-256_SSWU_NU_.json' assert { type: 'json' };
// bls-G1
import { default as g1_ro } from './hash-to-curve/BLS12381G1_XMD:SHA-256_SSWU_RO_.json' assert { type: 'json' };
import { default as g1_nu } from './hash-to-curve/BLS12381G1_XMD:SHA-256_SSWU_NU_.json' assert { type: 'json' };
import { default as g1_ro } from './hash-to-curve/BLS12381G1_XMD_SHA-256_SSWU_RO_.json' assert { type: 'json' };
import { default as g1_nu } from './hash-to-curve/BLS12381G1_XMD_SHA-256_SSWU_NU_.json' assert { type: 'json' };
// bls-G2
import { default as g2_ro } from './hash-to-curve/BLS12381G2_XMD:SHA-256_SSWU_RO_.json' assert { type: 'json' };
import { default as g2_nu } from './hash-to-curve/BLS12381G2_XMD:SHA-256_SSWU_NU_.json' assert { type: 'json' };
import { default as g2_ro } from './hash-to-curve/BLS12381G2_XMD_SHA-256_SSWU_RO_.json' assert { type: 'json' };
import { default as g2_nu } from './hash-to-curve/BLS12381G2_XMD_SHA-256_SSWU_NU_.json' assert { type: 'json' };
// ed25519
import { default as ed25519_ro } from './hash-to-curve/edwards25519_XMD:SHA-512_ELL2_RO_.json' assert { type: 'json' };
import { default as ed25519_nu } from './hash-to-curve/edwards25519_XMD:SHA-512_ELL2_NU_.json' assert { type: 'json' };
import { default as ed25519_ro } from './hash-to-curve/edwards25519_XMD_SHA-512_ELL2_RO_.json' assert { type: 'json' };
import { default as ed25519_nu } from './hash-to-curve/edwards25519_XMD_SHA-512_ELL2_NU_.json' assert { type: 'json' };
// ed448
import { default as ed448_ro } from './hash-to-curve/edwards448_XOF:SHAKE256_ELL2_RO_.json' assert { type: 'json' };
import { default as ed448_nu } from './hash-to-curve/edwards448_XOF:SHAKE256_ELL2_NU_.json' assert { type: 'json' };
import { default as ed448_ro } from './hash-to-curve/edwards448_XOF_SHAKE256_ELL2_RO_.json' assert { type: 'json' };
import { default as ed448_nu } from './hash-to-curve/edwards448_XOF_SHAKE256_ELL2_NU_.json' assert { type: 'json' };
function testExpandXMD(hash, vectors) {
describe(`${vectors.hash}/${vectors.DST.length}`, () => {

View File

@@ -9,6 +9,7 @@ import './ed25519.test.js';
import './ed25519-addons.test.js';
import './secp256k1.test.js';
import './secp256k1-schnorr.test.js';
import './secp256k1-bip-0324.test.js';
import './jubjub.test.js';
import './hash-to-curve.test.js';
import './poseidon.test.js';

View File

@@ -95,31 +95,51 @@ should('fields', () => {
for (const n in vectors) deepStrictEqual(NIST[n].CURVE.Fp.ORDER, vectors[n]);
});
// We don't support ASN.1 encoding of points. For tests we've implemented quick
// and dirty parser: take X last bytes of ASN.1 encoded sequence.
// If that doesn't work, we ignore such vector.
function verifyECDHVector(test, curve) {
if (test.flags.includes('InvalidAsn')) return; // Ignore invalid ASN
if (test.result === 'valid' || test.result === 'acceptable') {
const fnLen = curve.CURVE.nByteLength; // 32 for P256
const fpLen = curve.CURVE.Fp.BYTES; // 32 for P256
const encodedHexLen = fpLen * 2 * 2 + 2; // 130 (65 * 2) for P256
const pubB = test.public.slice(-encodedHexLen); // slice(-130) for P256
let privA = test.private;
// Some wycheproof vectors are padded with 00:
// 00c6cafb74e2a50c83b3d232c4585237f44d4c5433c4b3f50ce978e6aeda3a4f5d
// instead of
// c6cafb74e2a50c83b3d232c4585237f44d4c5433c4b3f50ce978e6aeda3a4f5d
if (privA.length / 2 === fnLen + 1 && privA.startsWith('00')) privA = privA.slice(2);
if (!curve.utils.isValidPrivateKey(privA)) return; // Ignore invalid private key size
try {
curve.ProjectivePoint.fromHex(pubB);
} catch (e) {
if (e.message.startsWith('Point of length')) return; // Ignore
throw e;
}
const shared = curve.getSharedSecret(privA, pubB).subarray(1);
deepStrictEqual(hex(shared), test.shared, 'valid');
} else if (test.result === 'invalid') {
let failed = false;
try {
curve.getSharedSecret(test.private, test.public);
} catch (error) {
failed = true;
}
deepStrictEqual(failed, true, 'invalid');
} else throw new Error('unknown test result');
}
describe('wycheproof ECDH', () => {
for (const group of ecdh.testGroups) {
const CURVE = NIST[group.curve];
if (!CURVE) continue;
const curve = NIST[group.curve];
if (!curve) continue;
should(group.curve, () => {
for (const test of group.tests) {
if (test.result === 'valid' || test.result === 'acceptable') {
try {
const pub = CURVE.ProjectivePoint.fromHex(test.public);
} catch (e) {
// Our strict validation filter doesn't let weird-length DER vectors
if (e.message.startsWith('Point of length')) continue;
throw e;
}
const shared = CURVE.getSharedSecret(test.private, test.public);
deepStrictEqual(shared, test.shared, 'valid');
} else if (test.result === 'invalid') {
let failed = false;
try {
CURVE.getSharedSecret(test.private, test.public);
} catch (error) {
failed = true;
}
deepStrictEqual(failed, true, 'invalid');
} else throw new Error('unknown test result');
verifyECDHVector(test, curve);
}
});
}
@@ -151,30 +171,12 @@ describe('wycheproof ECDH', () => {
for (const name in WYCHEPROOF_ECDH) {
const { curve, tests } = WYCHEPROOF_ECDH[name];
for (let i = 0; i < tests.length; i++) {
const test = tests[i];
for (let j = 0; j < test.testGroups.length; j++) {
const group = test.testGroups[j];
should(`additional ${name} (${i}/${j})`, () => {
const curveTests = tests[i];
for (let j = 0; j < curveTests.testGroups.length; j++) {
const group = curveTests.testGroups[j];
should(`additional ${name} (${group.tests.length})`, () => {
for (const test of group.tests) {
if (test.result === 'valid' || test.result === 'acceptable') {
try {
const pub = curve.ProjectivePoint.fromHex(test.public);
} catch (e) {
// Our strict validation filter doesn't let weird-length DER vectors
if (e.message.includes('Point of length')) continue;
throw e;
}
const shared = curve.getSharedSecret(test.private, test.public);
deepStrictEqual(hex(shared), test.shared, 'valid');
} else if (test.result === 'invalid') {
let failed = false;
try {
curve.getSharedSecret(test.private, test.public);
} catch (error) {
failed = true;
}
deepStrictEqual(failed, true, 'invalid');
} else throw new Error('unknown test result');
verifyECDHVector(test, curve);
}
});
}

View File

@@ -0,0 +1,122 @@
import { deepStrictEqual } from 'assert';
import { should, describe } from 'micro-should';
import * as fs from 'fs';
import {
hexToBytes,
hexToNumber,
concatBytes,
bytesToHex as toHex,
} from '../esm/abstract/utils.js';
// Generic tests for all curves in package
import { secp256k1, elligatorSwift } from '../esm/secp256k1.js';
// ESM is broken.
import { dirname } from 'path';
import { fileURLToPath } from 'url';
export const __dirname = dirname(fileURLToPath(import.meta.url));
// https://eprint.iacr.org/2022/759
const parseCSV = (path) => {
const data = fs.readFileSync(`${__dirname}/vectors/secp256k1/${path}`, 'utf8');
const lines = data.split('\n').filter((i) => !!i);
const rows = lines.map((i) => i.trim().split(','));
const lengths = new Set(rows.map((i) => i.length));
if (lengths.size !== 1) throw new Error('wrong dimensions');
if (rows.length < 2) throw new Error('wrong rows length');
const [head, ...rest] = rows;
return rest.map((row) => Object.fromEntries(row.map((cell, j) => [head[j], cell])));
};
describe('ElligatorSwift', () => {
should('packet_encoding_test_vectors', () => {
for (const t of parseCSV('bip-0324/packet_encoding_test_vectors.csv')) {
const inPriv = hexToNumber(t['in_priv_ours']);
const pubX = secp256k1.ProjectivePoint.BASE.multiply(inPriv)
.x.toString(16)
.padStart(2 * 32, '0');
deepStrictEqual(pubX, t['mid_x_ours']);
const bytesOurs = hexToBytes(t['in_ellswift_ours']);
const decoded = elligatorSwift.decode(bytesOurs);
deepStrictEqual(toHex(decoded), t['mid_x_ours']);
const bytesTheirs = hexToBytes(t['in_ellswift_theirs']);
deepStrictEqual(toHex(elligatorSwift.decode(bytesTheirs)), t['mid_x_theirs']);
const xShared = elligatorSwift.getSharedSecret(t['in_priv_ours'], bytesTheirs);
deepStrictEqual(toHex(xShared), t['mid_x_shared']);
const sharedSecret = elligatorSwift.getSharedSecretBip324(
t['in_priv_ours'],
t['in_ellswift_theirs'],
t['in_ellswift_ours'],
t['in_initiating'] === '1'
);
deepStrictEqual(toHex(sharedSecret), t['mid_shared_secret']);
}
});
should('xswiftec_inv_test_vectors', () => {
for (const t of parseCSV('bip-0324/xswiftec_inv_test_vectors.csv')) {
const Fp = secp256k1.CURVE.Fp;
const u = Fp.create(Fp.fromBytes(hexToBytes(t['u'])));
const x = Fp.create(Fp.fromBytes(hexToBytes(t['x'])));
for (let c = 0; c < 8; c++) {
const name = `case${c}_t`;
const ret = elligatorSwift._inv(x, u, c);
if (!ret) deepStrictEqual(t[name], '', 'empty case');
else {
deepStrictEqual(toHex(Fp.toBytes(ret)), t[name], 'real case');
deepStrictEqual(
elligatorSwift.decode(concatBytes(Fp.toBytes(u), Fp.toBytes(ret))),
Fp.toBytes(x)
);
}
}
}
});
should('ellswift_decode_test_vectors', () => {
for (const t of parseCSV('bip-0324/ellswift_decode_test_vectors.csv')) {
deepStrictEqual(toHex(elligatorSwift.decode(t['ellswift'])), t['x']);
}
});
should('Example', () => {
// random, so test more.
for (let i = 0; i < 100; i++) {
const alice = elligatorSwift.keygen();
const bob = elligatorSwift.keygen();
// ECDH
const sharedAlice = elligatorSwift.getSharedSecret(alice.privateKey, bob.publicKey);
const sharedBob = elligatorSwift.getSharedSecret(bob.privateKey, alice.publicKey);
deepStrictEqual(sharedAlice, sharedBob);
// ECDH BIP324
const sharedAlice2 = elligatorSwift.getSharedSecretBip324(
alice.privateKey,
bob.publicKey,
alice.publicKey,
true
);
const sharedBob2 = elligatorSwift.getSharedSecretBip324(
bob.privateKey,
alice.publicKey,
bob.publicKey,
false
);
deepStrictEqual(sharedAlice2, sharedBob2);
// pubKey decoding
for (const k of [alice, bob]) {
deepStrictEqual(
toHex(elligatorSwift.decode(k.publicKey)),
toHex(secp256k1.getPublicKey(k.privateKey, true).subarray(1))
);
}
}
});
});
// ESM is broken.
import url from 'url';
if (import.meta.url === url.pathToFileURL(process.argv[1]).href) {
should.run();
}

