Release 0.5.2.

README.md

@@ -201,8 +201,6 @@ export type CurveFn = {
   ExtendedPoint: ExtendedPointConstructor;
   Signature: SignatureConstructor;
   utils: {
-    mod: (a: bigint, b?: bigint) => bigint;
-    invert: (number: bigint, modulo?: bigint) => bigint;
     randomPrivateKey: () => Uint8Array;
     getExtendedPublicKey: (key: PrivKey) => {
       head: Uint8Array;
@@ -306,6 +304,7 @@ export type CurveFn = {
   getPublicKey: (privateKey: PrivKey, isCompressed?: boolean) => Uint8Array;
   getSharedSecret: (privateA: PrivKey, publicB: PubKey, isCompressed?: boolean) => Uint8Array;
   sign: (msgHash: Hex, privKey: PrivKey, opts?: SignOpts) => SignatureType;
+  signUnhashed: (msg: Uint8Array, privKey: PrivKey, opts?: SignOpts) => SignatureType;
   verify: (
     signature: Hex | SignatureType,
     msgHash: Hex,
@@ -316,8 +315,6 @@ export type CurveFn = {
   ProjectivePoint: ProjectivePointConstructor;
   Signature: SignatureConstructor;
   utils: {
-    mod: (a: bigint) => bigint;
-    invert: (number: bigint) => bigint;
     isValidPrivateKey(privateKey: PrivKey): boolean;
     hashToPrivateKey: (hash: Hex) => Uint8Array;
     randomPrivateKey: () => Uint8Array;

package.json

@@ -1,6 +1,6 @@
 {
   "name": "@noble/curves",
-  "version": "0.5.1",
+  "version": "0.5.2",
   "description": "Minimal, auditable JS implementation of elliptic curve cryptography",
   "files": [
     "lib"
@@ -31,7 +31,7 @@
     "@types/node": "18.11.3",
     "fast-check": "3.0.0",
     "micro-bmark": "0.2.0",
-    "micro-should": "0.2.0",
+    "micro-should": "0.3.0",
     "prettier": "2.6.2",
     "rollup": "2.75.5",
     "typescript": "4.7.3"

src/abstract/bls.ts

@@ -1,13 +1,26 @@
 /*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
-// Barreto-Lynn-Scott Curves. A family of pairing friendly curves, with embedding degree = 12 or 24
+/**
-// NOTE: only 12 supported for now
+ * BLS (Barreto-Lynn-Scott) family of pairing-friendly curves.
-// Constructed from pair of weierstrass curves, based pairing logic
+ * Implements BLS (Boneh-Lynn-Shacham) signatures.
+ * Consists of two curves: G1 and G2:
+ * - G1 is a subgroup of (x, y) E(Fq) over y² = x³ + 4.
+ * - G2 is a subgroup of ((x₁, x₂+i), (y₁, y₂+i)) E(Fq²) over y² = x³ + 4(1 + i) where i is √-1
+ * - Gt, created by bilinear (ate) pairing e(G1, G2), consists of p-th roots of unity in
+ *   Fq^k where k is embedding degree. Only degree 12 is currently supported, 24 is not.
+ * Pairing is used to aggregate and verify signatures.
+ * We are using Fp for private keys (shorter) and Fp₂ for signatures (longer).
+ * Some projects may prefer to swap this relation, it is not supported for now.
+ */
 import * as mod from './modular.js';
-import { ensureBytes, numberToBytesBE, bytesToNumberBE, bitLen, bitGet } from './utils.js';
+import * as ut from './utils.js';
-import * as utils from './utils.js';
+// Types require separate import
-// Types
+import { Hex, PrivKey } from './utils.js';
-import { hexToBytes, bytesToHex, Hex, PrivKey } from './utils.js';
+import {
-import { htfOpts, stringToBytes, hash_to_field, expand_message_xmd } from './hash-to-curve.js';
+  htfOpts,
+  stringToBytes,
+  hash_to_field as hashToField,
+  expand_message_xmd as expandMessageXMD,
+} from './hash-to-curve.js';
 import { CurvePointsType, PointType, CurvePointsRes, weierstrassPoints } from './weierstrass.js';
 
 type Fp = bigint; // Can be different field?
@@ -39,7 +52,7 @@ export type CurveType<Fp, Fp2, Fp6, Fp12> = {
     finalExponentiate(num: Fp12): Fp12;
   };
   htfDefaults: htfOpts;
-  hash: utils.CHash; // Because we need outputLen for DRBG
+  hash: ut.CHash; // Because we need outputLen for DRBG
   randomBytes: (bytesLength?: number) => Uint8Array;
 };
 
@@ -80,12 +93,9 @@ export type CurveFn<Fp, Fp2, Fp6, Fp12> = {
     publicKeys: (Hex | PointType<Fp>)[]
   ) => boolean;
   utils: {
-    bytesToHex: typeof utils.bytesToHex;
-    hexToBytes: typeof utils.hexToBytes;
     stringToBytes: typeof stringToBytes;
-    hashToField: typeof hash_to_field;
+    hashToField: typeof hashToField;
-    expandMessageXMD: typeof expand_message_xmd;
+    expandMessageXMD: typeof expandMessageXMD;
-    mod: typeof mod.mod;
     getDSTLabel: () => string;
     setDSTLabel(newLabel: string): void;
   };
@@ -95,12 +105,9 @@ export function bls<Fp2, Fp6, Fp12>(
   CURVE: CurveType<Fp, Fp2, Fp6, Fp12>
 ): CurveFn<Fp, Fp2, Fp6, Fp12> {
   // Fields looks pretty specific for curve, so for now we need to pass them with options
-  const Fp = CURVE.Fp;
+  const { Fp, Fr, Fp2, Fp6, Fp12 } = CURVE;
-  const Fr = CURVE.Fr;
+  const BLS_X_LEN = ut.bitLen(CURVE.x);
-  const Fp2 = CURVE.Fp2;
+  const groupLen = 32; // TODO: calculate; hardcoded for now
-  const Fp6 = CURVE.Fp6;
-  const Fp12 = CURVE.Fp12;
-  const BLS_X_LEN = bitLen(CURVE.x);
 
   // Pre-compute coefficients for sparse multiplication
   // Point addition and point double calculations is reused for coefficients
@@ -125,7 +132,7 @@ export function bls<Fp2, Fp6, Fp12>(
       Rx = Fp2.div(Fp2.mul(Fp2.mul(Fp2.sub(t0, t3), Rx), Ry), 2n); // ((T0 - T3) * Rx * Ry) / 2
       Ry = Fp2.sub(Fp2.square(Fp2.div(Fp2.add(t0, t3), 2n)), Fp2.mul(Fp2.square(t2), 3n)); // ((T0 + T3) / 2)² - 3 * T2²
       Rz = Fp2.mul(t0, t4); // T0 * T4
-      if (bitGet(CURVE.x, i)) {
+      if (ut.bitGet(CURVE.x, i)) {
         // Addition
         let t0 = Fp2.sub(Ry, Fp2.mul(Qy, Rz)); // Ry - Qy * Rz
         let t1 = Fp2.sub(Rx, Fp2.mul(Qx, Rz)); // Rx - Qx * Rz
@@ -147,13 +154,14 @@ export function bls<Fp2, Fp6, Fp12>(
   }
 
   function millerLoop(ell: [Fp2, Fp2, Fp2][], g1: [Fp, Fp]): Fp12 {
+    const { x } = CURVE;
     const Px = g1[0];
     const Py = g1[1];
     let f12 = Fp12.ONE;
     for (let j = 0, i = BLS_X_LEN - 2; i >= 0; i--, j++) {
       const E = ell[j];
       f12 = Fp12.multiplyBy014(f12, E[0], Fp2.mul(E[1], Px), Fp2.mul(E[2], Py));
-      if (bitGet(CURVE.x, i)) {
+      if (ut.bitGet(x, i)) {
         j += 1;
         const F = ell[j];
         f12 = Fp12.multiplyBy014(f12, F[0], Fp2.mul(F[1], Px), Fp2.mul(F[2], Py));
@@ -163,81 +171,31 @@ export function bls<Fp2, Fp6, Fp12>(
     return Fp12.conjugate(f12);
   }
 
-  // bls12-381 is a construction of two curves:
-  // 1. Fp: (x, y)
-  // 2. Fp₂: ((x₁, x₂+i), (y₁, y₂+i)) - (complex numbers)
-  //
-  // Bilinear Pairing (ate pairing) is used to combine both elements into a paired one:
-  //   Fp₁₂ = e(Fp, Fp2)
-  //   where Fp₁₂ = 12-degree polynomial
-  // Pairing is used to verify signatures.
-  //
-  // We are using Fp for private keys (shorter) and Fp2 for signatures (longer).
-  // Some projects may prefer to swap this relation, it is not supported for now.
-
-  const htfDefaults = { ...CURVE.htfDefaults };
-
-  function isWithinCurveOrder(num: bigint): boolean {
-    return 0 < num && num < CURVE.r;
-  }
-
   const utils = {
-    hexToBytes: hexToBytes,
+    hexToBytes: ut.hexToBytes,
-    bytesToHex: bytesToHex,
+    bytesToHex: ut.bytesToHex,
-    mod: mod.mod,
+    stringToBytes: stringToBytes,
-    stringToBytes,
     // TODO: do we need to export it here?
-    hashToField: (msg: Uint8Array, count: number, options: Partial<typeof htfDefaults> = {}) =>
+    hashToField: (
-      hash_to_field(msg, count, { ...CURVE.htfDefaults, ...options }),
+      msg: Uint8Array,
+      count: number,
+      options: Partial<typeof CURVE.htfDefaults> = {}
+    ) => hashToField(msg, count, { ...CURVE.htfDefaults, ...options }),
     expandMessageXMD: (msg: Uint8Array, DST: Uint8Array, lenInBytes: number, H = CURVE.hash) =>
-      expand_message_xmd(msg, DST, lenInBytes, H),
+      expandMessageXMD(msg, DST, lenInBytes, H),
-
+    hashToPrivateKey: (hash: Hex): Uint8Array => Fr.toBytes(ut.hashToPrivateScalar(hash, CURVE.r)),
-    /**
+    randomBytes: (bytesLength: number = groupLen): Uint8Array => CURVE.randomBytes(bytesLength),
-     * Can take 40 or more bytes of uniform input e.g. from CSPRNG or KDF
+    randomPrivateKey: (): Uint8Array => utils.hashToPrivateKey(utils.randomBytes(groupLen + 8)),
-     * and convert them into private key, with the modulo bias being negligible.
+    getDSTLabel: () => CURVE.htfDefaults.DST,
-     * As per FIPS 186 B.1.1.
-     * https://research.kudelskisecurity.com/2020/07/28/the-definitive-guide-to-modulo-bias-and-how-to-avoid-it/
-     * @param hash hash output from sha512, or a similar function
-     * @returns valid private key
-     */
-    hashToPrivateKey: (hash: Hex): Uint8Array => {
-      hash = ensureBytes(hash);
-      if (hash.length < 40 || hash.length > 1024)
-        throw new Error('Expected 40-1024 bytes of private key as per FIPS 186');
-      //     hashToPrivateScalar(hash, CURVE.r)
-      // NOTE: doesn't add +/-1
-      const num = mod.mod(bytesToNumberBE(hash), CURVE.r);
-      // This should never happen
-      if (num === 0n || num === 1n) throw new Error('Invalid private key');
-      return numberToBytesBE(num, 32);
-    },
-
-    randomBytes: (bytesLength: number = 32): Uint8Array => CURVE.randomBytes(bytesLength),
-    // NIST SP 800-56A rev 3, section 5.6.1.2.2
-    // https://research.kudelskisecurity.com/2020/07/28/the-definitive-guide-to-modulo-bias-and-how-to-avoid-it/
-    randomPrivateKey: (): Uint8Array => utils.hashToPrivateKey(utils.randomBytes(40)),
-    getDSTLabel: () => htfDefaults.DST,
     setDSTLabel(newLabel: string) {
       // https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-3.1
       if (typeof newLabel !== 'string' || newLabel.length > 2048 || newLabel.length === 0) {
         throw new TypeError('Invalid DST');
       }
-      htfDefaults.DST = newLabel;
+      CURVE.htfDefaults.DST = newLabel;
     },
   };
 
-  function normalizePrivKey(key: PrivKey): bigint {
-    let int: bigint;
-    if (key instanceof Uint8Array && key.length === 32) int = bytesToNumberBE(key);
-    else if (typeof key === 'string' && key.length === 64) int = BigInt(`0x${key}`);
-    else if (typeof key === 'number' && key > 0 && Number.isSafeInteger(key)) int = BigInt(key);
-    else if (typeof key === 'bigint' && key > 0n) int = key;
-    else throw new TypeError('Expected valid private key');
-    int = mod.mod(int, CURVE.r);
-    if (!isWithinCurveOrder(int)) throw new Error('Private key must be 0 < key < CURVE.r');
-    return int;
-  }
-
   // Point on G1 curve: (x, y)
   const G1 = weierstrassPoints({
     n: Fr.ORDER,
@@ -309,7 +267,7 @@ export function bls<Fp2, Fp6, Fp12>(
   function sign(message: G2Hex, privateKey: PrivKey): Uint8Array | G2 {
     const msgPoint = normP2Hash(message);
     msgPoint.assertValidity();
-    const sigPoint = msgPoint.multiply(normalizePrivKey(privateKey));
+    const sigPoint = msgPoint.multiply(G1.normalizePrivateKey(privateKey));
     if (message instanceof G2.Point) return sigPoint;
     return Signature.encode(sigPoint);
   }

src/abstract/edwards.ts

@@ -2,28 +2,18 @@
 // Twisted Edwards curve. The formula is: ax² + y² = 1 + dx²y²
 
 // Differences from @noble/ed25519 1.7:
-// 1. Different field element lengths in ed448:
+// 1. Variable field element lengths between EDDSA/ECDH:
 //   EDDSA (RFC8032) is 456 bits / 57 bytes, ECDH (RFC7748) is 448 bits / 56 bytes
 // 2. Different addition formula (doubling is same)
 // 3. uvRatio differs between curves (half-expected, not only pow fn changes)
-// 4. Point decompression code is different too (unexpected), now using generalized formula
+// 4. Point decompression code is different (unexpected), now using generalized formula
 // 5. Domain function was no-op for ed25519, but adds some data even with empty context for ed448
 
 import * as mod from './modular.js';
-import {
+import * as ut from './utils.js';
-  bytesToHex,
+import { ensureBytes, Hex, PrivKey } from './utils.js';
-  concatBytes,
-  ensureBytes,
-  numberToBytesLE,
-  bytesToNumberLE,
-  hashToPrivateScalar,
-  BasicCurve,
-  validateOpts as utilOpts,
-  Hex,
-  PrivKey,
-} from './utils.js'; // TODO: import * as u from './utils.js'?
 import { Group, GroupConstructor, wNAF } from './group.js';
-import { hash_to_field, htfOpts, validateHTFOpts } from './hash-to-curve.js';
+import { hash_to_field as hashToField, htfOpts, validateHTFOpts } from './hash-to-curve.js';
 
 // Be friendly to bad ECMAScript parsers by not using bigint literals like 123n
 const _0n = BigInt(0);
@@ -31,49 +21,41 @@ const _1n = BigInt(1);
 const _2n = BigInt(2);
 const _8n = BigInt(8);
 
-export type CHash = {
+// Edwards curves must declare params a & d.
-  (message: Uint8Array | string): Uint8Array;
+export type CurveType = ut.BasicCurve<bigint> & {
-  blockLen: number;
-  outputLen: number;
-  create(): any;
-};
-
-export type CurveType = BasicCurve<bigint> & {
   // Params: a, d
   a: bigint;
   d: bigint;
   // Hashes
-  hash: CHash; // Because we need outputLen for DRBG
+  // The interface, because we need outputLen for DRBG
+  hash: ut.CHash;
+  // CSPRNG
   randomBytes: (bytesLength?: number) => Uint8Array;
+  // Probably clears bits in a byte array to produce a valid field element
   adjustScalarBytes?: (bytes: Uint8Array) => Uint8Array;
+  // Used during hashing
   domain?: (data: Uint8Array, ctx: Uint8Array, phflag: boolean) => Uint8Array;
+  // Ratio √(u/v)
   uvRatio?: (u: bigint, v: bigint) => { isValid: boolean; value: bigint };
-  preHash?: CHash;
+  // RFC 8032 pre-hashing of messages to sign() / verify()
-  clearCofactor?: (c: ExtendedPointConstructor, point: ExtendedPointType) => ExtendedPointType;
+  preHash?: ut.CHash;
-  // Hash to field opts
+  // Hash to field options
   htfDefaults?: htfOpts;
   mapToCurve?: (scalar: bigint[]) => { x: bigint; y: bigint };
 };
 
-// Should be separate from overrides, since overrides can use information about curve (for example nBits)
 function validateOpts(curve: CurveType) {
-  const opts = utilOpts(curve);
+  const opts = ut.validateOpts(curve);
-  if (typeof opts.hash !== 'function' || !Number.isSafeInteger(opts.hash.outputLen))
+  if (typeof opts.hash !== 'function' || !ut.isPositiveInt(opts.hash.outputLen))
     throw new Error('Invalid hash function');
   for (const i of ['a', 'd'] as const) {
-    if (typeof opts[i] !== 'bigint')
+    const val = opts[i];
-      throw new Error(`Invalid curve param ${i}=${opts[i]} (${typeof opts[i]})`);
+    if (typeof val !== 'bigint') throw new Error(`Invalid curve param ${i}=${val} (${typeof val})`);
   }
   for (const fn of ['randomBytes'] as const) {
     if (typeof opts[fn] !== 'function') throw new Error(`Invalid ${fn} function`);
   }
-  for (const fn of [
+  for (const fn of ['adjustScalarBytes', 'domain', 'uvRatio', 'mapToCurve'] as const) {
-    'adjustScalarBytes',
-    'domain',
-    'uvRatio',
-    'mapToCurve',
-    'clearCofactor',
-  ] as const) {
     if (opts[fn] === undefined) continue; // Optional
     if (typeof opts[fn] !== 'function') throw new Error(`Invalid ${fn} function`);
   }
@@ -96,7 +78,7 @@ export type SignatureConstructor = {
   fromHex(hex: Hex): SignatureType;
 };
 
-// Instance
+// Instance of Extended Point with coordinates in X, Y, Z, T
 export interface ExtendedPointType extends Group<ExtendedPointType> {
   readonly x: bigint;
   readonly y: bigint;
@@ -109,7 +91,7 @@ export interface ExtendedPointType extends Group<ExtendedPointType> {
   toAffine(invZ?: bigint): PointType;
   clearCofactor(): ExtendedPointType;
 }
-// Static methods
+// Static methods of Extended Point with coordinates in X, Y, Z, T
 export interface ExtendedPointConstructor extends GroupConstructor<ExtendedPointType> {
   new (x: bigint, y: bigint, z: bigint, t: bigint): ExtendedPointType;
   fromAffine(p: PointType): ExtendedPointType;
@@ -117,7 +99,7 @@ export interface ExtendedPointConstructor extends GroupConstructor<ExtendedPoint
   normalizeZ(points: ExtendedPointType[]): ExtendedPointType[];
 }
 
-// Instance
+// Instance of Affine Point with coordinates in X, Y
 export interface PointType extends Group<PointType> {
   readonly x: bigint;
   readonly y: bigint;
@@ -127,7 +109,7 @@ export interface PointType extends Group<PointType> {
   isTorsionFree(): boolean;
   clearCofactor(): PointType;
 }
-// Static methods
+// Static methods of Affine Point with coordinates in X, Y
 export interface PointConstructor extends GroupConstructor<PointType> {
   new (x: bigint, y: bigint): PointType;
   fromHex(hex: Hex): PointType;
@@ -148,8 +130,6 @@ export type CurveFn = {
   ExtendedPoint: ExtendedPointConstructor;
   Signature: SignatureConstructor;
   utils: {
-    mod: (a: bigint) => bigint;
-    invert: (number: bigint) => bigint;
     randomPrivateKey: () => Uint8Array;
     getExtendedPublicKey: (key: PrivKey) => {
       head: Uint8Array;
@@ -164,36 +144,31 @@ export type CurveFn = {
 // NOTE: it is not generic twisted curve for now, but ed25519/ed448 generic implementation
 export function twistedEdwards(curveDef: CurveType): CurveFn {
   const CURVE = validateOpts(curveDef) as ReturnType<typeof validateOpts>;
-  const Fp = CURVE.Fp as mod.Field<bigint>;
+  const Fp = CURVE.Fp;
   const CURVE_ORDER = CURVE.n;
-  const fieldLen = Fp.BYTES; // 32 (length of one field element)
+  const maxGroupElement = _2n ** BigInt(CURVE.nByteLength * 8);
-  if (fieldLen > 2048) throw new Error('Field lengths over 2048 are not supported');
-  const groupLen = CURVE.nByteLength;
-  // (2n ** 256n).toString(16);
-  const maxGroupElement = _2n ** BigInt(groupLen * 8); // previous POW_2_256
 
   // Function overrides
   const { randomBytes } = CURVE;
   const modP = Fp.create;
 
   // sqrt(u/v)
-  function _uvRatio(u: bigint, v: bigint) {
+  const uvRatio =
+    CURVE.uvRatio ||
+    ((u: bigint, v: bigint) => {
       try {
-      const value = Fp.sqrt(u * Fp.invert(v));
+        return { isValid: true, value: Fp.sqrt(u * Fp.invert(v)) };
-      return { isValid: true, value };
       } catch (e) {
         return { isValid: false, value: _0n };
       }
-  }
+    });
-  const uvRatio = CURVE.uvRatio || _uvRatio;
+  const adjustScalarBytes = CURVE.adjustScalarBytes || ((bytes: Uint8Array) => bytes); // NOOP
-
+  const domain =
-  const _adjustScalarBytes = (bytes: Uint8Array) => bytes; // NOOP
+    CURVE.domain ||
-  const adjustScalarBytes = CURVE.adjustScalarBytes || _adjustScalarBytes;
+    ((data: Uint8Array, ctx: Uint8Array, phflag: boolean) => {
-  function _domain(data: Uint8Array, ctx: Uint8Array, phflag: boolean) {
       if (ctx.length || phflag) throw new Error('Contexts/pre-hash are not supported');
       return data;
-  }
+    }); // NOOP
-  const domain = CURVE.domain || _domain; // NOOP
 
   /**
    * Extended Point works in extended coordinates: (x, y, z, t) ∋ (x=x/z, y=y/z, t=xy).
@@ -336,25 +311,27 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
     // Non-constant-time multiplication. Uses double-and-add algorithm.
     // It's faster, but should only be used when you don't care about
     // an exposed private key e.g. sig verification.
-    // Allows scalar bigger than curve order, but less than 2^256
     multiplyUnsafe(scalar: number | bigint): ExtendedPoint {
       let n = normalizeScalar(scalar, CURVE_ORDER, false);
-      const G = ExtendedPoint.BASE;
       const P0 = ExtendedPoint.ZERO;
       if (n === _0n) return P0;
       if (this.equals(P0) || n === _1n) return this;
-      if (this.equals(G)) return this.wNAF(n);
+      if (this.equals(ExtendedPoint.BASE)) return this.wNAF(n);
       return wnaf.unsafeLadder(this, n);
     }
 
+    // Checks if point is of small order.
+    // If you add something to small order point, you will have "dirty"
+    // point with torsion component.
     // Multiplies point by cofactor and checks if the result is 0.
     isSmallOrder(): boolean {
       return this.multiplyUnsafe(CURVE.h).equals(ExtendedPoint.ZERO);
     }
 
-    // Multiplies point by a very big scalar n and checks if the result is 0.
+    // Multiplies point by curve order (very big scalar CURVE.n) and checks if the result is 0.
+    // Returns `false` is the point is dirty.
     isTorsionFree(): boolean {
-      return this.multiplyUnsafe(CURVE_ORDER).equals(ExtendedPoint.ZERO);
+      return wnaf.unsafeLadder(this, CURVE_ORDER).equals(ExtendedPoint.ZERO);
     }
 
     // Converts Extended point to default (x, y) coordinates.
@@ -371,14 +348,12 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
       return new Point(ax, ay);
     }
     clearCofactor(): ExtendedPoint {
-      if (CURVE.h === _1n) return this; // Fast-path
+      const { h: cofactor } = CURVE;
-      // clear_cofactor(P) := h_eff * P
+      if (cofactor === _1n) return this;
-      // hEff = h for ed25519/ed448. Maybe worth moving to params?
+      return this.multiplyUnsafe(cofactor);
-      if (CURVE.clearCofactor) return CURVE.clearCofactor(ExtendedPoint, this) as ExtendedPoint;
-      return this.multiplyUnsafe(CURVE.h);
     }
   }
-  const wnaf = wNAF(ExtendedPoint, groupLen * 8);
+  const wnaf = wNAF(ExtendedPoint, CURVE.nByteLength * 8);
 
   function assertExtPoint(other: unknown) {
     if (!(other instanceof ExtendedPoint)) throw new TypeError('ExtendedPoint expected');
@@ -413,19 +388,20 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
     // Uses algo from RFC8032 5.1.3.
     static fromHex(hex: Hex, strict = true) {
       const { d, a } = CURVE;
-      hex = ensureBytes(hex, fieldLen);
+      const len = Fp.BYTES;
+      hex = ensureBytes(hex, len);
       // 1.  First, interpret the string as an integer in little-endian
       // representation. Bit 255 of this number is the least significant
       // bit of the x-coordinate and denote this value x_0.  The
       // y-coordinate is recovered simply by clearing this bit.  If the
       // resulting value is >= p, decoding fails.
       const normed = hex.slice();
-      const lastByte = hex[fieldLen - 1];
+      const lastByte = hex[len - 1];
-      normed[fieldLen - 1] = lastByte & ~0x80;
+      normed[len - 1] = lastByte & ~0x80;
-      const y = bytesToNumberLE(normed);
+      const y = ut.bytesToNumberLE(normed);
 
       if (strict && y >= Fp.ORDER) throw new Error('Expected 0 < hex < P');
-      if (!strict && y >= maxGroupElement) throw new Error('Expected 0 < hex < 2**256');
+      if (!strict && y >= maxGroupElement) throw new Error('Expected 0 < hex < CURVE.n');
 