View File

@@ -0,0 +1,77 @@
ellswift,x,comment
00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000,edd1fd3e327ce90cc7a3542614289aee9682003e9cf7dcc9cf2ca9743be5aa0c,u%p=0;t%p=0;valid_x(x2)
000000000000000000000000000000000000000000000000000000000000000001d3475bf7655b0fb2d852921035b2ef607f49069b97454e6795251062741771,b5da00b73cd6560520e7c364086e7cd23a34bf60d0e707be9fc34d4cd5fdfa2c,u%p=0;valid_x(x1)
000000000000000000000000000000000000000000000000000000000000000082277c4a71f9d22e66ece523f8fa08741a7c0912c66a69ce68514bfd3515b49f,f482f2e241753ad0fb89150d8491dc1e34ff0b8acfbb442cfe999e2e5e6fd1d2,u%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
00000000000000000000000000000000000000000000000000000000000000008421cc930e77c9f514b6915c3dbe2a94c6d8f690b5b739864ba6789fb8a55dd0,9f59c40275f5085a006f05dae77eb98c6fd0db1ab4a72ac47eae90a4fc9e57e0,u%p=0;valid_x(x2)
0000000000000000000000000000000000000000000000000000000000000000bde70df51939b94c9c24979fa7dd04ebd9b3572da7802290438af2a681895441,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa9fffffd6b,u%p=0;(u'^3-t'^2+7)%p=0;valid_x(x3)
0000000000000000000000000000000000000000000000000000000000000000d19c182d2759cd99824228d94799f8c6557c38a1c0d6779b9d4b729c6f1ccc42,70720db7e238d04121f5b1afd8cc5ad9d18944c6bdc94881f502b7a3af3aecff,u%p=0;valid_x(x3)
0000000000000000000000000000000000000000000000000000000000000000fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,edd1fd3e327ce90cc7a3542614289aee9682003e9cf7dcc9cf2ca9743be5aa0c,u%p=0;t%p=0;valid_x(x2);t>=p
0000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffff2664bbd5,50873db31badcc71890e4f67753a65757f97aaa7dd5f1e82b753ace32219064b,u%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
0000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffff7028de7d,1eea9cc59cfcf2fa151ac6c274eea4110feb4f7b68c5965732e9992e976ef68e,u%p=0;valid_x(x2);t>=p
0000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffcbcfb7e7,12303941aedc208880735b1f1795c8e55be520ea93e103357b5d2adb7ed59b8e,u%p=0;valid_x(x1);t>=p
0000000000000000000000000000000000000000000000000000000000000000fffffffffffffffffffffffffffffffffffffffffffffffffffffffff3113ad9,7eed6b70e7b0767c7d7feac04e57aa2a12fef5e0f48f878fcbb88b3b6b5e0783,u%p=0;valid_x(x3);t>=p
0a2d2ba93507f1df233770c2a797962cc61f6d15da14ecd47d8d27ae1cd5f8530000000000000000000000000000000000000000000000000000000000000000,532167c11200b08c0e84a354e74dcc40f8b25f4fe686e30869526366278a0688,t%p=0;(u'^3+t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
0a2d2ba93507f1df233770c2a797962cc61f6d15da14ecd47d8d27ae1cd5f853fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,532167c11200b08c0e84a354e74dcc40f8b25f4fe686e30869526366278a0688,t%p=0;(u'^3+t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
0ffde9ca81d751e9cdaffc1a50779245320b28996dbaf32f822f20117c22fbd6c74d99efceaa550f1ad1c0f43f46e7ff1ee3bd0162b7bf55f2965da9c3450646,74e880b3ffd18fe3cddf7902522551ddf97fa4a35a3cfda8197f947081a57b8f,valid_x(x3)
0ffde9ca81d751e9cdaffc1a50779245320b28996dbaf32f822f20117c22fbd6ffffffffffffffffffffffffffffffffffffffffffffffffffffffff156ca896,377b643fce2271f64e5c8101566107c1be4980745091783804f654781ac9217c,valid_x(x2);t>=p
123658444f32be8f02ea2034afa7ef4bbe8adc918ceb49b12773b625f490b368ffffffffffffffffffffffffffffffffffffffffffffffffffffffff8dc5fe11,ed16d65cf3a9538fcb2c139f1ecbc143ee14827120cbc2659e667256800b8142,(u'^3-t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
146f92464d15d36e35382bd3ca5b0f976c95cb08acdcf2d5b3570617990839d7ffffffffffffffffffffffffffffffffffffffffffffffffffffffff3145e93b,0d5cd840427f941f65193079ab8e2e83024ef2ee7ca558d88879ffd879fb6657,(u'^3+t'^2+7)%p=0;valid_x(x3);t>=p
15fdf5cf09c90759add2272d574d2bb5fe1429f9f3c14c65e3194bf61b82aa73ffffffffffffffffffffffffffffffffffffffffffffffffffffffff04cfd906,16d0e43946aec93f62d57eb8cde68951af136cf4b307938dd1447411e07bffe1,(u'^3+t'^2+7)%p=0;valid_x(x2);t>=p
1f67edf779a8a649d6def60035f2fa22d022dd359079a1a144073d84f19b92d50000000000000000000000000000000000000000000000000000000000000000,025661f9aba9d15c3118456bbe980e3e1b8ba2e047c737a4eb48a040bb566f6c,t%p=0;valid_x(x2)
1f67edf779a8a649d6def60035f2fa22d022dd359079a1a144073d84f19b92d5fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,025661f9aba9d15c3118456bbe980e3e1b8ba2e047c737a4eb48a040bb566f6c,t%p=0;valid_x(x2);t>=p
1fe1e5ef3fceb5c135ab7741333ce5a6e80d68167653f6b2b24bcbcfaaaff507fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,98bec3b2a351fa96cfd191c1778351931b9e9ba9ad1149f6d9eadca80981b801,t%p=0;(u'^3-t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
4056a34a210eec7892e8820675c860099f857b26aad85470ee6d3cf1304a9dcf375e70374271f20b13c9986ed7d3c17799698cfc435dbed3a9f34b38c823c2b4,868aac2003b29dbcad1a3e803855e078a89d16543ac64392d122417298cec76e,(u'^3-t'^2+7)%p=0;valid_x(x3)
4197ec3723c654cfdd32ab075506648b2ff5070362d01a4fff14b336b78f963fffffffffffffffffffffffffffffffffffffffffffffffffffffffffb3ab1e95,ba5a6314502a8952b8f456e085928105f665377a8ce27726a5b0eb7ec1ac0286,(u'^3+t'^2+7)%p=0;valid_x(x1);t>=p
47eb3e208fedcdf8234c9421e9cd9a7ae873bfbdbc393723d1ba1e1e6a8e6b24ffffffffffffffffffffffffffffffffffffffffffffffffffffffff7cd12cb1,d192d52007e541c9807006ed0468df77fd214af0a795fe119359666fdcf08f7c,(u'^3+t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
5eb9696a2336fe2c3c666b02c755db4c0cfd62825c7b589a7b7bb442e141c1d693413f0052d49e64abec6d5831d66c43612830a17df1fe4383db896468100221,ef6e1da6d6c7627e80f7a7234cb08a022c1ee1cf29e4d0f9642ae924cef9eb38,(u'^3+t'^2+7)%p=0;valid_x(x1)
7bf96b7b6da15d3476a2b195934b690a3a3de3e8ab8474856863b0de3af90b0e0000000000000000000000000000000000000000000000000000000000000000,50851dfc9f418c314a437295b24feeea27af3d0cd2308348fda6e21c463e46ff,t%p=0;valid_x(x1)
7bf96b7b6da15d3476a2b195934b690a3a3de3e8ab8474856863b0de3af90b0efffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,50851dfc9f418c314a437295b24feeea27af3d0cd2308348fda6e21c463e46ff,t%p=0;valid_x(x1);t>=p
851b1ca94549371c4f1f7187321d39bf51c6b7fb61f7cbf027c9da62021b7a65fc54c96837fb22b362eda63ec52ec83d81bedd160c11b22d965d9f4a6d64d251,3e731051e12d33237eb324f2aa5b16bb868eb49a1aa1fadc19b6e8761b5a5f7b,(u'^3+t'^2+7)%p=0;valid_x(x2)
943c2f775108b737fe65a9531e19f2fc2a197f5603e3a2881d1d83e4008f91250000000000000000000000000000000000000000000000000000000000000000,311c61f0ab2f32b7b1f0223fa72f0a78752b8146e46107f8876dd9c4f92b2942,t%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
943c2f775108b737fe65a9531e19f2fc2a197f5603e3a2881d1d83e4008f9125fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,311c61f0ab2f32b7b1f0223fa72f0a78752b8146e46107f8876dd9c4f92b2942,t%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
a0f18492183e61e8063e573606591421b06bc3513631578a73a39c1c3306239f2f32904f0d2a33ecca8a5451705bb537d3bf44e071226025cdbfd249fe0f7ad6,97a09cf1a2eae7c494df3c6f8a9445bfb8c09d60832f9b0b9d5eabe25fbd14b9,valid_x(x1)
a1ed0a0bd79d8a23cfe4ec5fef5ba5cccfd844e4ff5cb4b0f2e71627341f1c5b17c499249e0ac08d5d11ea1c2c8ca7001616559a7994eadec9ca10fb4b8516dc,65a89640744192cdac64b2d21ddf989cdac7500725b645bef8e2200ae39691f2,valid_x(x2)
ba94594a432721aa3580b84c161d0d134bc354b690404d7cd4ec57c16d3fbe98ffffffffffffffffffffffffffffffffffffffffffffffffffffffffea507dd7,5e0d76564aae92cb347e01a62afd389a9aa401c76c8dd227543dc9cd0efe685a,valid_x(x1);t>=p
bcaf7219f2f6fbf55fe5e062dce0e48c18f68103f10b8198e974c184750e1be3932016cbf69c4471bd1f656c6a107f1973de4af7086db897277060e25677f19a,2d97f96cac882dfe73dc44db6ce0f1d31d6241358dd5d74eb3d3b50003d24c2b,valid_x(x3);valid_x(x2);valid_x(x1)
bcaf7219f2f6fbf55fe5e062dce0e48c18f68103f10b8198e974c184750e1be3ffffffffffffffffffffffffffffffffffffffffffffffffffffffff6507d09a,e7008afe6e8cbd5055df120bd748757c686dadb41cce75e4addcc5e02ec02b44,valid_x(x3);valid_x(x2);valid_x(x1);t>=p
c5981bae27fd84401c72a155e5707fbb811b2b620645d1028ea270cbe0ee225d4b62aa4dca6506c1acdbecc0552569b4b21436a5692e25d90d3bc2eb7ce24078,948b40e7181713bc018ec1702d3d054d15746c59a7020730dd13ecf985a010d7,(u'^3+t'^2+7)%p=0;valid_x(x3)
c894ce48bfec433014b931a6ad4226d7dbd8eaa7b6e3faa8d0ef94052bcf8cff336eeb3919e2b4efb746c7f71bbca7e9383230fbbc48ffafe77e8bcc69542471,f1c91acdc2525330f9b53158434a4d43a1c547cff29f15506f5da4eb4fe8fa5a,(u'^3-t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
cbb0deab125754f1fdb2038b0434ed9cb3fb53ab735391129994a535d925f6730000000000000000000000000000000000000000000000000000000000000000,872d81ed8831d9998b67cb7105243edbf86c10edfebb786c110b02d07b2e67cd,t%p=0;(u'^3-t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
d917b786dac35670c330c9c5ae5971dfb495c8ae523ed97ee2420117b171f41effffffffffffffffffffffffffffffffffffffffffffffffffffffff2001f6f6,e45b71e110b831f2bdad8651994526e58393fde4328b1ec04d59897142584691,valid_x(x3);t>=p
e28bd8f5929b467eb70e04332374ffb7e7180218ad16eaa46b7161aa679eb4260000000000000000000000000000000000000000000000000000000000000000,66b8c980a75c72e598d383a35a62879f844242ad1e73ff12edaa59f4e58632b5,t%p=0;valid_x(x3)
e28bd8f5929b467eb70e04332374ffb7e7180218ad16eaa46b7161aa679eb426fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,66b8c980a75c72e598d383a35a62879f844242ad1e73ff12edaa59f4e58632b5,t%p=0;valid_x(x3);t>=p
e7ee5814c1706bf8a89396a9b032bc014c2cac9c121127dbf6c99278f8bb53d1dfd04dbcda8e352466b6fcd5f2dea3e17d5e133115886eda20db8a12b54de71b,e842c6e3529b234270a5e97744edc34a04d7ba94e44b6d2523c9cf0195730a50,(u'^3+t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
f292e46825f9225ad23dc057c1d91c4f57fcb1386f29ef10481cb1d22518593fffffffffffffffffffffffffffffffffffffffffffffffffffffffff7011c989,3cea2c53b8b0170166ac7da67194694adacc84d56389225e330134dab85a4d55,(u'^3-t'^2+7)%p=0;valid_x(x3);t>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f0000000000000000000000000000000000000000000000000000000000000000,edd1fd3e327ce90cc7a3542614289aee9682003e9cf7dcc9cf2ca9743be5aa0c,u%p=0;t%p=0;valid_x(x2);u>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f01d3475bf7655b0fb2d852921035b2ef607f49069b97454e6795251062741771,b5da00b73cd6560520e7c364086e7cd23a34bf60d0e707be9fc34d4cd5fdfa2c,u%p=0;valid_x(x1);u>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f4218f20ae6c646b363db68605822fb14264ca8d2587fdd6fbc750d587e76a7ee,aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa9fffffd6b,u%p=0;(u'^3-t'^2+7)%p=0;valid_x(x3);u>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f82277c4a71f9d22e66ece523f8fa08741a7c0912c66a69ce68514bfd3515b49f,f482f2e241753ad0fb89150d8491dc1e34ff0b8acfbb442cfe999e2e5e6fd1d2,u%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f8421cc930e77c9f514b6915c3dbe2a94c6d8f690b5b739864ba6789fb8a55dd0,9f59c40275f5085a006f05dae77eb98c6fd0db1ab4a72ac47eae90a4fc9e57e0,u%p=0;valid_x(x2);u>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2fd19c182d2759cd99824228d94799f8c6557c38a1c0d6779b9d4b729c6f1ccc42,70720db7e238d04121f5b1afd8cc5ad9d18944c6bdc94881f502b7a3af3aecff,u%p=0;valid_x(x3);u>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2ffffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,edd1fd3e327ce90cc7a3542614289aee9682003e9cf7dcc9cf2ca9743be5aa0c,u%p=0;t%p=0;valid_x(x2);u>=p;t>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2fffffffffffffffffffffffffffffffffffffffffffffffffffffffff2664bbd5,50873db31badcc71890e4f67753a65757f97aaa7dd5f1e82b753ace32219064b,u%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p;t>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2fffffffffffffffffffffffffffffffffffffffffffffffffffffffff7028de7d,1eea9cc59cfcf2fa151ac6c274eea4110feb4f7b68c5965732e9992e976ef68e,u%p=0;valid_x(x2);u>=p;t>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2fffffffffffffffffffffffffffffffffffffffffffffffffffffffffcbcfb7e7,12303941aedc208880735b1f1795c8e55be520ea93e103357b5d2adb7ed59b8e,u%p=0;valid_x(x1);u>=p;t>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2ffffffffffffffffffffffffffffffffffffffffffffffffffffffffff3113ad9,7eed6b70e7b0767c7d7feac04e57aa2a12fef5e0f48f878fcbb88b3b6b5e0783,u%p=0;valid_x(x3);u>=p;t>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff13cea4a70000000000000000000000000000000000000000000000000000000000000000,649984435b62b4a25d40c6133e8d9ab8c53d4b059ee8a154a3be0fcf4e892edb,t%p=0;valid_x(x1);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff13cea4a7fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,649984435b62b4a25d40c6133e8d9ab8c53d4b059ee8a154a3be0fcf4e892edb,t%p=0;valid_x(x1);u>=p;t>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff15028c590063f64d5a7f1c14915cd61eac886ab295bebd91992504cf77edb028bdd6267f,3fde5713f8282eead7d39d4201f44a7c85a5ac8a0681f35e54085c6b69543374,(u'^3+t'^2+7)%p=0;valid_x(x2);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff2715de860000000000000000000000000000000000000000000000000000000000000000,3524f77fa3a6eb4389c3cb5d27f1f91462086429cd6c0cb0df43ea8f1e7b3fb4,t%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff2715de86fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,3524f77fa3a6eb4389c3cb5d27f1f91462086429cd6c0cb0df43ea8f1e7b3fb4,t%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p;t>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff2c2c5709e7156c417717f2feab147141ec3da19fb759575cc6e37b2ea5ac9309f26f0f66,d2469ab3e04acbb21c65a1809f39caafe7a77c13d10f9dd38f391c01dc499c52,(u'^3-t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff3a08cc1efffffffffffffffffffffffffffffffffffffffffffffffffffffffff760e9f0,38e2a5ce6a93e795e16d2c398bc99f0369202ce21e8f09d56777b40fc512bccc,valid_x(x3);u>=p;t>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff3e91257d932016cbf69c4471bd1f656c6a107f1973de4af7086db897277060e25677f19a,864b3dc902c376709c10a93ad4bbe29fce0012f3dc8672c6286bba28d7d6d6fc,valid_x(x3);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff795d6c1c322cadf599dbb86481522b3cc55f15a67932db2afa0111d9ed6981bcd124bf44,766dfe4a700d9bee288b903ad58870e3d4fe2f0ef780bcac5c823f320d9a9bef,(u'^3+t'^2+7)%p=0;valid_x(x1);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff8e426f0392389078c12b1a89e9542f0593bc96b6bfde8224f8654ef5d5cda935a3582194,faec7bc1987b63233fbc5f956edbf37d54404e7461c58ab8631bc68e451a0478,valid_x(x1);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff91192139ffffffffffffffffffffffffffffffffffffffffffffffffffffffff45f0f1eb,ec29a50bae138dbf7d8e24825006bb5fc1a2cc1243ba335bc6116fb9e498ec1f,valid_x(x2);u>=p;t>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff98eb9ab76e84499c483b3bf06214abfe065dddf43b8601de596d63b9e45a166a580541fe,1e0ff2dee9b09b136292a9e910f0d6ac3e552a644bba39e64e9dd3e3bbd3d4d4,(u'^3-t'^2+7)%p=0;valid_x(x3);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff9b77b7f2c74d99efceaa550f1ad1c0f43f46e7ff1ee3bd0162b7bf55f2965da9c3450646,8b7dd5c3edba9ee97b70eff438f22dca9849c8254a2f3345a0a572ffeaae0928,valid_x(x2);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffff9b77b7f2ffffffffffffffffffffffffffffffffffffffffffffffffffffffff156ca896,0881950c8f51d6b9a6387465d5f12609ef1bb25412a08a74cb2dfb200c74bfbf,valid_x(x3);valid_x(x2);valid_x(x1);u>=p;t>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffa2f5cd838816c16c4fe8a1661d606fdb13cf9af04b979a2e159a09409ebc8645d58fde02,2f083207b9fd9b550063c31cd62b8746bd543bdc5bbf10e3a35563e927f440c8,(u'^3+t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffb13f75c00000000000000000000000000000000000000000000000000000000000000000,4f51e0be078e0cddab2742156adba7e7a148e73157072fd618cd60942b146bd0,t%p=0;valid_x(x3);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffb13f75c0fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,4f51e0be078e0cddab2742156adba7e7a148e73157072fd618cd60942b146bd0,t%p=0;valid_x(x3);u>=p;t>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffe7bc1f8d0000000000000000000000000000000000000000000000000000000000000000,16c2ccb54352ff4bd794f6efd613c72197ab7082da5b563bdf9cb3edaafe74c2,t%p=0;valid_x(x2);u>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffe7bc1f8dfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,16c2ccb54352ff4bd794f6efd613c72197ab7082da5b563bdf9cb3edaafe74c2,t%p=0;valid_x(x2);u>=p;t>=p
ffffffffffffffffffffffffffffffffffffffffffffffffffffffffef64d162750546ce42b0431361e52d4f5242d8f24f33e6b1f99b591647cbc808f462af51,d41244d11ca4f65240687759f95ca9efbab767ededb38fd18c36e18cd3b6f6a9,(u'^3+t'^2+7)%p=0;valid_x(x3);u>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffff0e5be52372dd6e894b2a326fc3605a6e8f3c69c710bf27d630dfe2004988b78eb6eab36,64bf84dd5e03670fdb24c0f5d3c2c365736f51db6c92d95010716ad2d36134c8,valid_x(x3);valid_x(x2);valid_x(x1);u>=p
fffffffffffffffffffffffffffffffffffffffffffffffffffffffffefbb982fffffffffffffffffffffffffffffffffffffffffffffffffffffffff6d6db1f,1c92ccdfcf4ac550c28db57cff0c8515cb26936c786584a70114008d6c33a34b,valid_x(x1);u>=p;t>=p
1 ellswift x comment
2 00000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000 edd1fd3e327ce90cc7a3542614289aee9682003e9cf7dcc9cf2ca9743be5aa0c u%p=0;t%p=0;valid_x(x2)
3 000000000000000000000000000000000000000000000000000000000000000001d3475bf7655b0fb2d852921035b2ef607f49069b97454e6795251062741771 b5da00b73cd6560520e7c364086e7cd23a34bf60d0e707be9fc34d4cd5fdfa2c u%p=0;valid_x(x1)
4 000000000000000000000000000000000000000000000000000000000000000082277c4a71f9d22e66ece523f8fa08741a7c0912c66a69ce68514bfd3515b49f f482f2e241753ad0fb89150d8491dc1e34ff0b8acfbb442cfe999e2e5e6fd1d2 u%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
5 00000000000000000000000000000000000000000000000000000000000000008421cc930e77c9f514b6915c3dbe2a94c6d8f690b5b739864ba6789fb8a55dd0 9f59c40275f5085a006f05dae77eb98c6fd0db1ab4a72ac47eae90a4fc9e57e0 u%p=0;valid_x(x2)
6 0000000000000000000000000000000000000000000000000000000000000000bde70df51939b94c9c24979fa7dd04ebd9b3572da7802290438af2a681895441 aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa9fffffd6b u%p=0;(u'^3-t'^2+7)%p=0;valid_x(x3)
7 0000000000000000000000000000000000000000000000000000000000000000d19c182d2759cd99824228d94799f8c6557c38a1c0d6779b9d4b729c6f1ccc42 70720db7e238d04121f5b1afd8cc5ad9d18944c6bdc94881f502b7a3af3aecff u%p=0;valid_x(x3)
8 0000000000000000000000000000000000000000000000000000000000000000fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f edd1fd3e327ce90cc7a3542614289aee9682003e9cf7dcc9cf2ca9743be5aa0c u%p=0;t%p=0;valid_x(x2);t>=p
9 0000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffff2664bbd5 50873db31badcc71890e4f67753a65757f97aaa7dd5f1e82b753ace32219064b u%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
10 0000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffff7028de7d 1eea9cc59cfcf2fa151ac6c274eea4110feb4f7b68c5965732e9992e976ef68e u%p=0;valid_x(x2);t>=p
11 0000000000000000000000000000000000000000000000000000000000000000ffffffffffffffffffffffffffffffffffffffffffffffffffffffffcbcfb7e7 12303941aedc208880735b1f1795c8e55be520ea93e103357b5d2adb7ed59b8e u%p=0;valid_x(x1);t>=p
12 0000000000000000000000000000000000000000000000000000000000000000fffffffffffffffffffffffffffffffffffffffffffffffffffffffff3113ad9 7eed6b70e7b0767c7d7feac04e57aa2a12fef5e0f48f878fcbb88b3b6b5e0783 u%p=0;valid_x(x3);t>=p
13 0a2d2ba93507f1df233770c2a797962cc61f6d15da14ecd47d8d27ae1cd5f8530000000000000000000000000000000000000000000000000000000000000000 532167c11200b08c0e84a354e74dcc40f8b25f4fe686e30869526366278a0688 t%p=0;(u'^3+t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
14 0a2d2ba93507f1df233770c2a797962cc61f6d15da14ecd47d8d27ae1cd5f853fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f 532167c11200b08c0e84a354e74dcc40f8b25f4fe686e30869526366278a0688 t%p=0;(u'^3+t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
15 0ffde9ca81d751e9cdaffc1a50779245320b28996dbaf32f822f20117c22fbd6c74d99efceaa550f1ad1c0f43f46e7ff1ee3bd0162b7bf55f2965da9c3450646 74e880b3ffd18fe3cddf7902522551ddf97fa4a35a3cfda8197f947081a57b8f valid_x(x3)
16 0ffde9ca81d751e9cdaffc1a50779245320b28996dbaf32f822f20117c22fbd6ffffffffffffffffffffffffffffffffffffffffffffffffffffffff156ca896 377b643fce2271f64e5c8101566107c1be4980745091783804f654781ac9217c valid_x(x2);t>=p
17 123658444f32be8f02ea2034afa7ef4bbe8adc918ceb49b12773b625f490b368ffffffffffffffffffffffffffffffffffffffffffffffffffffffff8dc5fe11 ed16d65cf3a9538fcb2c139f1ecbc143ee14827120cbc2659e667256800b8142 (u'^3-t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
18 146f92464d15d36e35382bd3ca5b0f976c95cb08acdcf2d5b3570617990839d7ffffffffffffffffffffffffffffffffffffffffffffffffffffffff3145e93b 0d5cd840427f941f65193079ab8e2e83024ef2ee7ca558d88879ffd879fb6657 (u'^3+t'^2+7)%p=0;valid_x(x3);t>=p
19 15fdf5cf09c90759add2272d574d2bb5fe1429f9f3c14c65e3194bf61b82aa73ffffffffffffffffffffffffffffffffffffffffffffffffffffffff04cfd906 16d0e43946aec93f62d57eb8cde68951af136cf4b307938dd1447411e07bffe1 (u'^3+t'^2+7)%p=0;valid_x(x2);t>=p
20 1f67edf779a8a649d6def60035f2fa22d022dd359079a1a144073d84f19b92d50000000000000000000000000000000000000000000000000000000000000000 025661f9aba9d15c3118456bbe980e3e1b8ba2e047c737a4eb48a040bb566f6c t%p=0;valid_x(x2)
21 1f67edf779a8a649d6def60035f2fa22d022dd359079a1a144073d84f19b92d5fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f 025661f9aba9d15c3118456bbe980e3e1b8ba2e047c737a4eb48a040bb566f6c t%p=0;valid_x(x2);t>=p
22 1fe1e5ef3fceb5c135ab7741333ce5a6e80d68167653f6b2b24bcbcfaaaff507fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f 98bec3b2a351fa96cfd191c1778351931b9e9ba9ad1149f6d9eadca80981b801 t%p=0;(u'^3-t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
23 4056a34a210eec7892e8820675c860099f857b26aad85470ee6d3cf1304a9dcf375e70374271f20b13c9986ed7d3c17799698cfc435dbed3a9f34b38c823c2b4 868aac2003b29dbcad1a3e803855e078a89d16543ac64392d122417298cec76e (u'^3-t'^2+7)%p=0;valid_x(x3)
24 4197ec3723c654cfdd32ab075506648b2ff5070362d01a4fff14b336b78f963fffffffffffffffffffffffffffffffffffffffffffffffffffffffffb3ab1e95 ba5a6314502a8952b8f456e085928105f665377a8ce27726a5b0eb7ec1ac0286 (u'^3+t'^2+7)%p=0;valid_x(x1);t>=p
25 47eb3e208fedcdf8234c9421e9cd9a7ae873bfbdbc393723d1ba1e1e6a8e6b24ffffffffffffffffffffffffffffffffffffffffffffffffffffffff7cd12cb1 d192d52007e541c9807006ed0468df77fd214af0a795fe119359666fdcf08f7c (u'^3+t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
26 5eb9696a2336fe2c3c666b02c755db4c0cfd62825c7b589a7b7bb442e141c1d693413f0052d49e64abec6d5831d66c43612830a17df1fe4383db896468100221 ef6e1da6d6c7627e80f7a7234cb08a022c1ee1cf29e4d0f9642ae924cef9eb38 (u'^3+t'^2+7)%p=0;valid_x(x1)
27 7bf96b7b6da15d3476a2b195934b690a3a3de3e8ab8474856863b0de3af90b0e0000000000000000000000000000000000000000000000000000000000000000 50851dfc9f418c314a437295b24feeea27af3d0cd2308348fda6e21c463e46ff t%p=0;valid_x(x1)