       // 2.  To recover the x-coordinate, the curve equation implies
       // Ed25519: x² = (y² - 1) / (d y² + 1) (mod p).
@@ -459,16 +435,18 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
     // When compressing point, it's enough to only store its y coordinate
     // and use the last byte to encode sign of x.
     toRawBytes(): Uint8Array {
-      const bytes = numberToBytesLE(this.y, fieldLen);
+      const bytes = ut.numberToBytesLE(this.y, Fp.BYTES);
-      bytes[fieldLen - 1] |= this.x & _1n ? 0x80 : 0;
+      bytes[Fp.BYTES - 1] |= this.x & _1n ? 0x80 : 0;
       return bytes;
     }
 
     // Same as toRawBytes, but returns string.
     toHex(): string {
-      return bytesToHex(this.toRawBytes());
+      return ut.bytesToHex(this.toRawBytes());
     }
 
+    // Determines if point is in prime-order subgroup.
+    // Returns `false` is the point is dirty.
     isTorsionFree(): boolean {
       return ExtendedPoint.fromAffine(this).isTorsionFree();
     }
@@ -509,20 +487,20 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
     // Encodes byte string to elliptic curve
     // https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-3
     static hashToCurve(msg: Hex, options?: Partial<htfOpts>) {
-      if (!CURVE.mapToCurve) throw new Error('No mapToCurve defined for curve');
+      const { mapToCurve, htfDefaults } = CURVE;
-      msg = ensureBytes(msg);
+      if (!mapToCurve) throw new Error('No mapToCurve defined for curve');
-      const u = hash_to_field(msg, 2, { ...CURVE.htfDefaults, ...options } as htfOpts);
+      const u = hashToField(ensureBytes(msg), 2, { ...htfDefaults, ...options } as htfOpts);
-      const { x: x0, y: y0 } = CURVE.mapToCurve(u[0]);
+      const { x: x0, y: y0 } = mapToCurve(u[0]);
-      const { x: x1, y: y1 } = CURVE.mapToCurve(u[1]);
+      const { x: x1, y: y1 } = mapToCurve(u[1]);
       const p = new Point(x0, y0).add(new Point(x1, y1)).clearCofactor();
       return p;
     }
     // https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-16#section-3
     static encodeToCurve(msg: Hex, options?: Partial<htfOpts>) {
-      if (!CURVE.mapToCurve) throw new Error('No mapToCurve defined for curve');
+      const { mapToCurve, htfDefaults } = CURVE;
-      msg = ensureBytes(msg);
+      if (!mapToCurve) throw new Error('No mapToCurve defined for curve');
-      const u = hash_to_field(msg, 1, { ...CURVE.htfDefaults, ...options } as htfOpts);
+      const u = hashToField(ensureBytes(msg), 1, { ...htfDefaults, ...options } as htfOpts);
-      const { x, y } = CURVE.mapToCurve(u[0]);
+      const { x, y } = mapToCurve(u[0]);
       return new Point(x, y).clearCofactor();
     }
   }
@@ -536,9 +514,10 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
     }
 
     static fromHex(hex: Hex) {
-      const bytes = ensureBytes(hex, 2 * fieldLen);
+      const len = Fp.BYTES;
-      const r = Point.fromHex(bytes.slice(0, fieldLen), false);
+      const bytes = ensureBytes(hex, 2 * len);
-      const s = bytesToNumberLE(bytes.slice(fieldLen, 2 * fieldLen));
+      const r = Point.fromHex(bytes.slice(0, len), false);
+      const s = ut.bytesToNumberLE(bytes.slice(len, 2 * len));
       return new Signature(r, s);
     }
 
@@ -551,17 +530,17 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
     }
 
     toRawBytes() {
-      return concatBytes(this.r.toRawBytes(), numberToBytesLE(this.s, fieldLen));
+      return ut.concatBytes(this.r.toRawBytes(), ut.numberToBytesLE(this.s, Fp.BYTES));
     }
 
     toHex() {
-      return bytesToHex(this.toRawBytes());
+      return ut.bytesToHex(this.toRawBytes());
     }
   }
 
   // Little-endian SHA512 with modulo n
-  function modlLE(hash: Uint8Array): bigint {
+  function modnLE(hash: Uint8Array): bigint {
-    return mod.mod(bytesToNumberLE(hash), CURVE_ORDER);
+    return mod.mod(ut.bytesToNumberLE(hash), CURVE_ORDER);
   }
 
   /**
@@ -572,7 +551,7 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
    */
   function normalizeScalar(num: number | bigint, max: bigint, strict = true): bigint {
     if (!max) throw new TypeError('Specify max value');
-    if (typeof num === 'number' && Number.isSafeInteger(num)) num = BigInt(num);
+    if (ut.isPositiveInt(num)) num = BigInt(num);
     if (typeof num === 'bigint' && num < max) {
       if (strict) {
         if (_0n < num) return num;
@@ -580,37 +559,32 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
         if (_0n <= num) return num;
       }
     }
-    throw new TypeError('Expected valid scalar: 0 < scalar < max');
+    throw new TypeError(`Expected valid scalar: 0 < scalar < ${max}`);
-  }
-
-  function checkPrivateKey(key: PrivKey) {
-    // Normalize bigint / number / string to Uint8Array
-    key =
-      typeof key === 'bigint' || typeof key === 'number'
-        ? numberToBytesLE(normalizeScalar(key, maxGroupElement), groupLen)
-        : ensureBytes(key);
-    if (key.length !== groupLen) throw new Error(`Expected ${groupLen} bytes, got ${key.length}`);
-    return key;
-  }
-
-  // Takes 64 bytes
-  function getKeyFromHash(hashed: Uint8Array) {
-    // First 32 bytes of 64b uniformingly random input are taken,
-    // clears 3 bits of it to produce a random field element.
-    const head = adjustScalarBytes(hashed.slice(0, groupLen));
-    // Second 32 bytes is called key prefix (5.1.6)
-    const prefix = hashed.slice(groupLen, 2 * groupLen);
-    // The actual private scalar
-    const scalar = modlLE(head);
-    // Point on Edwards curve aka public key
-    const point = Point.BASE.multiply(scalar);
-    const pointBytes = point.toRawBytes();
-    return { head, prefix, scalar, point, pointBytes };
   }
 
   /** Convenience method that creates public key and other stuff. RFC8032 5.1.5 */
   function getExtendedPublicKey(key: PrivKey) {
-    return getKeyFromHash(CURVE.hash(checkPrivateKey(key)));
+    const groupLen = CURVE.nByteLength;
+    // Normalize bigint / number / string to Uint8Array
+    const keyb =
+      typeof key === 'bigint' || typeof key === 'number'
+        ? ut.numberToBytesLE(normalizeScalar(key, maxGroupElement), groupLen)
+        : key;
+    // Hash private key with curve's hash function to produce uniformingly random input
+    // We check byte lengths e.g.: ensureBytes(64, hash(ensureBytes(32, key)))
+    const hashed = ensureBytes(CURVE.hash(ensureBytes(keyb, groupLen)), 2 * groupLen);
+
+    // First half's bits are cleared to produce a random field element.
+    const head = adjustScalarBytes(hashed.slice(0, groupLen));
+    // Second half is called key prefix (5.1.6)
+    const prefix = hashed.slice(groupLen, 2 * groupLen);
+    // The actual private scalar
+    const scalar = modnLE(head);
+    // Point on Edwards curve aka public key
+    const point = Point.BASE.multiply(scalar);
+    // Uint8Array representation
+    const pointBytes = point.toRawBytes();
+    return { head, prefix, scalar, point, pointBytes };
   }
 
   /**
@@ -625,7 +599,7 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
   const EMPTY = new Uint8Array();
   function hashDomainToScalar(message: Uint8Array, context: Hex = EMPTY) {
     context = ensureBytes(context);
-    return modlLE(CURVE.hash(domain(message, context, !!CURVE.preHash)));
+    return modnLE(CURVE.hash(domain(message, context, !!CURVE.preHash)));
   }
 
   /** Signs message with privateKey. RFC8032 5.1.6 */
@@ -633,9 +607,9 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
     message = ensureBytes(message);
     if (CURVE.preHash) message = CURVE.preHash(message);
     const { prefix, scalar, pointBytes } = getExtendedPublicKey(privateKey);
-    const r = hashDomainToScalar(concatBytes(prefix, message), context);
+    const r = hashDomainToScalar(ut.concatBytes(prefix, message), context);
     const R = Point.BASE.multiply(r); // R = rG
-    const k = hashDomainToScalar(concatBytes(R.toRawBytes(), pointBytes, message), context); // k = hash(R+P+msg)
+    const k = hashDomainToScalar(ut.concatBytes(R.toRawBytes(), pointBytes, message), context); // k = hash(R+P+msg)
     const s = mod.mod(r + k * scalar, CURVE_ORDER); // s = r + kp
     return new Signature(R, s).toRawBytes();
   }
@@ -672,13 +646,13 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
     const { r, s } = sig;
     const SB = ExtendedPoint.BASE.multiplyUnsafe(s);
     const k = hashDomainToScalar(
-      concatBytes(r.toRawBytes(), publicKey.toRawBytes(), message),
+      ut.concatBytes(r.toRawBytes(), publicKey.toRawBytes(), message),
       context
     );
     const kA = ExtendedPoint.fromAffine(publicKey).multiplyUnsafe(k);
     const RkA = ExtendedPoint.fromAffine(r).add(kA);
     // [8][S]B = [8]R + [8][k]A'
-    return RkA.subtract(SB).multiplyUnsafe(CURVE.h).equals(ExtendedPoint.ZERO);
+    return RkA.subtract(SB).clearCofactor().equals(ExtendedPoint.ZERO);
   }
 
   // Enable precomputes. Slows down first publicKey computation by 20ms.
@@ -686,19 +660,16 @@ export function twistedEdwards(curveDef: CurveType): CurveFn {
 
   const utils = {
     getExtendedPublicKey,
-    mod: modP,
-    invert: Fp.invert,
-
     /**
      * Not needed for ed25519 private keys. Needed if you use scalars directly (rare).
      */
-    hashToPrivateScalar: (hash: Hex): bigint => hashToPrivateScalar(hash, CURVE_ORDER, true),
+    hashToPrivateScalar: (hash: Hex): bigint => ut.hashToPrivateScalar(hash, CURVE_ORDER, true),
 
     /**
      * ed25519 private keys are uniform 32-bit strings. We do not need to check for
      * modulo bias like we do in secp256k1 randomPrivateKey()
      */
-    randomPrivateKey: (): Uint8Array => randomBytes(fieldLen),
+    randomPrivateKey: (): Uint8Array => randomBytes(Fp.BYTES),
 
     /**
      * We're doing scalar multiplication (used in getPublicKey etc) with precomputed BASE_POINT

src/abstract/modular.ts

@@ -55,6 +55,7 @@ export function invert(number: bigint, modulo: bigint): bigint {
   // prettier-ignore
   let x = _0n, y = _1n, u = _1n, v = _0n;
   while (a !== _0n) {
+    // JIT applies optimization if those two lines follow each other
     const q = b / a;
     const r = b % a;
     const m = x - u * q;
@@ -68,7 +69,8 @@ export function invert(number: bigint, modulo: bigint): bigint {
 }
 
 // Tonelli-Shanks algorithm
-// https://eprint.iacr.org/2012/685.pdf (page 12)
+// Paper 1: https://eprint.iacr.org/2012/685.pdf (page 12)
+// Paper 2: Square Roots from 1; 24, 51, 10 to Dan Shanks
 export function tonelliShanks(P: bigint) {
   // Legendre constant: used to calculate Legendre symbol (a | p),
   // which denotes the value of a^((p-1)/2) (mod p).
@@ -79,7 +81,7 @@ export function tonelliShanks(P: bigint) {
 
   let Q: bigint, S: number, Z: bigint;
   // Step 1: By factoring out powers of 2 from p - 1,
-  // find q and s such that p - 1 = q2s with q odd
+  // find q and s such that p - 1 = q*(2^s) with q odd
   for (Q = P - _1n, S = 0; Q % _2n === _0n; Q /= _2n, S++);
 
   // Step 2: Select a non-square z such that (z | p) ≡ -1 and set c ≡ zq
@@ -98,31 +100,30 @@ export function tonelliShanks(P: bigint) {
   // Slow-path
   const Q1div2 = (Q + _1n) / _2n;
   return function tonelliSlow<T>(Fp: Field<T>, n: T): T {
-    // Step 0: Check that n is indeed a square: (n | p) must be ≡ 1
+    // Step 0: Check that n is indeed a square: (n | p) should not be ≡ -1
-    if (Fp.pow(n, legendreC) !== Fp.ONE) throw new Error('Cannot find square root');
+    if (Fp.pow(n, legendreC) === Fp.negate(Fp.ONE)) throw new Error('Cannot find square root');
-    let s = S;
+    let r = S;
+    // TODO: will fail at Fp2/etc
+    let g = Fp.pow(Fp.mul(Fp.ONE, Z), Q); // will update both x and b
+    let x = Fp.pow(n, Q1div2); // first guess at the square root
+    let b = Fp.pow(n, Q); // first guess at the fudge factor
 