28 7bf96b7b6da15d3476a2b195934b690a3a3de3e8ab8474856863b0de3af90b0efffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f 50851dfc9f418c314a437295b24feeea27af3d0cd2308348fda6e21c463e46ff t%p=0;valid_x(x1);t>=p
29 851b1ca94549371c4f1f7187321d39bf51c6b7fb61f7cbf027c9da62021b7a65fc54c96837fb22b362eda63ec52ec83d81bedd160c11b22d965d9f4a6d64d251 3e731051e12d33237eb324f2aa5b16bb868eb49a1aa1fadc19b6e8761b5a5f7b (u'^3+t'^2+7)%p=0;valid_x(x2)
30 943c2f775108b737fe65a9531e19f2fc2a197f5603e3a2881d1d83e4008f91250000000000000000000000000000000000000000000000000000000000000000 311c61f0ab2f32b7b1f0223fa72f0a78752b8146e46107f8876dd9c4f92b2942 t%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
31 943c2f775108b737fe65a9531e19f2fc2a197f5603e3a2881d1d83e4008f9125fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f 311c61f0ab2f32b7b1f0223fa72f0a78752b8146e46107f8876dd9c4f92b2942 t%p=0;valid_x(x3);valid_x(x2);valid_x(x1);t>=p
32 a0f18492183e61e8063e573606591421b06bc3513631578a73a39c1c3306239f2f32904f0d2a33ecca8a5451705bb537d3bf44e071226025cdbfd249fe0f7ad6 97a09cf1a2eae7c494df3c6f8a9445bfb8c09d60832f9b0b9d5eabe25fbd14b9 valid_x(x1)
33 a1ed0a0bd79d8a23cfe4ec5fef5ba5cccfd844e4ff5cb4b0f2e71627341f1c5b17c499249e0ac08d5d11ea1c2c8ca7001616559a7994eadec9ca10fb4b8516dc 65a89640744192cdac64b2d21ddf989cdac7500725b645bef8e2200ae39691f2 valid_x(x2)
34 ba94594a432721aa3580b84c161d0d134bc354b690404d7cd4ec57c16d3fbe98ffffffffffffffffffffffffffffffffffffffffffffffffffffffffea507dd7 5e0d76564aae92cb347e01a62afd389a9aa401c76c8dd227543dc9cd0efe685a valid_x(x1);t>=p
35 bcaf7219f2f6fbf55fe5e062dce0e48c18f68103f10b8198e974c184750e1be3932016cbf69c4471bd1f656c6a107f1973de4af7086db897277060e25677f19a 2d97f96cac882dfe73dc44db6ce0f1d31d6241358dd5d74eb3d3b50003d24c2b valid_x(x3);valid_x(x2);valid_x(x1)
36 bcaf7219f2f6fbf55fe5e062dce0e48c18f68103f10b8198e974c184750e1be3ffffffffffffffffffffffffffffffffffffffffffffffffffffffff6507d09a e7008afe6e8cbd5055df120bd748757c686dadb41cce75e4addcc5e02ec02b44 valid_x(x3);valid_x(x2);valid_x(x1);t>=p
37 c5981bae27fd84401c72a155e5707fbb811b2b620645d1028ea270cbe0ee225d4b62aa4dca6506c1acdbecc0552569b4b21436a5692e25d90d3bc2eb7ce24078 948b40e7181713bc018ec1702d3d054d15746c59a7020730dd13ecf985a010d7 (u'^3+t'^2+7)%p=0;valid_x(x3)
38 c894ce48bfec433014b931a6ad4226d7dbd8eaa7b6e3faa8d0ef94052bcf8cff336eeb3919e2b4efb746c7f71bbca7e9383230fbbc48ffafe77e8bcc69542471 f1c91acdc2525330f9b53158434a4d43a1c547cff29f15506f5da4eb4fe8fa5a (u'^3-t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
39 cbb0deab125754f1fdb2038b0434ed9cb3fb53ab735391129994a535d925f6730000000000000000000000000000000000000000000000000000000000000000 872d81ed8831d9998b67cb7105243edbf86c10edfebb786c110b02d07b2e67cd t%p=0;(u'^3-t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
40 d917b786dac35670c330c9c5ae5971dfb495c8ae523ed97ee2420117b171f41effffffffffffffffffffffffffffffffffffffffffffffffffffffff2001f6f6 e45b71e110b831f2bdad8651994526e58393fde4328b1ec04d59897142584691 valid_x(x3);t>=p
41 e28bd8f5929b467eb70e04332374ffb7e7180218ad16eaa46b7161aa679eb4260000000000000000000000000000000000000000000000000000000000000000 66b8c980a75c72e598d383a35a62879f844242ad1e73ff12edaa59f4e58632b5 t%p=0;valid_x(x3)
42 e28bd8f5929b467eb70e04332374ffb7e7180218ad16eaa46b7161aa679eb426fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f 66b8c980a75c72e598d383a35a62879f844242ad1e73ff12edaa59f4e58632b5 t%p=0;valid_x(x3);t>=p
43 e7ee5814c1706bf8a89396a9b032bc014c2cac9c121127dbf6c99278f8bb53d1dfd04dbcda8e352466b6fcd5f2dea3e17d5e133115886eda20db8a12b54de71b e842c6e3529b234270a5e97744edc34a04d7ba94e44b6d2523c9cf0195730a50 (u'^3+t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1)
44 f292e46825f9225ad23dc057c1d91c4f57fcb1386f29ef10481cb1d22518593fffffffffffffffffffffffffffffffffffffffffffffffffffffffff7011c989 3cea2c53b8b0170166ac7da67194694adacc84d56389225e330134dab85a4d55 (u'^3-t'^2+7)%p=0;valid_x(x3);t>=p
45 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f0000000000000000000000000000000000000000000000000000000000000000 edd1fd3e327ce90cc7a3542614289aee9682003e9cf7dcc9cf2ca9743be5aa0c u%p=0;t%p=0;valid_x(x2);u>=p
46 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f01d3475bf7655b0fb2d852921035b2ef607f49069b97454e6795251062741771 b5da00b73cd6560520e7c364086e7cd23a34bf60d0e707be9fc34d4cd5fdfa2c u%p=0;valid_x(x1);u>=p
47 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f4218f20ae6c646b363db68605822fb14264ca8d2587fdd6fbc750d587e76a7ee aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa9fffffd6b u%p=0;(u'^3-t'^2+7)%p=0;valid_x(x3);u>=p
48 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f82277c4a71f9d22e66ece523f8fa08741a7c0912c66a69ce68514bfd3515b49f f482f2e241753ad0fb89150d8491dc1e34ff0b8acfbb442cfe999e2e5e6fd1d2 u%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p
49 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f8421cc930e77c9f514b6915c3dbe2a94c6d8f690b5b739864ba6789fb8a55dd0 9f59c40275f5085a006f05dae77eb98c6fd0db1ab4a72ac47eae90a4fc9e57e0 u%p=0;valid_x(x2);u>=p
50 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2fd19c182d2759cd99824228d94799f8c6557c38a1c0d6779b9d4b729c6f1ccc42 70720db7e238d04121f5b1afd8cc5ad9d18944c6bdc94881f502b7a3af3aecff u%p=0;valid_x(x3);u>=p
51 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2ffffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f edd1fd3e327ce90cc7a3542614289aee9682003e9cf7dcc9cf2ca9743be5aa0c u%p=0;t%p=0;valid_x(x2);u>=p;t>=p
52 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2fffffffffffffffffffffffffffffffffffffffffffffffffffffffff2664bbd5 50873db31badcc71890e4f67753a65757f97aaa7dd5f1e82b753ace32219064b u%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p;t>=p
53 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2fffffffffffffffffffffffffffffffffffffffffffffffffffffffff7028de7d 1eea9cc59cfcf2fa151ac6c274eea4110feb4f7b68c5965732e9992e976ef68e u%p=0;valid_x(x2);u>=p;t>=p
54 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2fffffffffffffffffffffffffffffffffffffffffffffffffffffffffcbcfb7e7 12303941aedc208880735b1f1795c8e55be520ea93e103357b5d2adb7ed59b8e u%p=0;valid_x(x1);u>=p;t>=p
55 fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2ffffffffffffffffffffffffffffffffffffffffffffffffffffffffff3113ad9 7eed6b70e7b0767c7d7feac04e57aa2a12fef5e0f48f878fcbb88b3b6b5e0783 u%p=0;valid_x(x3);u>=p;t>=p
56 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff13cea4a70000000000000000000000000000000000000000000000000000000000000000 649984435b62b4a25d40c6133e8d9ab8c53d4b059ee8a154a3be0fcf4e892edb t%p=0;valid_x(x1);u>=p
57 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff13cea4a7fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f 649984435b62b4a25d40c6133e8d9ab8c53d4b059ee8a154a3be0fcf4e892edb t%p=0;valid_x(x1);u>=p;t>=p
58 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff15028c590063f64d5a7f1c14915cd61eac886ab295bebd91992504cf77edb028bdd6267f 3fde5713f8282eead7d39d4201f44a7c85a5ac8a0681f35e54085c6b69543374 (u'^3+t'^2+7)%p=0;valid_x(x2);u>=p
59 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff2715de860000000000000000000000000000000000000000000000000000000000000000 3524f77fa3a6eb4389c3cb5d27f1f91462086429cd6c0cb0df43ea8f1e7b3fb4 t%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p
60 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff2715de86fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f 3524f77fa3a6eb4389c3cb5d27f1f91462086429cd6c0cb0df43ea8f1e7b3fb4 t%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p;t>=p
61 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff2c2c5709e7156c417717f2feab147141ec3da19fb759575cc6e37b2ea5ac9309f26f0f66 d2469ab3e04acbb21c65a1809f39caafe7a77c13d10f9dd38f391c01dc499c52 (u'^3-t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p
62 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff3a08cc1efffffffffffffffffffffffffffffffffffffffffffffffffffffffff760e9f0 38e2a5ce6a93e795e16d2c398bc99f0369202ce21e8f09d56777b40fc512bccc valid_x(x3);u>=p;t>=p
63 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff3e91257d932016cbf69c4471bd1f656c6a107f1973de4af7086db897277060e25677f19a 864b3dc902c376709c10a93ad4bbe29fce0012f3dc8672c6286bba28d7d6d6fc valid_x(x3);u>=p
64 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff795d6c1c322cadf599dbb86481522b3cc55f15a67932db2afa0111d9ed6981bcd124bf44 766dfe4a700d9bee288b903ad58870e3d4fe2f0ef780bcac5c823f320d9a9bef (u'^3+t'^2+7)%p=0;valid_x(x1);u>=p
65 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff8e426f0392389078c12b1a89e9542f0593bc96b6bfde8224f8654ef5d5cda935a3582194 faec7bc1987b63233fbc5f956edbf37d54404e7461c58ab8631bc68e451a0478 valid_x(x1);u>=p
66 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff91192139ffffffffffffffffffffffffffffffffffffffffffffffffffffffff45f0f1eb ec29a50bae138dbf7d8e24825006bb5fc1a2cc1243ba335bc6116fb9e498ec1f valid_x(x2);u>=p;t>=p
67 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff98eb9ab76e84499c483b3bf06214abfe065dddf43b8601de596d63b9e45a166a580541fe 1e0ff2dee9b09b136292a9e910f0d6ac3e552a644bba39e64e9dd3e3bbd3d4d4 (u'^3-t'^2+7)%p=0;valid_x(x3);u>=p
68 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff9b77b7f2c74d99efceaa550f1ad1c0f43f46e7ff1ee3bd0162b7bf55f2965da9c3450646 8b7dd5c3edba9ee97b70eff438f22dca9849c8254a2f3345a0a572ffeaae0928 valid_x(x2);u>=p
69 ffffffffffffffffffffffffffffffffffffffffffffffffffffffff9b77b7f2ffffffffffffffffffffffffffffffffffffffffffffffffffffffff156ca896 0881950c8f51d6b9a6387465d5f12609ef1bb25412a08a74cb2dfb200c74bfbf valid_x(x3);valid_x(x2);valid_x(x1);u>=p;t>=p
70 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffa2f5cd838816c16c4fe8a1661d606fdb13cf9af04b979a2e159a09409ebc8645d58fde02 2f083207b9fd9b550063c31cd62b8746bd543bdc5bbf10e3a35563e927f440c8 (u'^3+t'^2+7)%p=0;valid_x(x3);valid_x(x2);valid_x(x1);u>=p
71 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffb13f75c00000000000000000000000000000000000000000000000000000000000000000 4f51e0be078e0cddab2742156adba7e7a148e73157072fd618cd60942b146bd0 t%p=0;valid_x(x3);u>=p
72 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffb13f75c0fffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f 4f51e0be078e0cddab2742156adba7e7a148e73157072fd618cd60942b146bd0 t%p=0;valid_x(x3);u>=p;t>=p
73 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffe7bc1f8d0000000000000000000000000000000000000000000000000000000000000000 16c2ccb54352ff4bd794f6efd613c72197ab7082da5b563bdf9cb3edaafe74c2 t%p=0;valid_x(x2);u>=p
74 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffe7bc1f8dfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f 16c2ccb54352ff4bd794f6efd613c72197ab7082da5b563bdf9cb3edaafe74c2 t%p=0;valid_x(x2);u>=p;t>=p
75 ffffffffffffffffffffffffffffffffffffffffffffffffffffffffef64d162750546ce42b0431361e52d4f5242d8f24f33e6b1f99b591647cbc808f462af51 d41244d11ca4f65240687759f95ca9efbab767ededb38fd18c36e18cd3b6f6a9 (u'^3+t'^2+7)%p=0;valid_x(x3);u>=p
76 fffffffffffffffffffffffffffffffffffffffffffffffffffffffff0e5be52372dd6e894b2a326fc3605a6e8f3c69c710bf27d630dfe2004988b78eb6eab36 64bf84dd5e03670fdb24c0f5d3c2c365736f51db6c92d95010716ad2d36134c8 valid_x(x3);valid_x(x2);valid_x(x1);u>=p
77 fffffffffffffffffffffffffffffffffffffffffffffffffffffffffefbb982fffffffffffffffffffffffffffffffffffffffffffffffffffffffff6d6db1f 1c92ccdfcf4ac550c28db57cff0c8515cb26936c786584a70114008d6c33a34b valid_x(x1);u>=p;t>=p

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u,x,case0_t,case1_t,case2_t,case3_t,case4_t,case5_t,case6_t,case7_t,comment
05ff6bdad900fc3261bc7fe34e2fb0f569f06e091ae437d3a52e9da0cbfb9590,80cdf63774ec7022c89a5a8558e373a279170285e0ab27412dbce510bdfe23fc,,,45654798ece071ba79286d04f7f3eb1c3f1d17dd883610f2ad2efd82a287466b,0aeaa886f6b76c7158452418cbf5033adc5747e9e9b5d3b2303db96936528557,,,ba9ab867131f8e4586d792fb080c14e3c0e2e82277c9ef0d52d1027c5d78b5c4,f51557790948938ea7badbe7340afcc523a8b816164a2c4dcfc24695c9ad76d8,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:info[v=0]&ok;case3:ok;case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:info[v=0]&ok;case7:ok
1737a85f4c8d146cec96e3ffdca76d9903dcf3bd53061868d478c78c63c2aa9e,39e48dd150d2f429be088dfd5b61882e7e8407483702ae9a5ab35927b15f85ea,1be8cc0b04be0c681d0c6a68f733f82c6c896e0c8a262fcd392918e303a7abf4,605b5814bf9b8cb066667c9e5480d22dc5b6c92f14b4af3ee0a9eb83b03685e3,,,e41733f4fb41f397e2f3959708cc07d3937691f375d9d032c6d6e71bfc58503b,9fa4a7eb4064734f99998361ab7f2dd23a4936d0eb4b50c11f56147b4fc9764c,,,case0:ok;case1:ok;case2:info[v=0]&bad[non_square(s)];case3:bad[non_square(s)];case4:ok;case5:ok;case6:info[v=0]&bad[non_square(s)];case7:bad[non_square(s)]
1aaa1ccebf9c724191033df366b36f691c4d902c228033ff4516d122b2564f68,c75541259d3ba98f207eaa30c69634d187d0b6da594e719e420f4898638fc5b0,,,,,,,,,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:bad[non_square(q)];case3:bad[non_square(q)];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:bad[non_square(q)];case7:bad[non_square(q)]
2323a1d079b0fd72fc8bb62ec34230a815cb0596c2bfac998bd6b84260f5dc26,239342dfb675500a34a196310b8d87d54f49dcac9da50c1743ceab41a7b249ff,f63580b8aa49c4846de56e39e1b3e73f171e881eba8c66f614e67e5c975dfc07,b6307b332e699f1cf77841d90af25365404deb7fed5edb3090db49e642a156b6,,,09ca7f4755b63b7b921a91c61e4c18c0e8e177e145739909eb1981a268a20028,49cf84ccd19660e30887be26f50dac9abfb2148012a124cf6f24b618bd5ea579,,,case0:ok;case1:ok;case2:bad[non_square(q)];case3:bad[non_square(q)];case4:ok;case5:ok;case6:bad[non_square(q)];case7:bad[non_square(q)]
2dc90e640cb646ae9164c0b5a9ef0169febe34dc4437d6e46acb0e27e219d1e8,d236f19bf349b9516e9b3f4a5610fe960141cb23bbc8291b9534f1d71de62a47,e69df7d9c026c36600ebdf588072675847c0c431c8eb730682533e964b6252c9,4f18bbdf7c2d6c5f818c18802fa35cd069eaa79fff74e4fc837c80d93fece2f8,,,196208263fd93c99ff1420a77f8d98a7b83f3bce37148cf97dacc168b49da966,b0e7442083d293a07e73e77fd05ca32f96155860008b1b037c837f25c0131937,,,case0:ok;case1:info[v=0]&ok;case2:bad[non_square(q)];case3:bad[non_square(q)];case4:ok;case5:info[v=0]&ok;case6:bad[non_square(q)];case7:bad[non_square(q)]
3edd7b3980e2f2f34d1409a207069f881fda5f96f08027ac4465b63dc278d672,053a98de4a27b1961155822b3a3121f03b2a14458bd80eb4a560c4c7a85c149c,,,b3dae4b7dcf858e4c6968057cef2b156465431526538199cf52dc1b2d62fda30,4aa77dd55d6b6d3cfa10cc9d0fe42f79232e4575661049ae36779c1d0c666d88,,,4c251b482307a71b39697fa8310d4ea9b9abcead9ac7e6630ad23e4c29d021ff,b558822aa29492c305ef3362f01bd086dcd1ba8a99efb651c98863e1f3998ea7,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:ok;case3:ok;case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:ok;case7:ok
4295737efcb1da6fb1d96b9ca7dcd1e320024b37a736c4948b62598173069f70,fa7ffe4f25f88362831c087afe2e8a9b0713e2cac1ddca6a383205a266f14307,,,,,,,,,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:bad[non_square(s)];case3:bad[non_square(s)];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:bad[non_square(s)];case7:bad[non_square(s)]
587c1a0cee91939e7f784d23b963004a3bf44f5d4e32a0081995ba20b0fca59e,2ea988530715e8d10363907ff25124524d471ba2454d5ce3be3f04194dfd3a3c,cfd5a094aa0b9b8891b76c6ab9438f66aa1c095a65f9f70135e8171292245e74,a89057d7c6563f0d6efa19ae84412b8a7b47e791a191ecdfdf2af84fd97bc339,475d0ae9ef46920df07b34117be5a0817de1023e3cc32689e9be145b406b0aef,a0759178ad80232454f827ef05ea3e72ad8d75418e6d4cc1cd4f5306c5e7c453,302a5f6b55f464776e48939546bc709955e3f6a59a0608feca17e8ec6ddb9dbb,576fa82839a9c0f29105e6517bbed47584b8186e5e6e132020d507af268438f6,b8a2f51610b96df20f84cbee841a5f7e821efdc1c33cd9761641eba3bf94f140,5f8a6e87527fdcdbab07d810fa15c18d52728abe7192b33e32b0acf83a1837dc,case0:ok;case1:ok;case2:ok;case3:ok;case4:ok;case5:ok;case6:ok;case7:ok
5fa88b3365a635cbbcee003cce9ef51dd1a310de277e441abccdb7be1e4ba249,79461ff62bfcbcac4249ba84dd040f2cec3c63f725204dc7f464c16bf0ff3170,,,6bb700e1f4d7e236e8d193ff4a76c1b3bcd4e2b25acac3d51c8dac653fe909a0,f4c73410633da7f63a4f1d55aec6dd32c4c6d89ee74075edb5515ed90da9e683,,,9448ff1e0b281dc9172e6c00b5893e4c432b1d4da5353c2ae3725399c016f28f,0b38cbef9cc25809c5b0e2aa513922cd3b39276118bf8a124aaea125f25615ac,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:ok;case3:info[v=0]&ok;case4:bad[non_square(s)];case5:bad[non_square(s)];case6:ok;case7:info[v=0]&ok
6fb31c7531f03130b42b155b952779efbb46087dd9807d241a48eac63c3d96d6,56f81be753e8d4ae4940ea6f46f6ec9fda66a6f96cc95f506cb2b57490e94260,,,59059774795bdb7a837fbe1140a5fa59984f48af8df95d57dd6d1c05437dcec1,22a644db79376ad4e7b3a009e58b3f13137c54fdf911122cc93667c47077d784,,,a6fa688b86a424857c8041eebf5a05a667b0b7507206a2a82292e3f9bc822d6e,dd59bb2486c8952b184c5ff61a74c0ecec83ab0206eeedd336c9983a8f8824ab,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:ok;case3:info[v=0]&ok;case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:ok;case7:info[v=0]&ok
704cd226e71cb6826a590e80dac90f2d2f5830f0fdf135a3eae3965bff25ff12,138e0afa68936ee670bd2b8db53aedbb7bea2a8597388b24d0518edd22ad66ec,,,,,,,,,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:bad[non_square(q)];case3:bad[non_square(q)];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:bad[non_square(q)];case7:bad[non_square(q)]
725e914792cb8c8949e7e1168b7cdd8a8094c91c6ec2202ccd53a6a18771edeb,8da16eb86d347376b6181ee9748322757f6b36e3913ddfd332ac595d788e0e44,dd357786b9f6873330391aa5625809654e43116e82a5a5d82ffd1d6624101fc4,a0b7efca01814594c59c9aae8e49700186ca5d95e88bcc80399044d9c2d8613d,,,22ca8879460978cccfc6e55a9da7f69ab1bcee917d5a5a27d002e298dbefdc6b,5f481035fe7eba6b3a63655171b68ffe7935a26a1774337fc66fbb253d279af2,,,case0:ok;case1:info[v=0]&ok;case2:bad[non_square(s)];case3:bad[non_square(s)];case4:ok;case5:info[v=0]&ok;case6:bad[non_square(s)];case7:bad[non_square(s)]
78fe6b717f2ea4a32708d79c151bf503a5312a18c0963437e865cc6ed3f6ae97,8701948e80d15b5cd8f72863eae40afc5aced5e73f69cbc8179a33902c094d98,,,,,,,,,case0:bad[non_square(s)];case1:info[v=0]&bad[non_square(s)];case2:bad[non_square(q)];case3:bad[non_square(q)];case4:bad[non_square(s)];case5:info[v=0]&bad[non_square(s)];case6:bad[non_square(q)];case7:bad[non_square(q)]
7c37bb9c5061dc07413f11acd5a34006e64c5c457fdb9a438f217255a961f50d,5c1a76b44568eb59d6789a7442d9ed7cdc6226b7752b4ff8eaf8e1a95736e507,,,b94d30cd7dbff60b64620c17ca0fafaa40b3d1f52d077a60a2e0cafd145086c2,,,,46b2cf32824009f49b9df3e835f05055bf4c2e0ad2f8859f5d1f3501ebaf756d,,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:info[q=0]&info[X=0]&ok;case3:info[q=0]&bad[r=0];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:info[q=0]&info[X=0]&ok;case7:info[q=0]&bad[r=0]
82388888967f82a6b444438a7d44838e13c0d478b9ca060da95a41fb94303de6,29e9654170628fec8b4972898b113cf98807f4609274f4f3140d0674157c90a0,,,,,,,,,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:bad[non_square(s)];case3:info[v=0]&bad[non_square(s)];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:bad[non_square(s)];case7:info[v=0]&bad[non_square(s)]
91298f5770af7a27f0a47188d24c3b7bf98ab2990d84b0b898507e3c561d6472,144f4ccbd9a74698a88cbf6fd00ad886d339d29ea19448f2c572cac0a07d5562,e6a0ffa3807f09dadbe71e0f4be4725f2832e76cad8dc1d943ce839375eff248,837b8e68d4917544764ad0903cb11f8615d2823cefbb06d89049dbabc69befda,,,195f005c7f80f6252418e1f0b41b8da0d7cd189352723e26bc317c6b8a1009e7,7c8471972b6e8abb89b52f6fc34ee079ea2d7dc31044f9276fb6245339640c55,,,case0:ok;case1:ok;case2:bad[non_square(s)];case3:info[v=0]&bad[non_square(s)];case4:ok;case5:ok;case6:bad[non_square(s)];case7:info[v=0]&bad[non_square(s)]
b682f3d03bbb5dee4f54b5ebfba931b4f52f6a191e5c2f483c73c66e9ace97e1,904717bf0bc0cb7873fcdc38aa97f19e3a62630972acff92b24cc6dda197cb96,,,,,,,,,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:bad[non_square(s)];case3:bad[non_square(s)];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:bad[non_square(s)];case7:bad[non_square(s)]
c17ec69e665f0fb0dbab48d9c2f94d12ec8a9d7eacb58084833091801eb0b80b,147756e66d96e31c426d3cc85ed0c4cfbef6341dd8b285585aa574ea0204b55e,6f4aea431a0043bdd03134d6d9159119ce034b88c32e50e8e36c4ee45eac7ae9,fd5be16d4ffa2690126c67c3ef7cb9d29b74d397c78b06b3605fda34dc9696a6,5e9c60792a2f000e45c6250f296f875e174efc0e9703e628706103a9dd2d82c7,,90b515bce5ffbc422fcecb2926ea6ee631fcb4773cd1af171c93b11aa1538146,02a41e92b005d96fed93983c1083462d648b2c683874f94c9fa025ca23696589,a1639f86d5d0fff1ba39daf0d69078a1e8b103f168fc19d78f9efc5522d27968,,case0:ok;case1:ok;case2:info[q=0]&info[X=0]&ok;case3:info[q=0]&bad[r=0];case4:ok;case5:ok;case6:info[q=0]&info[X=0]&ok;case7:info[q=0]&bad[r=0]
c25172fc3f29b6fc4a1155b8575233155486b27464b74b8b260b499a3f53cb14,1ea9cbdb35cf6e0329aa31b0bb0a702a65123ed008655a93b7dcd5280e52e1ab,,,7422edc7843136af0053bb8854448a8299994f9ddcefd3a9a92d45462c59298a,78c7774a266f8b97ea23d05d064f033c77319f923f6b78bce4e20bf05fa5398d,,,8bdd12387bcec950ffac4477abbb757d6666b06223102c5656d2bab8d3a6d2a5,873888b5d990746815dc2fa2f9b0fcc388ce606dc09487431b1df40ea05ac2a2,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:ok;case3:ok;case4:bad[non_square(s)];case5:bad[non_square(s)];case6:ok;case7:ok
cab6626f832a4b1280ba7add2fc5322ff011caededf7ff4db6735d5026dc0367,2b2bef0852c6f7c95d72ac99a23802b875029cd573b248d1f1b3fc8033788eb6,,,,,,,,,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:info[v=0]&bad[non_square(s)];case3:bad[non_square(s)];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:info[v=0]&bad[non_square(s)];case7:bad[non_square(s)]
d8621b4ffc85b9ed56e99d8dd1dd24aedcecb14763b861a17112dc771a104fd2,812cabe972a22aa67c7da0c94d8a936296eb9949d70c37cb2b2487574cb3ce58,fbc5febc6fdbc9ae3eb88a93b982196e8b6275a6d5a73c17387e000c711bd0e3,8724c96bd4e5527f2dd195a51c468d2d211ba2fac7cbe0b4b3434253409fb42d,,,043a014390243651c147756c467de691749d8a592a58c3e8c781fff28ee42b4c,78db36942b1aad80d22e6a5ae3b972d2dee45d0538341f4b4cbcbdabbf604802,,,case0:ok;case1:ok;case2:bad[non_square(s)];case3:bad[non_square(s)];case4:ok;case5:ok;case6:bad[non_square(s)];case7:bad[non_square(s)]
da463164c6f4bf7129ee5f0ec00f65a675a8adf1bd931b39b64806afdcda9a22,25b9ce9b390b408ed611a0f13ff09a598a57520e426ce4c649b7f94f2325620d,,,,,,,,,case0:bad[non_square(s)];case1:info[v=0]&bad[non_square(s)];case2:bad[non_square(s)];case3:bad[non_square(s)];case4:bad[non_square(s)];case5:info[v=0]&bad[non_square(s)];case6:bad[non_square(s)];case7:bad[non_square(s)]