-    let c = pow(Z, Q, P);
+    while (!Fp.equals(b, Fp.ONE)) {
-    let r = Fp.pow(n, Q1div2);
+      if (Fp.equals(b, Fp.ZERO)) return Fp.ZERO; // https://en.wikipedia.org/wiki/Tonelli%E2%80%93Shanks_algorithm (4. If t = 0, return r = 0)
-    let t = Fp.pow(n, Q);
+      // Find m such b^(2^m)==1
-
+      let m = 1;
-    let t2 = Fp.ZERO;
+      for (let t2 = Fp.square(b); m < r; m++) {
-    while (!Fp.equals(Fp.sub(t, Fp.ONE), Fp.ZERO)) {
+        if (Fp.equals(t2, Fp.ONE)) break;
-      t2 = Fp.square(t);
+        t2 = Fp.square(t2); // t2 *= t2
-      let i;
-      for (i = 1; i < s; i++) {
-        // stop if t2-1 == 0
-        if (Fp.equals(Fp.sub(t2, Fp.ONE), Fp.ZERO)) break;
-        // t2 *= t2
-        t2 = Fp.square(t2);
       }
-      let b = pow(c, BigInt(1 << (s - i - 1)), P);
+      // NOTE: r-m-1 can be bigger than 32, need to convert to bigint before shift, otherwise there will be overflow
-      r = Fp.mul(r, b);
+      const ge = Fp.pow(g, _1n << BigInt(r - m - 1)); // ge = 2^(r-m-1)
-      c = mod(b * b, P);
+      g = Fp.square(ge); // g = ge * ge
-      t = Fp.mul(t, c);
+      x = Fp.mul(x, ge); // x *= ge
-      s = i;
+      b = Fp.mul(b, g); // b *= g
+      r = m;
     }
-    return r;
+    return x;
   };
 }
 

src/abstract/montgomery.ts

@@ -1,11 +1,6 @@
 /*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
 import * as mod from './modular.js';
-import {
+import { ensureBytes, numberToBytesLE, bytesToNumberLE, isPositiveInt } from './utils.js';
-  ensureBytes,
-  numberToBytesLE,
-  bytesToNumberLE,
-  // nLength,
-} from './utils.js';
 
 const _0n = BigInt(0);
 const _1n = BigInt(1);
@@ -38,7 +33,7 @@ function validateOpts(curve: CurveType) {
   }
   for (const i of ['montgomeryBits', 'nByteLength'] as const) {
     if (curve[i] === undefined) continue; // Optional
-    if (!Number.isSafeInteger(curve[i]))
+    if (!isPositiveInt(curve[i]))
       throw new Error(`Invalid curve param ${i}=${curve[i]} (${typeof curve[i]})`);
   }
   for (const fn of ['adjustScalarBytes', 'domain', 'powPminus2'] as const) {

src/abstract/utils.ts

@@ -40,19 +40,24 @@ export type BasicCurve<T> = {
   allowInfinityPoint?: boolean;
 };
 
+// Bans floats and integers above 2^53-1
+export function isPositiveInt(num: any): num is number {
+  return typeof num === 'number' && Number.isSafeInteger(num) && num > 0;
+}
+
 export function validateOpts<FP, T>(curve: BasicCurve<FP> & T) {
   mod.validateField(curve.Fp);
   for (const i of ['n', 'h'] as const) {
-    if (typeof curve[i] !== 'bigint')
+    const val = curve[i];
-      throw new Error(`Invalid curve param ${i}=${curve[i]} (${typeof curve[i]})`);
+    if (typeof val !== 'bigint') throw new Error(`Invalid curve param ${i}=${val} (${typeof val})`);
   }
   if (!curve.Fp.isValid(curve.Gx)) throw new Error('Invalid generator X coordinate Fp element');
   if (!curve.Fp.isValid(curve.Gy)) throw new Error('Invalid generator Y coordinate Fp element');
 
   for (const i of ['nBitLength', 'nByteLength'] as const) {
-    if (curve[i] === undefined) continue; // Optional
+    const val = curve[i];
-    if (!Number.isSafeInteger(curve[i]))
+    if (val === undefined) continue; // Optional
-      throw new Error(`Invalid curve param ${i}=${curve[i]} (${typeof curve[i]})`);
+    if (!isPositiveInt(val)) throw new Error(`Invalid curve param ${i}=${val} (${typeof val})`);
   }
   // Set defaults
   return Object.freeze({ ...nLength(curve.n, curve.nBitLength), ...curve } as const);
@@ -144,21 +149,22 @@ export function nLength(n: bigint, nBitLength?: number) {
 }
 
 /**
+ * FIPS 186 B.4.1-compliant "constant-time" private key generation utility.
  * Can take (n+8) or more bytes of uniform input e.g. from CSPRNG or KDF
  * and convert them into private scalar, with the modulo bias being neglible.
- * As per FIPS 186 B.4.1.
+ * Needs at least 40 bytes of input for 32-byte private key.
  * https://research.kudelskisecurity.com/2020/07/28/the-definitive-guide-to-modulo-bias-and-how-to-avoid-it/
- * @param hash hash output from sha512, or a similar function
+ * @param hash hash output from SHA3 or a similar function
  * @returns valid private scalar
  */
-export function hashToPrivateScalar(hash: Hex, CURVE_ORDER: bigint, isLE = false): bigint {
+export function hashToPrivateScalar(hash: Hex, groupOrder: bigint, isLE = false): bigint {
   hash = ensureBytes(hash);
-  const orderLen = nLength(CURVE_ORDER).nByteLength;
+  const hashLen = hash.length;
-  const minLen = orderLen + 8;
+  const minLen = nLength(groupOrder).nByteLength + 8;
-  if (orderLen < 16 || hash.length < minLen || hash.length > 1024)
+  if (minLen < 24 || hashLen < minLen || hashLen > 1024)
-    throw new Error('Expected valid bytes of private key as per FIPS 186');
+    throw new Error(`hashToPrivateScalar: expected ${minLen}-1024 bytes of input, got ${hashLen}`);
   const num = isLE ? bytesToNumberLE(hash) : bytesToNumberBE(hash);
-  return mod.mod(num, CURVE_ORDER - _1n) + _1n;
+  return mod.mod(num, groupOrder - _1n) + _1n;
 }
 
 export function equalBytes(b1: Uint8Array, b2: Uint8Array) {

src/abstract/weierstrass.ts

@@ -1,28 +1,16 @@
 /*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
 // Short Weierstrass curve. The formula is: y² = x³ + ax + b
 
-// TODO: sync vs async naming
-// TODO: default randomBytes
 // Differences from @noble/secp256k1 1.7:
 // 1. Different double() formula (but same addition)
 // 2. Different sqrt() function
 // 3. truncateHash() truncateOnly mode
 // 4. DRBG supports outputLen bigger than outputLen of hmac
+// 5. Support for different hash functions
 
 import * as mod from './modular.js';
-import {
+import * as ut from './utils.js';
-  bytesToHex,
+import { bytesToHex, Hex, PrivKey } from './utils.js';
-  bytesToNumberBE,
-  concatBytes,
-  ensureBytes,
-  hexToBytes,
-  hexToNumber,
-  numberToHexUnpadded,
-  hashToPrivateScalar,
-  Hex,
-  PrivKey,
-} from './utils.js';
-import * as utils from './utils.js';
 import { hash_to_field, htfOpts, validateHTFOpts } from './hash-to-curve.js';
 import { Group, GroupConstructor, wNAF } from './group.js';
 
@@ -31,39 +19,45 @@ type EndomorphismOpts = {
   beta: bigint;
   splitScalar: (k: bigint) => { k1neg: boolean; k1: bigint; k2neg: boolean; k2: bigint };
 };
-export type BasicCurve<T> = utils.BasicCurve<T> & {
+export type BasicCurve<T> = ut.BasicCurve<T> & {
   // Params: a, b
   a: T;
   b: T;
-  // TODO: move into options?
 
+  // Optional params
+  // Executed before privkey validation. Useful for P521 with var-length priv key
   normalizePrivateKey?: (key: PrivKey) => PrivKey;
+  // Whether to execute modular division on a private key, useful for bls curves with cofactor > 1
+  wrapPrivateKey?: boolean;
   // Endomorphism options for Koblitz curves
   endo?: EndomorphismOpts;
-  // Torsions, can be optimized via endomorphisms
+  // When a cofactor != 1, there can be an effective methods to:
+  // 1. Determine whether a point is torsion-free
   isTorsionFree?: (c: ProjectiveConstructor<T>, point: ProjectivePointType<T>) => boolean;
+  // 2. Clear torsion component
   clearCofactor?: (
     c: ProjectiveConstructor<T>,
     point: ProjectivePointType<T>
   ) => ProjectivePointType<T>;
-  // Hash to field opts
+  // Hash to field options
   htfDefaults?: htfOpts;
   mapToCurve?: (scalar: bigint[]) => { x: T; y: T };
 };
-// DER encoding utilities
+
+// ASN.1 DER encoding utilities
 class DERError extends Error {
   constructor(message: string) {
     super(message);
   }
 }
 
-function sliceDER(s: string): string {
+const DER = {
+  slice(s: string): string {
     // Proof: any([(i>=0x80) == (int(hex(i).replace('0x', '').zfill(2)[0], 16)>=8)  for i in range(0, 256)])
     // Padding done by numberToHex
     return Number.parseInt(s[0], 16) >= 8 ? '00' + s : s;
-}
+  },
-
+  parseInt(data: Uint8Array): { data: bigint; left: Uint8Array } {
-function parseDERInt(data: Uint8Array) {
     if (data.length < 2 || data[0] !== 0x02) {
       throw new DERError(`Invalid signature integer tag: ${bytesToHex(data)}`);
     }
@@ -76,31 +70,26 @@ function parseDERInt(data: Uint8Array) {
     if (res[0] === 0x00 && res[1] <= 0x7f) {
       throw new DERError('Invalid signature integer: trailing length');
     }
-  return { data: bytesToNumberBE(res), left: data.subarray(len + 2) };
+    return { data: ut.bytesToNumberBE(res), left: data.subarray(len + 2) };
-}
+  },
-
+  parseSig(data: Uint8Array): { r: bigint; s: bigint } {
-function parseDERSignature(data: Uint8Array) {
     if (data.length < 2 || data[0] != 0x30) {
       throw new DERError(`Invalid signature tag: ${bytesToHex(data)}`);
     }
     if (data[1] !== data.length - 2) {
       throw new DERError('Invalid signature: incorrect length');
     }
-  const { data: r, left: sBytes } = parseDERInt(data.subarray(2));
+    const { data: r, left: sBytes } = DER.parseInt(data.subarray(2));
-  const { data: s, left: rBytesLeft } = parseDERInt(sBytes);
+    const { data: s, left: rBytesLeft } = DER.parseInt(sBytes);
     if (rBytesLeft.length) {
       throw new DERError(`Invalid signature: left bytes after parsing: ${bytesToHex(rBytesLeft)}`);
     }
     return { r, s };
-}
+  },
-
+};
-// Be friendly to bad ECMAScript parsers by not using bigint literals like 123n
-const _0n = BigInt(0);
-const _1n = BigInt(1);
-const _3n = BigInt(3);
 
 type Entropy = Hex | true;
-type SignOpts = { lowS?: boolean; extraEntropy?: Entropy };
+export type SignOpts = { lowS?: boolean; extraEntropy?: Entropy };
 
 /**
  * ### Design rationale for types
@@ -124,7 +113,7 @@ type SignOpts = { lowS?: boolean; extraEntropy?: Entropy };
  * TODO: https://www.typescriptlang.org/docs/handbook/release-notes/typescript-2-7.html#unique-symbol
  */
 
-// Instance
+// Instance for 3d XYZ points
 export interface ProjectivePointType<T> extends Group<ProjectivePointType<T>> {
   readonly x: T;
   readonly y: T;
@@ -133,14 +122,14 @@ export interface ProjectivePointType<T> extends Group<ProjectivePointType<T>> {
   multiplyUnsafe(scalar: bigint): ProjectivePointType<T>;
   toAffine(invZ?: T): PointType<T>;
 }
-// Static methods
+// Static methods for 3d XYZ points
 export interface ProjectiveConstructor<T> extends GroupConstructor<ProjectivePointType<T>> {
   new (x: T, y: T, z: T): ProjectivePointType<T>;
   fromAffine(p: PointType<T>): ProjectivePointType<T>;
   toAffineBatch(points: ProjectivePointType<T>[]): PointType<T>[];
   normalizeZ(points: ProjectivePointType<T>[]): ProjectivePointType<T>[];
 }
-// Instance
+// Instance for 2d XY points
 export interface PointType<T> extends Group<PointType<T>> {
   readonly x: T;
   readonly y: T;
@@ -151,7 +140,7 @@ export interface PointType<T> extends Group<PointType<T>> {
   assertValidity(): void;
   multiplyAndAddUnsafe(Q: PointType<T>, a: bigint, b: bigint): PointType<T> | undefined;
 }
-// Static methods
+// Static methods for 2d XY points
 export interface PointConstructor<T> extends GroupConstructor<PointType<T>> {
   new (x: T, y: T): PointType<T>;
   fromHex(hex: Hex): PointType<T>;
@@ -167,7 +156,7 @@ export type CurvePointsType<T> = BasicCurve<T> & {
 };
 
 function validatePointOpts<T>(curve: CurvePointsType<T>) {
-  const opts = utils.validateOpts(curve);
+  const opts = ut.validateOpts(curve);
   const Fp = opts.Fp;
   for (const i of ['a', 'b'] as const) {
     if (!Fp.isValid(curve[i]))
@@ -206,22 +195,14 @@ export type CurvePointsRes<T> = {
   isWithinCurveOrder: (num: bigint) => boolean;
 };
 