dafc971e4a3a7b6dcfb42a08d9692d82ad9e7838523fcbda1d4827e14481ae2d,250368e1b5c58492304bd5f72696d27d526187c7adc03425e2b7d81dbb7e4e02,,,370c28f1be665efacde6aa436bf86fe21e6e314c1e53dd040e6c73a46b4c8c49,cd8acee98ffe56531a84d7eb3e48fa4034206ce825ace907d0edf0eaeb5e9ca2,,,c8f3d70e4199a105321955bc9407901de191ceb3e1ac22fbf1938c5a94b36fe6,327531167001a9ace57b2814c1b705bfcbdf9317da5316f82f120f1414a15f8d,case0:bad[non_square(s)];case1:info[v=0]&bad[non_square(s)];case2:ok;case3:ok;case4:bad[non_square(s)];case5:info[v=0]&bad[non_square(s)];case6:ok;case7:ok
e0294c8bc1a36b4166ee92bfa70a5c34976fa9829405efea8f9cd54dcb29b99e,ae9690d13b8d20a0fbbf37bed8474f67a04e142f56efd78770a76b359165d8a1,,,dcd45d935613916af167b029058ba3a700d37150b9df34728cb05412c16d4182,,,,232ba26ca9ec6e950e984fd6fa745c58ff2c8eaf4620cb8d734fabec3e92baad,,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:info[q=0]&info[X=0]&ok;case3:info[q=0]&bad[r=0];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:info[q=0]&info[X=0]&ok;case7:info[q=0]&bad[r=0]
e148441cd7b92b8b0e4fa3bd68712cfd0d709ad198cace611493c10e97f5394e,164a639794d74c53afc4d3294e79cdb3cd25f99f6df45c000f758aba54d699c0,,,,,,,,,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:bad[non_square(s)];case3:info[v=0]&bad[non_square(s)];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:bad[non_square(s)];case7:info[v=0]&bad[non_square(s)]
e4b00ec97aadcca97644d3b0c8a931b14ce7bcf7bc8779546d6e35aa5937381c,94e9588d41647b3fcc772dc8d83c67ce3be003538517c834103d2cd49d62ef4d,c88d25f41407376bb2c03a7fffeb3ec7811cc43491a0c3aac0378cdc78357bee,51c02636ce00c2345ecd89adb6089fe4d5e18ac924e3145e6669501cd37a00d4,205b3512db40521cb200952e67b46f67e09e7839e0de44004138329ebd9138c5,58aab390ab6fb55c1d1b80897a207ce94a78fa5b4aa61a33398bcae9adb20d3e,3772da0bebf8c8944d3fc5800014c1387ee33bcb6e5f3c553fc8732287ca8041,ae3fd9c931ff3dcba132765249f7601b2a1e7536db1ceba19996afe22c85fb5b,dfa4caed24bfade34dff6ad1984b90981f6187c61f21bbffbec7cd60426ec36a,a7554c6f54904aa3e2e47f7685df8316b58705a4b559e5ccc6743515524deef1,case0:ok;case1:ok;case2:ok;case3:info[v=0]&ok;case4:ok;case5:ok;case6:ok;case7:info[v=0]&ok
e5bbb9ef360d0a501618f0067d36dceb75f5be9a620232aa9fd5139d0863fde5,e5bbb9ef360d0a501618f0067d36dceb75f5be9a620232aa9fd5139d0863fde5,,,,,,,,,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:bad[s=0];case3:bad[s=0];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:bad[s=0];case7:bad[s=0]
e6bcb5c3d63467d490bfa54fbbc6092a7248c25e11b248dc2964a6e15edb1457,19434a3c29cb982b6f405ab04439f6d58db73da1ee4db723d69b591da124e7d8,67119877832ab8f459a821656d8261f544a553b89ae4f25c52a97134b70f3426,ffee02f5e649c07f0560eff1867ec7b32d0e595e9b1c0ea6e2a4fc70c97cd71f,b5e0c189eb5b4bacd025b7444d74178be8d5246cfa4a9a207964a057ee969992,5746e4591bf7f4c3044609ea372e908603975d279fdef8349f0b08d32f07619d,98ee67887cd5470ba657de9a927d9e0abb5aac47651b0da3ad568eca48f0c809,0011fd0a19b63f80fa9f100e7981384cd2f1a6a164e3f1591d5b038e36832510,4a1f3e7614a4b4532fda48bbb28be874172adb9305b565df869b5fa71169629d,a8b91ba6e4080b3cfbb9f615c8d16f79fc68a2d8602107cb60f4f72bd0f89a92,case0:ok;case1:info[v=0]&ok;case2:ok;case3:ok;case4:ok;case5:info[v=0]&ok;case6:ok;case7:ok
f28fba64af766845eb2f4302456e2b9f8d80affe57e7aae42738d7cddb1c2ce6,f28fba64af766845eb2f4302456e2b9f8d80affe57e7aae42738d7cddb1c2ce6,4f867ad8bb3d840409d26b67307e62100153273f72fa4b7484becfa14ebe7408,5bbc4f59e452cc5f22a99144b10ce8989a89a995ec3cea1c91ae10e8f721bb5d,,,b079852744c27bfbf62d9498cf819deffeacd8c08d05b48b7b41305db1418827,a443b0a61bad33a0dd566ebb4ef317676576566a13c315e36e51ef1608de40d2,,,case0:ok;case1:ok;case2:bad[s=0];case3:bad[s=0];case4:ok;case5:ok;case6:bad[s=0];case7:bad[s=0]
f455605bc85bf48e3a908c31023faf98381504c6c6d3aeb9ede55f8dd528924d,d31fbcd5cdb798f6c00db6692f8fe8967fa9c79dd10958f4a194f01374905e99,,,0c00c5715b56fe632d814ad8a77f8e66628ea47a6116834f8c1218f3a03cbd50,df88e44fac84fa52df4d59f48819f18f6a8cd4151d162afaf773166f57c7ff46,,,f3ff3a8ea4a9019cd27eb527588071999d715b859ee97cb073ede70b5fc33edf,20771bb0537b05ad20b2a60b77e60e7095732beae2e9d505088ce98fa837fce9,case0:bad[non_square(s)];case1:bad[non_square(s)];case2:info[v=0]&ok;case3:ok;case4:bad[non_square(s)];case5:bad[non_square(s)];case6:info[v=0]&ok;case7:ok
f58cd4d9830bad322699035e8246007d4be27e19b6f53621317b4f309b3daa9d,78ec2b3dc0948de560148bbc7c6dc9633ad5df70a5a5750cbed721804f082a3b,6c4c580b76c7594043569f9dae16dc2801c16a1fbe12860881b75f8ef929bce5,94231355e7385c5f25ca436aa64191471aea4393d6e86ab7a35fe2afacaefd0d,dff2a1951ada6db574df834048149da3397a75b829abf58c7e69db1b41ac0989,a52b66d3c907035548028bf804711bf422aba95f1a666fc86f4648e05f29caae,93b3a7f48938a6bfbca9606251e923d7fe3e95e041ed79f77e48a07006d63f4a,6bdcecaa18c7a3a0da35bc9559be6eb8e515bc6c291795485ca01d4f5350ff22,200d5e6ae525924a8b207cbfb7eb625cc6858a47d6540a73819624e3be53f2a6,5ad4992c36f8fcaab7fd7407fb8ee40bdd5456a0e599903790b9b71ea0d63181,case0:ok;case1:ok;case2:info[v=0]&ok;case3:ok;case4:ok;case5:ok;case6:info[v=0]&ok;case7:ok
fd7d912a40f182a3588800d69ebfb5048766da206fd7ebc8d2436c81cbef6421,8d37c862054debe731694536ff46b273ec122b35a9bf1445ac3c4ff9f262c952,,,,,,,,,case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:info[v=0]&bad[non_square(s)];case3:bad[non_square(s)];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:info[v=0]&bad[non_square(s)];case7:bad[non_square(s)]
1 u x case0_t case1_t case2_t case3_t case4_t case5_t case6_t case7_t comment
2 05ff6bdad900fc3261bc7fe34e2fb0f569f06e091ae437d3a52e9da0cbfb9590 80cdf63774ec7022c89a5a8558e373a279170285e0ab27412dbce510bdfe23fc 45654798ece071ba79286d04f7f3eb1c3f1d17dd883610f2ad2efd82a287466b 0aeaa886f6b76c7158452418cbf5033adc5747e9e9b5d3b2303db96936528557 ba9ab867131f8e4586d792fb080c14e3c0e2e82277c9ef0d52d1027c5d78b5c4 f51557790948938ea7badbe7340afcc523a8b816164a2c4dcfc24695c9ad76d8 case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:info[v=0]&ok;case3:ok;case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:info[v=0]&ok;case7:ok
3 1737a85f4c8d146cec96e3ffdca76d9903dcf3bd53061868d478c78c63c2aa9e 39e48dd150d2f429be088dfd5b61882e7e8407483702ae9a5ab35927b15f85ea 1be8cc0b04be0c681d0c6a68f733f82c6c896e0c8a262fcd392918e303a7abf4 605b5814bf9b8cb066667c9e5480d22dc5b6c92f14b4af3ee0a9eb83b03685e3 e41733f4fb41f397e2f3959708cc07d3937691f375d9d032c6d6e71bfc58503b 9fa4a7eb4064734f99998361ab7f2dd23a4936d0eb4b50c11f56147b4fc9764c case0:ok;case1:ok;case2:info[v=0]&bad[non_square(s)];case3:bad[non_square(s)];case4:ok;case5:ok;case6:info[v=0]&bad[non_square(s)];case7:bad[non_square(s)]
4 1aaa1ccebf9c724191033df366b36f691c4d902c228033ff4516d122b2564f68 c75541259d3ba98f207eaa30c69634d187d0b6da594e719e420f4898638fc5b0 case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:bad[non_square(q)];case3:bad[non_square(q)];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:bad[non_square(q)];case7:bad[non_square(q)]
5 2323a1d079b0fd72fc8bb62ec34230a815cb0596c2bfac998bd6b84260f5dc26 239342dfb675500a34a196310b8d87d54f49dcac9da50c1743ceab41a7b249ff f63580b8aa49c4846de56e39e1b3e73f171e881eba8c66f614e67e5c975dfc07 b6307b332e699f1cf77841d90af25365404deb7fed5edb3090db49e642a156b6 09ca7f4755b63b7b921a91c61e4c18c0e8e177e145739909eb1981a268a20028 49cf84ccd19660e30887be26f50dac9abfb2148012a124cf6f24b618bd5ea579 case0:ok;case1:ok;case2:bad[non_square(q)];case3:bad[non_square(q)];case4:ok;case5:ok;case6:bad[non_square(q)];case7:bad[non_square(q)]
6 2dc90e640cb646ae9164c0b5a9ef0169febe34dc4437d6e46acb0e27e219d1e8 d236f19bf349b9516e9b3f4a5610fe960141cb23bbc8291b9534f1d71de62a47 e69df7d9c026c36600ebdf588072675847c0c431c8eb730682533e964b6252c9 4f18bbdf7c2d6c5f818c18802fa35cd069eaa79fff74e4fc837c80d93fece2f8 196208263fd93c99ff1420a77f8d98a7b83f3bce37148cf97dacc168b49da966 b0e7442083d293a07e73e77fd05ca32f96155860008b1b037c837f25c0131937 case0:ok;case1:info[v=0]&ok;case2:bad[non_square(q)];case3:bad[non_square(q)];case4:ok;case5:info[v=0]&ok;case6:bad[non_square(q)];case7:bad[non_square(q)]
7 3edd7b3980e2f2f34d1409a207069f881fda5f96f08027ac4465b63dc278d672 053a98de4a27b1961155822b3a3121f03b2a14458bd80eb4a560c4c7a85c149c b3dae4b7dcf858e4c6968057cef2b156465431526538199cf52dc1b2d62fda30 4aa77dd55d6b6d3cfa10cc9d0fe42f79232e4575661049ae36779c1d0c666d88 4c251b482307a71b39697fa8310d4ea9b9abcead9ac7e6630ad23e4c29d021ff b558822aa29492c305ef3362f01bd086dcd1ba8a99efb651c98863e1f3998ea7 case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:ok;case3:ok;case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:ok;case7:ok
8 4295737efcb1da6fb1d96b9ca7dcd1e320024b37a736c4948b62598173069f70 fa7ffe4f25f88362831c087afe2e8a9b0713e2cac1ddca6a383205a266f14307 case0:bad[non_square(s)];case1:bad[non_square(s)];case2:bad[non_square(s)];case3:bad[non_square(s)];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:bad[non_square(s)];case7:bad[non_square(s)]
9 587c1a0cee91939e7f784d23b963004a3bf44f5d4e32a0081995ba20b0fca59e 2ea988530715e8d10363907ff25124524d471ba2454d5ce3be3f04194dfd3a3c cfd5a094aa0b9b8891b76c6ab9438f66aa1c095a65f9f70135e8171292245e74 a89057d7c6563f0d6efa19ae84412b8a7b47e791a191ecdfdf2af84fd97bc339 475d0ae9ef46920df07b34117be5a0817de1023e3cc32689e9be145b406b0aef a0759178ad80232454f827ef05ea3e72ad8d75418e6d4cc1cd4f5306c5e7c453 302a5f6b55f464776e48939546bc709955e3f6a59a0608feca17e8ec6ddb9dbb 576fa82839a9c0f29105e6517bbed47584b8186e5e6e132020d507af268438f6 b8a2f51610b96df20f84cbee841a5f7e821efdc1c33cd9761641eba3bf94f140 5f8a6e87527fdcdbab07d810fa15c18d52728abe7192b33e32b0acf83a1837dc case0:ok;case1:ok;case2:ok;case3:ok;case4:ok;case5:ok;case6:ok;case7:ok
10 5fa88b3365a635cbbcee003cce9ef51dd1a310de277e441abccdb7be1e4ba249 79461ff62bfcbcac4249ba84dd040f2cec3c63f725204dc7f464c16bf0ff3170 6bb700e1f4d7e236e8d193ff4a76c1b3bcd4e2b25acac3d51c8dac653fe909a0 f4c73410633da7f63a4f1d55aec6dd32c4c6d89ee74075edb5515ed90da9e683 9448ff1e0b281dc9172e6c00b5893e4c432b1d4da5353c2ae3725399c016f28f 0b38cbef9cc25809c5b0e2aa513922cd3b39276118bf8a124aaea125f25615ac case0:bad[non_square(s)];case1:bad[non_square(s)];case2:ok;case3:info[v=0]&ok;case4:bad[non_square(s)];case5:bad[non_square(s)];case6:ok;case7:info[v=0]&ok
11 6fb31c7531f03130b42b155b952779efbb46087dd9807d241a48eac63c3d96d6 56f81be753e8d4ae4940ea6f46f6ec9fda66a6f96cc95f506cb2b57490e94260 59059774795bdb7a837fbe1140a5fa59984f48af8df95d57dd6d1c05437dcec1 22a644db79376ad4e7b3a009e58b3f13137c54fdf911122cc93667c47077d784 a6fa688b86a424857c8041eebf5a05a667b0b7507206a2a82292e3f9bc822d6e dd59bb2486c8952b184c5ff61a74c0ecec83ab0206eeedd336c9983a8f8824ab case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:ok;case3:info[v=0]&ok;case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:ok;case7:info[v=0]&ok