+// Be friendly to bad ECMAScript parsers by not using bigint literals like 123n
+const _0n = BigInt(0);
+const _1n = BigInt(1);
+const _3n = BigInt(3);
+
 export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
   const CURVE = validatePointOpts(opts);
-  const Fp = CURVE.Fp;
+  const { Fp } = CURVE; // All curves has same field / group length as for now, but they can differ
-  // Lengths
-  // All curves has same field / group length as for now, but it can be different for other curves
-  const { nByteLength, nBitLength } = CURVE;
-  const groupLen = nByteLength;
-
-  // Not using ** operator with bigints for old engines.
-  // 2n ** (8n * 32n) == 2n << (8n * 32n - 1n)
-  //const FIELD_MASK = _2n << (_8n * BigInt(fieldLen) - _1n);
-  // function numToFieldStr(num: bigint): string {
-  //   if (typeof num !== 'bigint') throw new Error('Expected bigint');
-  //   if (!(_0n <= num && num < FIELD_MASK)) throw new Error(`Expected number < 2^${fieldLen * 8}`);
-  //   return num.toString(16).padStart(2 * fieldLen, '0');
-  // }
 
   /**
    * y² = x³ + ax + b: Short weierstrass curve formula
@@ -234,35 +215,48 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
     return Fp.add(Fp.add(x3, Fp.mul(x, a)), b); // x3 + a * x + b
   }
 
+  // Valid group elements reside in range 1..n-1
   function isWithinCurveOrder(num: bigint): boolean {
     return _0n < num && num < CURVE.n;
   }
 
+  /**
+   * Validates if a private key is valid and converts it to bigint form.
+   * Supports two options, that are passed when CURVE is initialized:
+   * - `normalizePrivateKey()` executed before all checks
+   * - `wrapPrivateKey` when true, executed after most checks, but before `0 < key < n`
+   */
   function normalizePrivateKey(key: PrivKey): bigint {
-    if (typeof CURVE.normalizePrivateKey === 'function') {
+    const { normalizePrivateKey: custom, nByteLength: groupLen, wrapPrivateKey, n: order } = CURVE;
-      key = CURVE.normalizePrivateKey(key);
+    if (typeof custom === 'function') key = custom(key);
-    }
     let num: bigint;
     if (typeof key === 'bigint') {
+      // Curve order check is done below
       num = key;
-    } else if (typeof key === 'number' && Number.isSafeInteger(key) && key > 0) {
+    } else if (ut.isPositiveInt(key)) {
       num = BigInt(key);
     } else if (typeof key === 'string') {
       if (key.length !== 2 * groupLen) throw new Error(`Expected ${groupLen} bytes of private key`);
-      num = hexToNumber(key);
+      // Validates individual octets
+      num = ut.hexToNumber(key);
     } else if (key instanceof Uint8Array) {
       if (key.length !== groupLen) throw new Error(`Expected ${groupLen} bytes of private key`);
-      num = bytesToNumberBE(key);
+      num = ut.bytesToNumberBE(key);
     } else {
       throw new TypeError('Expected valid private key');
     }
-    if (CURVE.wrapPrivateKey) num = mod.mod(num, CURVE.n);
+    // Useful for curves with cofactor != 1
+    if (wrapPrivateKey) num = mod.mod(num, order);
     if (!isWithinCurveOrder(num)) throw new Error('Expected private key: 0 < key < n');
     return num;
   }
 
+  /**
+   * Validates if a scalar ("private number") is valid.
+   * Scalars are valid only if they are less than curve order.
+   */
   function normalizeScalar(num: number | bigint): bigint {
-    if (typeof num === 'number' && Number.isSafeInteger(num) && num > 0) return BigInt(num);
+    if (ut.isPositiveInt(num)) return BigInt(num);
     if (typeof num === 'bigint' && isWithinCurveOrder(num)) return num;
     throw new TypeError('Expected valid private scalar: 0 < scalar < curve.n');
   }
@@ -289,7 +283,7 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
 
     /**
      * Takes a bunch of Projective Points but executes only one
-     * invert on all of them. invert is very slow operation,
+     * inversion on all of them. Inversion is very slow operation,
      * so this improves performance massively.
      */
     static toAffineBatch(points: ProjectivePoint[]): Point[] {
@@ -297,6 +291,9 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
       return points.map((p, i) => p.toAffine(toInv[i]));
     }
 
+    /**
+     * Optimization: converts a list of projective points to a list of identical points with Z=1.
+     */
     static normalizeZ(points: ProjectivePoint[]): ProjectivePoint[] {
       return ProjectivePoint.toAffineBatch(points).map(ProjectivePoint.fromAffine);
     }
@@ -320,10 +317,6 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
       return new ProjectivePoint(this.x, Fp.negate(this.y), this.z);
     }
 
-    doubleAdd(): ProjectivePoint {
-      return this.add(this);
-    }
-
     // Renes-Costello-Batina exception-free doubling formula.
     // There is 30% faster Jacobian formula, but it is not complete.
     // https://eprint.iacr.org/2015/1060, algorithm 3
@@ -525,24 +518,25 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
       return new Point(ax, ay);
     }
     isTorsionFree(): boolean {
-      if (CURVE.h === _1n) return true; // No subgroups, always torsion fee
+      const { h: cofactor, isTorsionFree } = CURVE;
-      if (CURVE.isTorsionFree) return CURVE.isTorsionFree(ProjectivePoint, this);
+      if (cofactor === _1n) return true; // No subgroups, always torsion-free
-      // is multiplyUnsafe(CURVE.n) is always ok, same as for edwards?
+      if (isTorsionFree) return isTorsionFree(ProjectivePoint, this);
-      throw new Error('Unsupported!');
+      throw new Error('isTorsionFree() has not been declared for the elliptic curve');
     }
-    // Clear cofactor of G1
-    // https://eprint.iacr.org/2019/403
     clearCofactor(): ProjectivePoint {
-      if (CURVE.h === _1n) return this; // Fast-path
+      const { h: cofactor, clearCofactor } = CURVE;
-      if (CURVE.clearCofactor) return CURVE.clearCofactor(ProjectivePoint, this) as ProjectivePoint;
+      if (cofactor === _1n) return this; // Fast-path
+      if (clearCofactor) return clearCofactor(ProjectivePoint, this) as ProjectivePoint;
       return this.multiplyUnsafe(CURVE.h);
     }
   }
-  const wnaf = wNAF(ProjectivePoint, CURVE.endo ? nBitLength / 2 : nBitLength);
+  const _bits = CURVE.nBitLength;
+  const wnaf = wNAF(ProjectivePoint, CURVE.endo ? Math.ceil(_bits / 2) : _bits);
 
   function assertPrjPoint(other: unknown) {
     if (!(other instanceof ProjectivePoint)) throw new TypeError('ProjectivePoint expected');
   }
+
   // Stores precomputed values for points.
   const pointPrecomputes = new WeakMap<Point, ProjectivePoint[]>();
 
@@ -551,11 +545,11 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
    */
   class Point implements PointType<T> {
     /**
-     * Base point aka generator. public_key = Point.BASE * private_key
+     * Base point aka generator. Any public_key = Point.BASE * private_key
      */
     static BASE: Point = new Point(CURVE.Gx, CURVE.Gy);
     /**
-     * Identity point aka point at infinity. point = point + zero_point
+     * Identity point aka point at infinity. p - p = zero_p; p + zero_p = p
      */
     static ZERO: Point = new Point(Fp.ZERO, Fp.ZERO);
 
@@ -583,7 +577,7 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
      * @param hex short/long ECDSA hex
      */
     static fromHex(hex: Hex): Point {
-      const { x, y } = CURVE.fromBytes(ensureBytes(hex));
+      const { x, y } = CURVE.fromBytes(ut.ensureBytes(hex));
       const point = new Point(x, y);
       point.assertValidity();
       return point;
@@ -607,16 +601,16 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
       // Zero is valid point too!
       if (this.equals(Point.ZERO)) {
         if (CURVE.allowInfinityPoint) return;
-        throw new Error('Point is infinity');
+        throw new Error('Point at infinity');
       }
       // Some 3rd-party test vectors require different wording between here & `fromCompressedHex`
       const msg = 'Point is not on elliptic curve';
       const { x, y } = this;
+      // Check if x, y are valid field elements
       if (!Fp.isValid(x) || !Fp.isValid(y)) throw new Error(msg);
-      const left = Fp.square(y);
+      const left = Fp.square(y); // y²
-      const right = weierstrassEquation(x);
+      const right = weierstrassEquation(x); // x³ + ax + b
       if (!Fp.equals(left, right)) throw new Error(msg);
-      // TODO: flag to disable this?
       if (!this.isTorsionFree()) throw new Error('Point must be of prime-order subgroup');
     }
 
@@ -630,34 +624,37 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
       return new Point(this.x, Fp.negate(this.y));
     }
 
+    protected toProj() {
+      return ProjectivePoint.fromAffine(this);
+    }
+
     // Adds point to itself
     double() {
-      return ProjectivePoint.fromAffine(this).double().toAffine();
+      return this.toProj().double().toAffine();
     }
 
-    // Adds point to other point
     add(other: Point) {
-      return ProjectivePoint.fromAffine(this).add(ProjectivePoint.fromAffine(other)).toAffine();
+      return this.toProj().add(ProjectivePoint.fromAffine(other)).toAffine();
     }
 
-    // Subtracts other point from the point
     subtract(other: Point) {
       return this.add(other.negate());
     }
 
     multiply(scalar: number | bigint) {
-      return ProjectivePoint.fromAffine(this).multiply(scalar, this).toAffine();
+      return this.toProj().multiply(scalar, this).toAffine();
     }
 
     multiplyUnsafe(scalar: bigint) {
-      return ProjectivePoint.fromAffine(this).multiplyUnsafe(scalar).toAffine();
+      return this.toProj().multiplyUnsafe(scalar).toAffine();
     }
+
     clearCofactor() {
-      return ProjectivePoint.fromAffine(this).clearCofactor().toAffine();
+      return this.toProj().clearCofactor().toAffine();
     }
 
     isTorsionFree(): boolean {
-      return ProjectivePoint.fromAffine(this).isTorsionFree();
+      return this.toProj().isTorsionFree();
     }
 
     /**
@@ -667,7 +664,7 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
      * @returns non-zero affine point
      */
     multiplyAndAddUnsafe(Q: Point, a: bigint, b: bigint): Point | undefined {
-      const P = ProjectivePoint.fromAffine(this);
+      const P = this.toProj();
       const aP =
         a === _0n || a === _1n || this !== Point.BASE ? P.multiplyUnsafe(a) : P.multiply(a);
       const bQ = ProjectivePoint.fromAffine(Q).multiplyUnsafe(b);
@@ -678,23 +675,26 @@ export function weierstrassPoints<T>(opts: CurvePointsType<T>) {
     // Encodes byte string to elliptic curve
     // https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-11#section-3
     static hashToCurve(msg: Hex, options?: Partial<htfOpts>) {
-      if (!CURVE.mapToCurve) throw new Error('No mapToCurve defined for curve');
+      const { mapToCurve } = CURVE;
-      msg = ensureBytes(msg);
+      if (!mapToCurve) throw new Error('CURVE.mapToCurve() has not been defined');
+      msg = ut.ensureBytes(msg);
       const u = hash_to_field(msg, 2, { ...CURVE.htfDefaults, ...options } as htfOpts);
-      const { x: x0, y: y0 } = CURVE.mapToCurve(u[0]);
+      const { x: x0, y: y0 } = mapToCurve(u[0]);
-      const { x: x1, y: y1 } = CURVE.mapToCurve(u[1]);
+      const { x: x1, y: y1 } = mapToCurve(u[1]);
-      const p = new Point(x0, y0).add(new Point(x1, y1)).clearCofactor();
+      return new Point(x0, y0).add(new Point(x1, y1)).clearCofactor();
-      return p;
     }
+
     // https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-16#section-3
     static encodeToCurve(msg: Hex, options?: Partial<htfOpts>) {
-      if (!CURVE.mapToCurve) throw new Error('No mapToCurve defined for curve');
+      const { mapToCurve } = CURVE;
-      msg = ensureBytes(msg);
+      if (!mapToCurve) throw new Error('CURVE.mapToCurve() has not been defined');
+      msg = ut.ensureBytes(msg);
       const u = hash_to_field(msg, 1, { ...CURVE.htfDefaults, ...options } as htfOpts);
-      const { x, y } = CURVE.mapToCurve(u[0]);
+      const { x, y } = mapToCurve(u[0]);
       return new Point(x, y).clearCofactor();
     }
   }
+
   return {
     Point: Point as PointConstructor<T>,
     ProjectivePoint: ProjectivePoint as ProjectiveConstructor<T>,
@@ -733,15 +733,15 @@ export type CurveType = BasicCurve<bigint> & {
   // Default options
   lowS?: boolean;
   // Hashes
-  hash: utils.CHash; // Because we need outputLen for DRBG
+  hash: ut.CHash; // Because we need outputLen for DRBG
   hmac: HmacFnSync;
   randomBytes: (bytesLength?: number) => Uint8Array;
   truncateHash?: (hash: Uint8Array, truncateOnly?: boolean) => bigint;
 };
 
 function validateOpts(curve: CurveType) {
-  const opts = utils.validateOpts(curve);
+  const opts = ut.validateOpts(curve);
-  if (typeof opts.hash !== 'function' || !Number.isSafeInteger(opts.hash.outputLen))
+  if (typeof opts.hash !== 'function' || !ut.isPositiveInt(opts.hash.outputLen))
     throw new Error('Invalid hash function');
   if (typeof opts.hmac !== 'function') throw new Error('Invalid hmac function');
   if (typeof opts.randomBytes !== 'function') throw new Error('Invalid randomBytes function');
@@ -754,6 +754,7 @@ export type CurveFn = {
   getPublicKey: (privateKey: PrivKey, isCompressed?: boolean) => Uint8Array;
   getSharedSecret: (privateA: PrivKey, publicB: PubKey, isCompressed?: boolean) => Uint8Array;
   sign: (msgHash: Hex, privKey: PrivKey, opts?: SignOpts) => SignatureType;
+  signUnhashed: (msg: Uint8Array, privKey: PrivKey, opts?: SignOpts) => SignatureType;
   verify: (
     signature: Hex | SignatureType,
     msgHash: Hex,
@@ -766,8 +767,6 @@ export type CurveFn = {
   ProjectivePoint: ProjectiveConstructor<bigint>;
   Signature: SignatureConstructor;
   utils: {
-    mod: (a: bigint, b?: bigint) => bigint;
-    invert: (number: bigint, modulo?: bigint) => bigint;
     _bigintToBytes: (num: bigint) => Uint8Array;
     _bigintToString: (num: bigint) => string;
     _normalizePrivateKey: (key: PrivKey) => bigint;
@@ -824,19 +823,17 @@ class HmacDrbg {
       out.push(sl);
       len += this.v.length;
     }
-    return concatBytes(...out);
+    return ut.concatBytes(...out);
   }
-  // There is no need in clean() method
+  // There are no guarantees with JS GC whether bigints are removed even if you clean Uint8Arrays.
-  // It's useless, there are no guarantees with JS GC
-  // whether bigints are removed even if you clean Uint8Arrays.
 }
 
 export function weierstrass(curveDef: CurveType): CurveFn {
   const CURVE = validateOpts(curveDef) as ReturnType<typeof validateOpts>;
   const CURVE_ORDER = CURVE.n;
   const Fp = CURVE.Fp;
-  const compressedLen = Fp.BYTES + 1; // 33
+  const compressedLen = Fp.BYTES + 1; // e.g. 33 for 32
-  const uncompressedLen = 2 * Fp.BYTES + 1; // 65
+  const uncompressedLen = 2 * Fp.BYTES + 1; // e.g. 65 for 32
 
   function isValidFieldElement(num: bigint): boolean {
     // 0 is disallowed by arbitrary reasons. Probably because infinity point?
@@ -847,10 +844,12 @@ export function weierstrass(curveDef: CurveType): CurveFn {
     weierstrassPoints({
       ...CURVE,
       toBytes(c, point, isCompressed: boolean): Uint8Array {
+        const x = Fp.toBytes(point.x);
+        const cat = ut.concatBytes;
         if (isCompressed) {
-          return concatBytes(new Uint8Array([point.hasEvenY() ? 0x02 : 0x03]), Fp.toBytes(point.x));
+          return cat(Uint8Array.from([point.hasEvenY() ? 0x02 : 0x03]), x);
         } else {
-          return concatBytes(new Uint8Array([0x04]), Fp.toBytes(point.x), Fp.toBytes(point.y));
+          return cat(Uint8Array.from([0x04]), x, Fp.toBytes(point.y));
         }
       },
       fromBytes(bytes: Uint8Array) {
@@ -858,7 +857,7 @@ export function weierstrass(curveDef: CurveType): CurveFn {
         const header = bytes[0];
         // this.assertValidity() is done inside of fromHex
         if (len === compressedLen && (header === 0x02 || header === 0x03)) {
-          const x = bytesToNumberBE(bytes.subarray(1));
+          const x = ut.bytesToNumberBE(bytes.subarray(1));
           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
@@ -910,15 +909,18 @@ export function weierstrass(curveDef: CurveType): CurveFn {
     return isBiggerThanHalfOrder(s) ? mod.mod(-s, CURVE_ORDER) : s;
   }
 
+  function bits2int_2(bytes: Uint8Array): bigint {
+    const delta = bytes.length * 8 - CURVE.nBitLength;
+    const num = ut.bytesToNumberBE(bytes);
+    return delta > 0 ? num >> BigInt(delta) : num;
+  }
+
   // Ensures ECDSA message hashes are 32 bytes and < curve order
   function _truncateHash(hash: Uint8Array, truncateOnly = false): bigint {
-    const { n, nBitLength } = CURVE;
+    const h = bits2int_2(hash);
-    const byteLength = hash.length;
+    if (truncateOnly) return h;
-    const delta = byteLength * 8 - nBitLength; // size of curve.n (252 bits)
+    const { n } = CURVE;
-    let h = bytesToNumberBE(hash);
+    return h >= n ? h - n : h;
-    if (delta > 0) h = h >> BigInt(delta);
-    if (!truncateOnly && h >= n) h -= n;
-    return h;
   }
   const truncateHash = CURVE.truncateHash || _truncateHash;
 
@@ -930,15 +932,17 @@ export function weierstrass(curveDef: CurveType): CurveFn {
       this.assertValidity();
     }
 
-    // pair (32 bytes of r, 32 bytes of s)
+    // pair (bytes of r, bytes of s)
     static fromCompact(hex: Hex) {
       const arr = hex instanceof Uint8Array;
       const name = 'Signature.fromCompact';
       if (typeof hex !== 'string' && !arr)
         throw new TypeError(`${name}: Expected string or Uint8Array`);
       const str = arr ? bytesToHex(hex) : hex;
-      if (str.length !== 128) throw new Error(`${name}: Expected 64-byte hex`);
+      const gl = CURVE.nByteLength * 2; // group length in hex, not ui8a
-      return new Signature(hexToNumber(str.slice(0, 64)), hexToNumber(str.slice(64, 128)));
+      if (str.length !== 2 * gl) throw new Error(`${name}: Expected ${gl / 2}-byte hex`);
+      const slice = (from: number, to: number) => ut.hexToNumber(str.slice(from, to));
+      return new Signature(slice(0, gl), slice(gl, 2 * gl));
     }
 
     // DER encoded ECDSA signature
@@ -947,7 +951,7 @@ export function weierstrass(curveDef: CurveType): CurveFn {
       const arr = hex instanceof Uint8Array;
       if (typeof hex !== 'string' && !arr)
         throw new TypeError(`Signature.fromDER: Expected string or Uint8Array`);
-      const { r, s } = parseDERSignature(arr ? hex : hexToBytes(hex));
+      const { r, s } = DER.parseSig(arr ? hex : ut.hexToBytes(hex));
       return new Signature(r, s);
     }
 
@@ -964,6 +968,7 @@ export function weierstrass(curveDef: CurveType): CurveFn {
     /**
      * Recovers public key from signature with recovery bit. Throws on invalid hash.
      * https://en.wikipedia.org/wiki/Elliptic_Curve_Digital_Signature_Algorithm#Public_key_recovery
+     * It's also possible to recover key without bit: try all 4 bit values and check for sig match.
      *
      * ```
      * recover(r, s, h) where
@@ -978,16 +983,18 @@ export function weierstrass(curveDef: CurveType): CurveFn {
     recoverPublicKey(msgHash: Hex): Point {
       const { r, s, recovery } = this;
       if (recovery == null) throw new Error('Cannot recover: recovery bit is not present');
-      if (recovery !== 0 && recovery !== 1) throw new Error('Cannot recover: invalid recovery bit');
+      if (![0, 1, 2, 3].includes(recovery)) throw new Error('Cannot recover: invalid recovery bit');
-      const h = truncateHash(ensureBytes(msgHash));
+      const h = truncateHash(ut.ensureBytes(msgHash));
       const { n } = CURVE;
-      const rinv = mod.invert(r, n);
+      const radj = recovery === 2 || recovery === 3 ? r + n : r;
+      if (radj >= Fp.ORDER) throw new Error('Cannot recover: bit 2/3 is invalid with current r');
+      const rinv = mod.invert(radj, n);
       // Q = u1⋅G + u2⋅R
       const u1 = mod.mod(-h * rinv, n);
       const u2 = mod.mod(s * rinv, n);
       const prefix = recovery & 1 ? '03' : '02';
-      const R = Point.fromHex(prefix + numToFieldStr(r));
+      const R = Point.fromHex(prefix + numToFieldStr(radj));
-      const Q = Point.BASE.multiplyAndAddUnsafe(R, u1, u2);
+      const Q = Point.BASE.multiplyAndAddUnsafe(R, u1, u2); // unsafe is fine: no priv data leaked
       if (!Q) throw new Error('Cannot recover: point at infinify');
       Q.assertValidity();
       return Q;
@@ -1009,22 +1016,24 @@ export function weierstrass(curveDef: CurveType): CurveFn {
     }
 
     // DER-encoded
-    toDERRawBytes(isCompressed = false) {
+    toDERRawBytes() {
-      return hexToBytes(this.toDERHex(isCompressed));
+      return ut.hexToBytes(this.toDERHex());
     }
-    toDERHex(isCompressed = false) {
+    toDERHex() {
-      const sHex = sliceDER(numberToHexUnpadded(this.s));
+      const { numberToHexUnpadded: toHex } = ut;
-      if (isCompressed) return sHex;
+      const sHex = DER.slice(toHex(this.s));
-      const rHex = sliceDER(numberToHexUnpadded(this.r));
+      const rHex = DER.slice(toHex(this.r));
-      const rLen = numberToHexUnpadded(rHex.length / 2);
+      const sHexL = sHex.length / 2;
-      const sLen = numberToHexUnpadded(sHex.length / 2);
+      const rHexL = rHex.length / 2;
-      const length = numberToHexUnpadded(rHex.length / 2 + sHex.length / 2 + 4);
+      const sLen = toHex(sHexL);
+      const rLen = toHex(rHexL);
+      const length = toHex(rHexL + sHexL + 4);
       return `30${length}02${rLen}${rHex}02${sLen}${sHex}`;
     }
 
-    // 32 bytes of r, then 32 bytes of s
+    // padded bytes of r, then padded bytes of s
     toCompactRawBytes() {
-      return hexToBytes(this.toCompactHex());
+      return ut.hexToBytes(this.toCompactHex());
     }
     toCompactHex() {
       return numToFieldStr(this.r) + numToFieldStr(this.s);
@@ -1032,8 +1041,6 @@ export function weierstrass(curveDef: CurveType): CurveFn {
   }
 
   const utils = {
-    mod: (n: bigint, modulo = Fp.ORDER) => mod.mod(n, modulo),
-    invert: Fp.invert,
     isValidPrivateKey(privateKey: PrivKey) {
       try {
         normalizePrivateKey(privateKey);
@@ -1053,7 +1060,8 @@ export function weierstrass(curveDef: CurveType): CurveFn {
     /**
      * Converts some bytes to a valid private key. Needs at least (nBitLength+64) bytes.
      */
-    hashToPrivateKey: (hash: Hex): Uint8Array => numToField(hashToPrivateScalar(hash, CURVE_ORDER)),
+    hashToPrivateKey: (hash: Hex): Uint8Array =>
+      numToField(ut.hashToPrivateScalar(hash, CURVE_ORDER)),
 
     /**
      * Produces cryptographically secure private key from random of size (nBitLength+64)
@@ -1078,10 +1086,10 @@ export function weierstrass(curveDef: CurveType): CurveFn {
   };
 
   /**
-   * Computes public key for a private key.
+   * Computes public key for a private key. Checks for validity of the private key.
    * @param privateKey private key
-   * @param isCompressed whether to return compact, or full key
+   * @param isCompressed whether to return compact (default), or full key
-   * @returns Public key, full by default; short when isCompressed=true
+   * @returns Public key, full when isCompressed=false; short when isCompressed=true
    */
   function getPublicKey(privateKey: PrivKey, isCompressed = false): Uint8Array {
     return Point.fromPrivateKey(privateKey).toRawBytes(isCompressed);
@@ -1101,12 +1109,12 @@ export function weierstrass(curveDef: CurveType): CurveFn {
   }
 
   /**
-   * ECDH (Elliptic Curve Diffie Hellman) implementation.
+   * ECDH (Elliptic Curve Diffie Hellman).
-   * 1. Checks for validity of private key
+   * Computes shared public key from private key and public key.
-   * 2. Checks for the public key of being on-curve
+   * Checks: 1) private key validity 2) shared key is on-curve
    * @param privateA private key
    * @param publicB different public key
-   * @param isCompressed whether to return compact (33-byte), or full (65-byte) key
+   * @param isCompressed whether to return compact (default), or full key
    * @returns shared public key
    */
   function getSharedSecret(privateA: PrivKey, publicB: PubKey, isCompressed = false): Uint8Array {
@@ -1118,9 +1126,21 @@ export function weierstrass(curveDef: CurveType): CurveFn {
   }
 