12 704cd226e71cb6826a590e80dac90f2d2f5830f0fdf135a3eae3965bff25ff12 138e0afa68936ee670bd2b8db53aedbb7bea2a8597388b24d0518edd22ad66ec case0:bad[non_square(s)];case1:bad[non_square(s)];case2:bad[non_square(q)];case3:bad[non_square(q)];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:bad[non_square(q)];case7:bad[non_square(q)]
13 725e914792cb8c8949e7e1168b7cdd8a8094c91c6ec2202ccd53a6a18771edeb 8da16eb86d347376b6181ee9748322757f6b36e3913ddfd332ac595d788e0e44 dd357786b9f6873330391aa5625809654e43116e82a5a5d82ffd1d6624101fc4 a0b7efca01814594c59c9aae8e49700186ca5d95e88bcc80399044d9c2d8613d 22ca8879460978cccfc6e55a9da7f69ab1bcee917d5a5a27d002e298dbefdc6b 5f481035fe7eba6b3a63655171b68ffe7935a26a1774337fc66fbb253d279af2 case0:ok;case1:info[v=0]&ok;case2:bad[non_square(s)];case3:bad[non_square(s)];case4:ok;case5:info[v=0]&ok;case6:bad[non_square(s)];case7:bad[non_square(s)]
14 78fe6b717f2ea4a32708d79c151bf503a5312a18c0963437e865cc6ed3f6ae97 8701948e80d15b5cd8f72863eae40afc5aced5e73f69cbc8179a33902c094d98 case0:bad[non_square(s)];case1:info[v=0]&bad[non_square(s)];case2:bad[non_square(q)];case3:bad[non_square(q)];case4:bad[non_square(s)];case5:info[v=0]&bad[non_square(s)];case6:bad[non_square(q)];case7:bad[non_square(q)]
15 7c37bb9c5061dc07413f11acd5a34006e64c5c457fdb9a438f217255a961f50d 5c1a76b44568eb59d6789a7442d9ed7cdc6226b7752b4ff8eaf8e1a95736e507 b94d30cd7dbff60b64620c17ca0fafaa40b3d1f52d077a60a2e0cafd145086c2 46b2cf32824009f49b9df3e835f05055bf4c2e0ad2f8859f5d1f3501ebaf756d case0:bad[non_square(s)];case1:bad[non_square(s)];case2:info[q=0]&info[X=0]&ok;case3:info[q=0]&bad[r=0];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:info[q=0]&info[X=0]&ok;case7:info[q=0]&bad[r=0]
16 82388888967f82a6b444438a7d44838e13c0d478b9ca060da95a41fb94303de6 29e9654170628fec8b4972898b113cf98807f4609274f4f3140d0674157c90a0 case0:bad[non_square(s)];case1:bad[non_square(s)];case2:bad[non_square(s)];case3:info[v=0]&bad[non_square(s)];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:bad[non_square(s)];case7:info[v=0]&bad[non_square(s)]
17 91298f5770af7a27f0a47188d24c3b7bf98ab2990d84b0b898507e3c561d6472 144f4ccbd9a74698a88cbf6fd00ad886d339d29ea19448f2c572cac0a07d5562 e6a0ffa3807f09dadbe71e0f4be4725f2832e76cad8dc1d943ce839375eff248 837b8e68d4917544764ad0903cb11f8615d2823cefbb06d89049dbabc69befda 195f005c7f80f6252418e1f0b41b8da0d7cd189352723e26bc317c6b8a1009e7 7c8471972b6e8abb89b52f6fc34ee079ea2d7dc31044f9276fb6245339640c55 case0:ok;case1:ok;case2:bad[non_square(s)];case3:info[v=0]&bad[non_square(s)];case4:ok;case5:ok;case6:bad[non_square(s)];case7:info[v=0]&bad[non_square(s)]
18 b682f3d03bbb5dee4f54b5ebfba931b4f52f6a191e5c2f483c73c66e9ace97e1 904717bf0bc0cb7873fcdc38aa97f19e3a62630972acff92b24cc6dda197cb96 case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:bad[non_square(s)];case3:bad[non_square(s)];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:bad[non_square(s)];case7:bad[non_square(s)]
19 c17ec69e665f0fb0dbab48d9c2f94d12ec8a9d7eacb58084833091801eb0b80b 147756e66d96e31c426d3cc85ed0c4cfbef6341dd8b285585aa574ea0204b55e 6f4aea431a0043bdd03134d6d9159119ce034b88c32e50e8e36c4ee45eac7ae9 fd5be16d4ffa2690126c67c3ef7cb9d29b74d397c78b06b3605fda34dc9696a6 5e9c60792a2f000e45c6250f296f875e174efc0e9703e628706103a9dd2d82c7 90b515bce5ffbc422fcecb2926ea6ee631fcb4773cd1af171c93b11aa1538146 02a41e92b005d96fed93983c1083462d648b2c683874f94c9fa025ca23696589 a1639f86d5d0fff1ba39daf0d69078a1e8b103f168fc19d78f9efc5522d27968 case0:ok;case1:ok;case2:info[q=0]&info[X=0]&ok;case3:info[q=0]&bad[r=0];case4:ok;case5:ok;case6:info[q=0]&info[X=0]&ok;case7:info[q=0]&bad[r=0]
20 c25172fc3f29b6fc4a1155b8575233155486b27464b74b8b260b499a3f53cb14 1ea9cbdb35cf6e0329aa31b0bb0a702a65123ed008655a93b7dcd5280e52e1ab 7422edc7843136af0053bb8854448a8299994f9ddcefd3a9a92d45462c59298a 78c7774a266f8b97ea23d05d064f033c77319f923f6b78bce4e20bf05fa5398d 8bdd12387bcec950ffac4477abbb757d6666b06223102c5656d2bab8d3a6d2a5 873888b5d990746815dc2fa2f9b0fcc388ce606dc09487431b1df40ea05ac2a2 case0:bad[non_square(s)];case1:bad[non_square(s)];case2:ok;case3:ok;case4:bad[non_square(s)];case5:bad[non_square(s)];case6:ok;case7:ok
21 cab6626f832a4b1280ba7add2fc5322ff011caededf7ff4db6735d5026dc0367 2b2bef0852c6f7c95d72ac99a23802b875029cd573b248d1f1b3fc8033788eb6 case0:bad[non_square(s)];case1:bad[non_square(s)];case2:info[v=0]&bad[non_square(s)];case3:bad[non_square(s)];case4:bad[non_square(s)];case5:bad[non_square(s)];case6:info[v=0]&bad[non_square(s)];case7:bad[non_square(s)]
22 d8621b4ffc85b9ed56e99d8dd1dd24aedcecb14763b861a17112dc771a104fd2 812cabe972a22aa67c7da0c94d8a936296eb9949d70c37cb2b2487574cb3ce58 fbc5febc6fdbc9ae3eb88a93b982196e8b6275a6d5a73c17387e000c711bd0e3 8724c96bd4e5527f2dd195a51c468d2d211ba2fac7cbe0b4b3434253409fb42d 043a014390243651c147756c467de691749d8a592a58c3e8c781fff28ee42b4c 78db36942b1aad80d22e6a5ae3b972d2dee45d0538341f4b4cbcbdabbf604802 case0:ok;case1:ok;case2:bad[non_square(s)];case3:bad[non_square(s)];case4:ok;case5:ok;case6:bad[non_square(s)];case7:bad[non_square(s)]
23 da463164c6f4bf7129ee5f0ec00f65a675a8adf1bd931b39b64806afdcda9a22 25b9ce9b390b408ed611a0f13ff09a598a57520e426ce4c649b7f94f2325620d case0:bad[non_square(s)];case1:info[v=0]&bad[non_square(s)];case2:bad[non_square(s)];case3:bad[non_square(s)];case4:bad[non_square(s)];case5:info[v=0]&bad[non_square(s)];case6:bad[non_square(s)];case7:bad[non_square(s)]
24 dafc971e4a3a7b6dcfb42a08d9692d82ad9e7838523fcbda1d4827e14481ae2d 250368e1b5c58492304bd5f72696d27d526187c7adc03425e2b7d81dbb7e4e02 370c28f1be665efacde6aa436bf86fe21e6e314c1e53dd040e6c73a46b4c8c49 cd8acee98ffe56531a84d7eb3e48fa4034206ce825ace907d0edf0eaeb5e9ca2 c8f3d70e4199a105321955bc9407901de191ceb3e1ac22fbf1938c5a94b36fe6 327531167001a9ace57b2814c1b705bfcbdf9317da5316f82f120f1414a15f8d case0:bad[non_square(s)];case1:info[v=0]&bad[non_square(s)];case2:ok;case3:ok;case4:bad[non_square(s)];case5:info[v=0]&bad[non_square(s)];case6:ok;case7:ok
25 e0294c8bc1a36b4166ee92bfa70a5c34976fa9829405efea8f9cd54dcb29b99e ae9690d13b8d20a0fbbf37bed8474f67a04e142f56efd78770a76b359165d8a1 dcd45d935613916af167b029058ba3a700d37150b9df34728cb05412c16d4182 232ba26ca9ec6e950e984fd6fa745c58ff2c8eaf4620cb8d734fabec3e92baad case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:info[q=0]&info[X=0]&ok;case3:info[q=0]&bad[r=0];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:info[q=0]&info[X=0]&ok;case7:info[q=0]&bad[r=0]
26 e148441cd7b92b8b0e4fa3bd68712cfd0d709ad198cace611493c10e97f5394e 164a639794d74c53afc4d3294e79cdb3cd25f99f6df45c000f758aba54d699c0 case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:bad[non_square(s)];case3:info[v=0]&bad[non_square(s)];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:bad[non_square(s)];case7:info[v=0]&bad[non_square(s)]
27 e4b00ec97aadcca97644d3b0c8a931b14ce7bcf7bc8779546d6e35aa5937381c 94e9588d41647b3fcc772dc8d83c67ce3be003538517c834103d2cd49d62ef4d c88d25f41407376bb2c03a7fffeb3ec7811cc43491a0c3aac0378cdc78357bee 51c02636ce00c2345ecd89adb6089fe4d5e18ac924e3145e6669501cd37a00d4 205b3512db40521cb200952e67b46f67e09e7839e0de44004138329ebd9138c5 58aab390ab6fb55c1d1b80897a207ce94a78fa5b4aa61a33398bcae9adb20d3e 3772da0bebf8c8944d3fc5800014c1387ee33bcb6e5f3c553fc8732287ca8041 ae3fd9c931ff3dcba132765249f7601b2a1e7536db1ceba19996afe22c85fb5b dfa4caed24bfade34dff6ad1984b90981f6187c61f21bbffbec7cd60426ec36a a7554c6f54904aa3e2e47f7685df8316b58705a4b559e5ccc6743515524deef1 case0:ok;case1:ok;case2:ok;case3:info[v=0]&ok;case4:ok;case5:ok;case6:ok;case7:info[v=0]&ok
28 e5bbb9ef360d0a501618f0067d36dceb75f5be9a620232aa9fd5139d0863fde5 e5bbb9ef360d0a501618f0067d36dceb75f5be9a620232aa9fd5139d0863fde5 case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:bad[s=0];case3:bad[s=0];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:bad[s=0];case7:bad[s=0]
29 e6bcb5c3d63467d490bfa54fbbc6092a7248c25e11b248dc2964a6e15edb1457 19434a3c29cb982b6f405ab04439f6d58db73da1ee4db723d69b591da124e7d8 67119877832ab8f459a821656d8261f544a553b89ae4f25c52a97134b70f3426 ffee02f5e649c07f0560eff1867ec7b32d0e595e9b1c0ea6e2a4fc70c97cd71f b5e0c189eb5b4bacd025b7444d74178be8d5246cfa4a9a207964a057ee969992 5746e4591bf7f4c3044609ea372e908603975d279fdef8349f0b08d32f07619d 98ee67887cd5470ba657de9a927d9e0abb5aac47651b0da3ad568eca48f0c809 0011fd0a19b63f80fa9f100e7981384cd2f1a6a164e3f1591d5b038e36832510 4a1f3e7614a4b4532fda48bbb28be874172adb9305b565df869b5fa71169629d a8b91ba6e4080b3cfbb9f615c8d16f79fc68a2d8602107cb60f4f72bd0f89a92 case0:ok;case1:info[v=0]&ok;case2:ok;case3:ok;case4:ok;case5:info[v=0]&ok;case6:ok;case7:ok
30 f28fba64af766845eb2f4302456e2b9f8d80affe57e7aae42738d7cddb1c2ce6 f28fba64af766845eb2f4302456e2b9f8d80affe57e7aae42738d7cddb1c2ce6 4f867ad8bb3d840409d26b67307e62100153273f72fa4b7484becfa14ebe7408 5bbc4f59e452cc5f22a99144b10ce8989a89a995ec3cea1c91ae10e8f721bb5d b079852744c27bfbf62d9498cf819deffeacd8c08d05b48b7b41305db1418827 a443b0a61bad33a0dd566ebb4ef317676576566a13c315e36e51ef1608de40d2 case0:ok;case1:ok;case2:bad[s=0];case3:bad[s=0];case4:ok;case5:ok;case6:bad[s=0];case7:bad[s=0]
31 f455605bc85bf48e3a908c31023faf98381504c6c6d3aeb9ede55f8dd528924d d31fbcd5cdb798f6c00db6692f8fe8967fa9c79dd10958f4a194f01374905e99 0c00c5715b56fe632d814ad8a77f8e66628ea47a6116834f8c1218f3a03cbd50 df88e44fac84fa52df4d59f48819f18f6a8cd4151d162afaf773166f57c7ff46 f3ff3a8ea4a9019cd27eb527588071999d715b859ee97cb073ede70b5fc33edf 20771bb0537b05ad20b2a60b77e60e7095732beae2e9d505088ce98fa837fce9 case0:bad[non_square(s)];case1:bad[non_square(s)];case2:info[v=0]&ok;case3:ok;case4:bad[non_square(s)];case5:bad[non_square(s)];case6:info[v=0]&ok;case7:ok
32 f58cd4d9830bad322699035e8246007d4be27e19b6f53621317b4f309b3daa9d 78ec2b3dc0948de560148bbc7c6dc9633ad5df70a5a5750cbed721804f082a3b 6c4c580b76c7594043569f9dae16dc2801c16a1fbe12860881b75f8ef929bce5 94231355e7385c5f25ca436aa64191471aea4393d6e86ab7a35fe2afacaefd0d dff2a1951ada6db574df834048149da3397a75b829abf58c7e69db1b41ac0989 a52b66d3c907035548028bf804711bf422aba95f1a666fc86f4648e05f29caae 93b3a7f48938a6bfbca9606251e923d7fe3e95e041ed79f77e48a07006d63f4a 6bdcecaa18c7a3a0da35bc9559be6eb8e515bc6c291795485ca01d4f5350ff22 200d5e6ae525924a8b207cbfb7eb625cc6858a47d6540a73819624e3be53f2a6 5ad4992c36f8fcaab7fd7407fb8ee40bdd5456a0e599903790b9b71ea0d63181 case0:ok;case1:ok;case2:info[v=0]&ok;case3:ok;case4:ok;case5:ok;case6:info[v=0]&ok;case7:ok
33 fd7d912a40f182a3588800d69ebfb5048766da206fd7ebc8d2436c81cbef6421 8d37c862054debe731694536ff46b273ec122b35a9bf1445ac3c4ff9f262c952 case0:bad[valid_x(-x-u)];case1:bad[valid_x(-x-u)];case2:info[v=0]&bad[non_square(s)];case3:bad[non_square(s)];case4:bad[valid_x(-x-u)];case5:bad[valid_x(-x-u)];case6:info[v=0]&bad[non_square(s)];case7:bad[non_square(s)]