   // RFC6979 methods
-  function bits2int(bytes: Uint8Array) {
+  function bits2int(bytes: Uint8Array): bigint {
-    const slice = bytes.length > Fp.BYTES ? bytes.slice(0, Fp.BYTES) : bytes;
+    const { nByteLength } = CURVE;
-    return bytesToNumberBE(slice);
+    if (!(bytes instanceof Uint8Array)) throw new Error('Expected Uint8Array');
+    const slice = bytes.length > nByteLength ? bytes.slice(0, nByteLength) : bytes;
+    // const slice = bytes; nByteLength; nBitLength;
+    let num = ut.bytesToNumberBE(slice);
+    // const { nBitLength } = CURVE;
+    // const delta = (bytes.length * 8) - nBitLength;
+    // if (delta > 0) {
+    //   // console.log('bits=', bytes.length*8, 'CURVE n=', nBitLength, 'delta=', delta);
+    //   // console.log(bytes.length, nBitLength, delta);
+    //   // console.log(bytes, new Error().stack);
+    //   num >>= BigInt(delta);
+    // }
+    return num;
   }
   function bits2octets(bytes: Uint8Array): Uint8Array {
     const z1 = bits2int(bytes);
@@ -1128,28 +1148,28 @@ export function weierstrass(curveDef: CurveType): CurveFn {
     return int2octets(z2 < _0n ? z1 : z2);
   }
   function int2octets(num: bigint): Uint8Array {
-    return numToField(num); // prohibits >32 bytes
+    return numToField(num); // prohibits >nByteLength bytes
   }
   // Steps A, D of RFC6979 3.2
   // Creates RFC6979 seed; converts msg/privKey to numbers.
   function initSigArgs(msgHash: Hex, privateKey: PrivKey, extraEntropy?: Entropy) {
     if (msgHash == null) throw new Error(`sign: expected valid message hash, not "${msgHash}"`);
     // Step A is ignored, since we already provide hash instead of msg
-    const h1 = numToField(truncateHash(ensureBytes(msgHash)));
+    const h1 = numToField(truncateHash(ut.ensureBytes(msgHash)));
     const d = normalizePrivateKey(privateKey);
     // K = HMAC_K(V || 0x00 || int2octets(x) || bits2octets(h1) || k')
     const seedArgs = [int2octets(d), bits2octets(h1)];
     // RFC6979 3.6: additional k' could be provided
     if (extraEntropy != null) {
       if (extraEntropy === true) extraEntropy = CURVE.randomBytes(Fp.BYTES);
-      const e = ensureBytes(extraEntropy);
+      const e = ut.ensureBytes(extraEntropy);
       if (e.length !== Fp.BYTES) throw new Error(`sign: Expected ${Fp.BYTES} bytes of extra data`);
       seedArgs.push(e);
     }
     // seed is constructed from private key and message
     // Step D
     // V, 0x00 are done in HmacDRBG constructor.
-    const seed = concatBytes(...seedArgs);
+    const seed = ut.concatBytes(...seedArgs);
     const m = bits2int(h1);
     return { seed, m, d };
   }
@@ -1172,9 +1192,10 @@ export function weierstrass(curveDef: CurveType): CurveFn {
     // r = x mod n
     const r = mod.mod(q.x, n);
     if (r === _0n) return;
-    // s = (1/k * (m + dr) mod n
+    // s = (m + dr)/k mod n where x/k == x*inv(k)
     const s = mod.mod(kinv * mod.mod(m + mod.mod(d * r, n), n), n);
     if (s === _0n) return;
+    // recovery bit is usually 0 or 1; rarely it's 2 or 3, when q.x > n
     let recovery = (q.x === r ? 0 : 2) | Number(q.y & _1n);
     let normS = s;
     if (lowS && isBiggerThanHalfOrder(s)) {
@@ -1184,11 +1205,19 @@ export function weierstrass(curveDef: CurveType): CurveFn {
     return new Signature(r, normS, recovery);
   }
 
+  const defaultSigOpts: SignOpts = { lowS: CURVE.lowS };
+
   /**
    * Signs message hash (not message: you need to hash it by yourself).
+   * ```
+   * sign(m, d, k) where
+   *   (x, y) = G × k
+   *   r = x mod n
+   *   s = (m + dr)/k mod n
+   * ```
    * @param opts `lowS, extraEntropy`
    */
-  function sign(msgHash: Hex, privKey: PrivKey, opts: SignOpts = { lowS: CURVE.lowS }): Signature {
+  function sign(msgHash: Hex, privKey: PrivKey, opts = defaultSigOpts): Signature {
     // Steps A, D of RFC6979 3.2.
     const { seed, m, d } = initSigArgs(msgHash, privKey, opts.extraEntropy);
     // Steps B, C, D, E, F, G
@@ -1199,6 +1228,14 @@ export function weierstrass(curveDef: CurveType): CurveFn {
     while (!(sig = kmdToSig(drbg.generateSync(), m, d, opts.lowS))) drbg.reseedSync();
     return sig;
   }
+
+  /**
+   * Signs a message (not message hash).
+   */
+  function signUnhashed(msg: Uint8Array, privKey: PrivKey, opts = defaultSigOpts): Signature {
+    return sign(CURVE.hash(ut.ensureBytes(msg)), privKey, opts);
+  }
+
   // Enable precomputes. Slows down first publicKey computation by 20ms.
   Point.BASE._setWindowSize(8);
 
@@ -1234,7 +1271,7 @@ export function weierstrass(curveDef: CurveType): CurveFn {
           signature = Signature.fromCompact(signature as Hex);
         }
       }
-      msgHash = ensureBytes(msgHash);
+      msgHash = ut.ensureBytes(msgHash);
     } catch (error) {
       return false;
     }
@@ -1265,6 +1302,7 @@ export function weierstrass(curveDef: CurveType): CurveFn {
     getPublicKey,
     getSharedSecret,
     sign,
+    signUnhashed,
     verify,
     Point,
     ProjectivePoint,

src/bls12-381.ts

@@ -1,4 +1,16 @@
 /*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
+
+// The 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.
+// Differences from @noble/bls12-381 1.4:
+// - PointG1 -> G1.Point
+// - PointG2 -> G2.Point
+// - PointG2.fromSignature -> Signature.decode
+// - PointG2.toSignature ->  Signature.encode
+// - Fixed Fp2 ORDER
+// - Points now have only two coordinates
+
 import { sha256 } from '@noble/hashes/sha256';
 import { randomBytes } from '@noble/hashes/utils';
 import { bls, CurveFn } from './abstract/bls.js';
@@ -23,14 +35,6 @@ import {
 } from './abstract/weierstrass.js';
 import { isogenyMap } from './abstract/hash-to-curve.js';
 
-// Differences from bls12-381:
-// - PointG1 -> G1.Point
-// - PointG2 -> G2.Point
-// - PointG2.fromSignature -> Signature.decode
-// - PointG2.toSignature ->  Signature.encode
-// - Fixed Fp2 ORDER
-// Points now have only two coordinates
-
 // CURVE FIELDS
 // Finite field over p.
 const Fp =
@@ -131,6 +135,7 @@ const Fp2: mod.Field<Fp2> & Fp2Utils = {
     return { c0: Fp.mul(factor, Fp.create(a)), c1: Fp.mul(factor, Fp.create(-b)) };
   },
   sqrt: (num) => {
+    if (Fp2.equals(num, Fp2.ZERO)) return Fp2.ZERO; // Algo doesn't handles this case
     // TODO: Optimize this line. It's extremely slow.
     // Speeding this up would boost aggregateSignatures.
     // https://eprint.iacr.org/2012/685.pdf applicable?

src/ed25519.ts

@@ -260,7 +260,7 @@ const invertSqrt = (number: bigint) => uvRatio(_1n, number);
 
 const MAX_255B = BigInt('0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff');
 const bytes255ToNumberLE = (bytes: Uint8Array) =>
-  ed25519.utils.mod(bytesToNumberLE(bytes) & MAX_255B);
+  ed25519.CURVE.Fp.create(bytesToNumberLE(bytes) & MAX_255B);
 
 type ExtendedPoint = ExtendedPointType;
 
@@ -269,7 +269,7 @@ type ExtendedPoint = ExtendedPointType;
 function calcElligatorRistrettoMap(r0: bigint): ExtendedPoint {
   const { d } = ed25519.CURVE;
   const P = ed25519.CURVE.Fp.ORDER;
-  const { mod } = ed25519.utils;
+  const mod = ed25519.CURVE.Fp.create;
   const r = mod(SQRT_M1 * r0 * r0); // 1
   const Ns = mod((r + _1n) * ONE_MINUS_D_SQ); // 2
   let c = BigInt(-1); // 3
@@ -327,7 +327,7 @@ export class RistrettoPoint {
     hex = ensureBytes(hex, 32);
     const { a, d } = ed25519.CURVE;
     const P = ed25519.CURVE.Fp.ORDER;
-    const { mod } = ed25519.utils;
+    const mod = ed25519.CURVE.Fp.create;
     const emsg = 'RistrettoPoint.fromHex: the hex is not valid encoding of RistrettoPoint';
     const s = bytes255ToNumberLE(hex);
     // 1. Check that s_bytes is the canonical encoding of a field element, or else abort.
@@ -357,7 +357,7 @@ export class RistrettoPoint {
   toRawBytes(): Uint8Array {
     let { x, y, z, t } = this.ep;
     const P = ed25519.CURVE.Fp.ORDER;
-    const { mod } = ed25519.utils;
+    const mod = ed25519.CURVE.Fp.create;
     const u1 = mod(mod(z + y) * mod(z - y)); // 1
     const u2 = mod(x * y); // 2
     // Square root always exists
@@ -395,7 +395,7 @@ export class RistrettoPoint {
     assertRstPoint(other);
     const a = this.ep;
     const b = other.ep;
-    const { mod } = ed25519.utils;
+    const mod = ed25519.CURVE.Fp.create;
     // (x1 * y2 == y1 * x2) | (y1 * y2 == x1 * x2)
     const one = mod(a.x * b.y) === mod(a.y * b.x);
     const two = mod(a.y * b.y) === mod(a.x * b.x);

src/ed448.ts

@@ -138,7 +138,8 @@ const ED448_DEF = {
   ),
   // Finite field 𝔽p over which we'll do calculations; 2n ** 448n - 2n ** 224n - 1n
   Fp,
-  // Subgroup order: how many points ed448 has; 2n**446n - 13818066809895115352007386748515426880336692474882178609894547503885n
+  // Subgroup order: how many points curve has;
+  // 2n**446n - 13818066809895115352007386748515426880336692474882178609894547503885n
   n: BigInt(
     '181709681073901722637330951972001133588410340171829515070372549795146003961539585716195755291692375963310293709091662304773755859649779'
   ),

src/jubjub.ts

@@ -1,5 +1,5 @@
 /*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
-import { sha256 } from '@noble/hashes/sha256';
+import { sha512 } from '@noble/hashes/sha512';
 import { concatBytes, randomBytes, utf8ToBytes } from '@noble/hashes/utils';
 import { twistedEdwards } from './abstract/edwards.js';
 import { blake2s } from '@noble/hashes/blake2s';
@@ -8,6 +8,7 @@ import { Fp } from './abstract/modular.js';
 /**
  * jubjub Twisted Edwards curve.
  * https://neuromancer.sk/std/other/JubJub
+ * jubjub does not use EdDSA, so `hash`/sha512 params are passed because interface expects them.
  */
 
 export const jubjub = twistedEdwards({
@@ -15,16 +16,16 @@ export const jubjub = twistedEdwards({
   a: BigInt('0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000000'),
   d: BigInt('0x2a9318e74bfa2b48f5fd9207e6bd7fd4292d7f6d37579d2601065fd6d6343eb1'),
   // Finite field 𝔽p over which we'll do calculations
+  // Same value as bls12-381 Fr (not Fp)
   Fp: Fp(BigInt('0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001')),
-  // Subgroup order: how many points ed25519 has
+  // Subgroup order: how many points curve has
-  // 2n ** 252n + 27742317777372353535851937790883648493n;
   n: BigInt('0xe7db4ea6533afa906673b0101343b00a6682093ccc81082d0970e5ed6f72cb7'),
   // Cofactor
   h: BigInt(8),
   // Base point (x, y) aka generator point
   Gx: BigInt('0x11dafe5d23e1218086a365b99fbf3d3be72f6afd7d1f72623e6b071492d1122b'),
   Gy: BigInt('0x1d523cf1ddab1a1793132e78c866c0c33e26ba5cc220fed7cc3f870e59d292aa'),
-  hash: sha256,
+  hash: sha512,
   randomBytes,
 } as const);
 

src/secp256k1.ts

@@ -15,10 +15,8 @@ import { randomBytes } from '@noble/hashes/utils';
 import { isogenyMap } from './abstract/hash-to-curve.js';
 