View File

@@ -0,0 +1,33 @@
u,x,case0_t,case1_t,case2_t,case3_t,case4_t,case5_t,case6_t,case7_t
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33 f566cc6fccc657365c0197accf3a7d6f80f85209ff666ff774f4dcbc524aa842 0a9933903339a8c9a3fe685330c582907f07adf6009990088b0b2342adb553ed 3ab8dc4ecbc0441c685436ac0d76f16393769c353be6092bd6ec4ce094106bd8 3bd189b4ef3d1baa5610f2b14cb4a2b377eb171511e6f36ef6a05a2c7c52e368 1594764c6296402aadd123675d81f3505d35f2a52c52881568eadb7b675b53f0 c64fbf71138e66de8ce0abdf3b6f51d151ca8e1037ab5b979e62b2faa15be81c c54723b1343fbbe397abc953f2890e9c6c8963cac419f6d42913b31e6bef9057 c42e764b10c2e455a9ef0d4eb34b5d4c8814e8eaee190c91095fa5d283ad18c7 ea6b89b39d69bfd5522edc98a27e0cafa2ca0d5ad3ad77ea9715248398a4a83f 39b0408eec719921731f5420c490ae2eae3571efc854a468619d4d045ea41413

View File

@@ -1,15 +1,9 @@
{
"compilerOptions": {
"outDir": "esm",
"target": "es2020",
"module": "es6",
"moduleResolution": "node16",
"baseUrl": ".",
"paths": {
"@noble/hashes/crypto": ["src/crypto"]
},
"sourceMap": true,
"lib": ["es2020"],
"strict": true,
"sourceMap": true,
"allowSyntheticDefaultImports": false,
"allowUnreachableCode": false,
"esModuleInterop": false,
@@ -18,6 +12,13 @@
"noUncheckedIndexedAccess": false,
"noUnusedLocals": true,
"noUnusedParameters": true,
"baseUrl": ".",
"paths": {
"@noble/hashes/crypto": ["src/crypto"]
},
"module": "es2020",
"outDir": "esm",
"moduleResolution": "bundler"
},
"include": ["src"],
"exclude": ["node_modules", "lib"]

View File

@@ -1,24 +1,10 @@
{
"extends": "./tsconfig.esm.json",
"compilerOptions": {
"outDir": ".",
"target": "es2020",
"lib": ["es2020"], // Set explicitly to remove DOM
"module": "commonjs",
"moduleResolution": "node",
"baseUrl": ".",
"sourceMap": true,
"moduleResolution": "node10",
"outDir": ".",
"declaration": true,
"declarationMap": true,
"strict": true,
"allowSyntheticDefaultImports": false,
"allowUnreachableCode": false,
"esModuleInterop": false,
"noFallthroughCasesInSwitch": true,
"noImplicitReturns": true,
"noUncheckedIndexedAccess": false,
"noUnusedLocals": true,
"noUnusedParameters": true
},
"include": ["src"],
"exclude": ["node_modules", "*.d.ts"]
"declarationMap": true
}
}