 /**
- * secp256k1 belongs to Koblitz curves: it has
+ * secp256k1 belongs to Koblitz curves: it has efficiently computable endomorphism.
- * efficiently computable Frobenius endomorphism.
+ * Endomorphism uses 2x less RAM, speeds up precomputation by 2x and ECDH / key recovery by 20%.
- * Endomorphism improves efficiency:
- * Uses 2x less RAM, speeds up precomputation by 2x and ECDH / sign key recovery by 20%.
  * Should always be used for Projective's double-and-add multiplication.
  * For affines cached multiplication, it trades off 1/2 init time & 1/3 ram for 20% perf hit.
  * https://gist.github.com/paulmillr/eb670806793e84df628a7c434a873066
@@ -129,7 +127,7 @@ export const secp256k1 = createCurve(
         const b1 = -_1n * BigInt('0xe4437ed6010e88286f547fa90abfe4c3');
         const a2 = BigInt('0x114ca50f7a8e2f3f657c1108d9d44cfd8');
         const b2 = a1;
-        const POW_2_128 = BigInt('0x100000000000000000000000000000000');
+        const POW_2_128 = BigInt('0x100000000000000000000000000000000'); // (2n**128n).toString(16)
 
         const c1 = divNearest(b2 * k, n);
         const c2 = divNearest(-b1 * k, n);
@@ -175,7 +173,7 @@ function normalizePublicKey(publicKey: Hex | PointType<bigint>): PointType<bigin
   } else {
     const bytes = ensureBytes(publicKey);
     // Schnorr is 32 bytes
-    if (bytes.length === 32) {
+    if (bytes.length !== 32) throw new Error('Schnorr pubkeys must be 32 bytes');
     const x = bytesToNumberBE(bytes);
     if (!isValidFieldElement(x)) throw new Error('Point is not on curve');
     const y2 = secp256k1.utils._weierstrassEquation(x); // y² = x³ + ax + b
@@ -187,9 +185,6 @@ function normalizePublicKey(publicKey: Hex | PointType<bigint>): PointType<bigin
     point.assertValidity();
     return point;
   }
-    // Do we need that in schnorr at all?
-    return secp256k1.Point.fromHex(publicKey);
-  }
 }
 
 const isWithinCurveOrder = secp256k1.utils._isWithinCurveOrder;
@@ -227,10 +222,13 @@ class SchnorrSignature {
   }
   static fromHex(hex: Hex) {
     const bytes = ensureBytes(hex);
-    if (bytes.length !== 64)
+    const len = 32; // group length
-      throw new TypeError(`SchnorrSignature.fromHex: expected 64 bytes, not ${bytes.length}`);
+    if (bytes.length !== 2 * len)
-    const r = bytesToNumberBE(bytes.subarray(0, 32));
+      throw new TypeError(
-    const s = bytesToNumberBE(bytes.subarray(32, 64));
+        `SchnorrSignature.fromHex: expected ${2 * len} bytes, not ${bytes.length}`
+      );
+    const r = bytesToNumberBE(bytes.subarray(0, len));
+    const s = bytesToNumberBE(bytes.subarray(len, 2 * len));
     return new SchnorrSignature(r, s);
   }
   assertValidity() {

src/stark.ts

@@ -138,11 +138,12 @@ function hashKeyWithIndex(key: Uint8Array, index: number) {
 export function grindKey(seed: Hex) {
   const _seed = ensureBytes0x(seed);
   const sha256mask = 2n ** 256n;
-  const limit = sha256mask - starkCurve.utils.mod(sha256mask, starkCurve.CURVE.n);
+  const Fn = Fp(CURVE.n);
+  const limit = sha256mask - Fn.create(sha256mask);
   for (let i = 0; ; i++) {
     const key = hashKeyWithIndex(_seed, i);
     // key should be in [0, limit)
-    if (key < limit) return starkCurve.utils.mod(key, starkCurve.CURVE.n).toString(16);
+    if (key < limit) return Fn.create(key).toString(16);
   }
 }
 

test/basic.test.js

@@ -1,7 +1,8 @@
 import { deepStrictEqual, throws } from 'assert';
-import { should } from 'micro-should';
+import { should, describe } from 'micro-should';
 import * as fc from 'fast-check';
 import * as mod from '../lib/esm/abstract/modular.js';
+import { bytesToHex as toHex } from '../lib/esm/abstract/utils.js';
 // Generic tests for all curves in package
 import { secp192r1 } from '../lib/esm/p192.js';
 import { secp224r1 } from '../lib/esm/p224.js';
@@ -61,7 +62,8 @@ for (const c in FIELDS) {
     const FC_BIGINT = curve[f][1] ? curve[f][1] : fc.bigInt(1n, Fp.ORDER - 1n);
 
     const create = curve[f][2] ? curve[f][2].bind(null, Fp) : (num) => Fp.create(num);
-    should(`${name} equality`, () => {
+    describe(name, () => {
+      should('equality', () => {
         fc.assert(
           fc.property(FC_BIGINT, (num) => {
             const a = create(num);
@@ -71,7 +73,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} non-equality`, () => {
+      should('non-equality', () => {
         fc.assert(
           fc.property(FC_BIGINT, FC_BIGINT, (num1, num2) => {
             const a = create(num1);
@@ -81,7 +83,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} add/subtract/commutativity`, () => {
+      should('add/subtract/commutativity', () => {
         fc.assert(
           fc.property(FC_BIGINT, FC_BIGINT, (num1, num2) => {
             const a = create(num1);
@@ -90,7 +92,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} add/subtract/associativity`, () => {
+      should('add/subtract/associativity', () => {
         fc.assert(
           fc.property(FC_BIGINT, FC_BIGINT, FC_BIGINT, (num1, num2, num3) => {
             const a = create(num1);
@@ -100,7 +102,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} add/subtract/x+0=x`, () => {
+      should('add/subtract/x+0=x', () => {
         fc.assert(
           fc.property(FC_BIGINT, (num) => {
             const a = create(num);
@@ -108,7 +110,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} add/subtract/x-0=x`, () => {
+      should('add/subtract/x-0=x', () => {
         fc.assert(
           fc.property(FC_BIGINT, (num) => {
             const a = create(num);
@@ -117,7 +119,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} add/subtract/negate equality`, () => {
+      should('add/subtract/negate equality', () => {
         fc.assert(
           fc.property(FC_BIGINT, (num1) => {
             const a = create(num1);
@@ -128,7 +130,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} add/subtract/negate`, () => {
+      should('add/subtract/negate', () => {
         fc.assert(
           fc.property(FC_BIGINT, (num) => {
             const a = create(num);
@@ -137,8 +139,11 @@ for (const c in FIELDS) {
           })
         );
       });
+      should('negate(0)', () => {
+        deepStrictEqual(Fp.negate(Fp.ZERO), Fp.ZERO);
+      });
 
-    should(`${name} multiply/commutativity`, () => {
+      should('multiply/commutativity', () => {
         fc.assert(
           fc.property(FC_BIGINT, FC_BIGINT, (num1, num2) => {
             const a = create(num1);
@@ -147,7 +152,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} multiply/associativity`, () => {
+      should('multiply/associativity', () => {
         fc.assert(
           fc.property(FC_BIGINT, FC_BIGINT, FC_BIGINT, (num1, num2, num3) => {
             const a = create(num1);
@@ -157,7 +162,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} multiply/distributivity`, () => {
+      should('multiply/distributivity', () => {
         fc.assert(
           fc.property(FC_BIGINT, FC_BIGINT, FC_BIGINT, (num1, num2, num3) => {
             const a = create(num1);
@@ -167,7 +172,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} multiply/add equality`, () => {
+      should('multiply/add equality', () => {
         fc.assert(
           fc.property(FC_BIGINT, (num) => {
             const a = create(num);
@@ -181,7 +186,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} multiply/square equality`, () => {
+      should('multiply/square equality', () => {
         fc.assert(
           fc.property(FC_BIGINT, (num) => {
             const a = create(num);
@@ -189,7 +194,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} multiply/pow equality`, () => {
+      should('multiply/pow equality', () => {
         fc.assert(
           fc.property(FC_BIGINT, (num) => {
             const a = create(num);
@@ -200,9 +205,27 @@ for (const c in FIELDS) {
           })
         );
       });
+
+      should('square(0)', () => {
+        deepStrictEqual(Fp.square(Fp.ZERO), Fp.ZERO);
+        deepStrictEqual(Fp.mul(Fp.ZERO, Fp.ZERO), Fp.ZERO);
+      });
+
+      should('square(1)', () => {
+        deepStrictEqual(Fp.square(Fp.ONE), Fp.ONE);
+        deepStrictEqual(Fp.mul(Fp.ONE, Fp.ONE), Fp.ONE);
+      });
+
+      should('square(-1)', () => {
+        const minus1 = Fp.negate(Fp.ONE);
+        deepStrictEqual(Fp.square(minus1), Fp.ONE);
+        deepStrictEqual(Fp.mul(minus1, minus1), Fp.ONE);
+      });
+
       const isSquare = mod.FpIsSquare(Fp);
-    should(`${name} multiply/sqrt`, () => {
+      // Not implemented
-      if (Fp === bls12_381.CURVE.Fp12) return; // Not implemented
+      if (Fp !== bls12_381.CURVE.Fp12) {
+        should('multiply/sqrt', () => {
           fc.assert(
             fc.property(FC_BIGINT, (num) => {
               const a = create(num);
@@ -220,7 +243,18 @@ for (const c in FIELDS) {
           );
         });
 
-    should(`${name} div/division by one equality`, () => {
+        should('sqrt(0)', () => {
+          deepStrictEqual(Fp.sqrt(Fp.ZERO), Fp.ZERO);
+          const sqrt1 = Fp.sqrt(Fp.ONE);
+          deepStrictEqual(
+            Fp.equals(sqrt1, Fp.ONE) || Fp.equals(sqrt1, Fp.negate(Fp.ONE)),
+            true,
+            'sqrt(1) = 1 or -1'
+          );
+        });
+      }
+
+      should('div/division by one equality', () => {
         fc.assert(
           fc.property(FC_BIGINT, (num) => {
             const a = create(num);
@@ -230,7 +264,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} zero division equality`, () => {
+      should('zero division equality', () => {
         fc.assert(
           fc.property(FC_BIGINT, (num) => {
             const a = create(num);
@@ -238,7 +272,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} div/division distributivity`, () => {
+      should('div/division distributivity', () => {
         fc.assert(
           fc.property(FC_BIGINT, FC_BIGINT, FC_BIGINT, (num1, num2, num3) => {
             const a = create(num1);
@@ -248,7 +282,7 @@ for (const c in FIELDS) {
           })
         );
       });
-    should(`${name} div/division and multiplication equality`, () => {
+      should('div/division and multiplication equality', () => {
         fc.assert(
           fc.property(FC_BIGINT, FC_BIGINT, (num1, num2) => {
             const a = create(num1);
@@ -257,6 +291,7 @@ for (const c in FIELDS) {
           })
         );
       });
+    });
   }
 }
 
@@ -278,12 +313,12 @@ const NUM_RUNS = 5;
 const getXY = (p) => ({ x: p.x, y: p.y });
 
 function equal(a, b, comment) {
-  deepStrictEqual(a.equals(b), true, `eq(${comment})`);
+  deepStrictEqual(a.equals(b), true, 'eq(${comment})');
   if (a.toAffine && b.toAffine) {
-    deepStrictEqual(getXY(a.toAffine()), getXY(b.toAffine()), `eqToAffine(${comment})`);
+    deepStrictEqual(getXY(a.toAffine()), getXY(b.toAffine()), 'eqToAffine(${comment})');
   } else if (!a.toAffine && !b.toAffine) {
     // Already affine
-    deepStrictEqual(getXY(a), getXY(b), `eqAffine(${comment})`);
+    deepStrictEqual(getXY(a), getXY(b), 'eqAffine(${comment})');
   } else throw new Error('Different point types');
 }
 
@@ -308,46 +343,49 @@ for (const name in CURVES) {
 
     const G = [p.ZERO, p.BASE];
     for (let i = 2; i < 10; i++) G.push(G[1].multiply(i));
+    const title = `${name}/${pointName}`;
+    describe(title, () => {
+      describe('basic group laws', () => {
         // Here we check basic group laws, to verify that points works as group
-    should(`${name}/${pointName}/Basic group laws (zero)`, () => {
+        should('(zero)', () => {
           equal(G[0].double(), G[0], '(0*G).double() = 0');
           equal(G[0].add(G[0]), G[0], '0*G + 0*G = 0');
           equal(G[0].subtract(G[0]), G[0], '0*G - 0*G = 0');
           equal(G[0].negate(), G[0], '-0 = 0');
           for (let i = 0; i < G.length; i++) {
             const p = G[i];
-        equal(p, p.add(G[0]), `${i}*G + 0 = ${i}*G`);
+            equal(p, p.add(G[0]), '${i}*G + 0 = ${i}*G');
-        equal(G[0].multiply(i + 1), G[0], `${i + 1}*0 = 0`);
+            equal(G[0].multiply(i + 1), G[0], '${i + 1}*0 = 0');
           }
         });
-    should(`${name}/${pointName}/Basic group laws (one)`, () => {
+        should('(one)', () => {
           equal(G[1].double(), G[2], '(1*G).double() = 2*G');
           equal(G[1].subtract(G[1]), G[0], '1*G - 1*G = 0');
           equal(G[1].add(G[1]), G[2], '1*G + 1*G = 2*G');
         });
-    should(`${name}/${pointName}/Basic group laws (sanity tests)`, () => {
+        should('(sanity tests)', () => {
-      equal(G[2].double(), G[4], `(2*G).double() = 4*G`);
+          equal(G[2].double(), G[4], '(2*G).double() = 4*G');
-      equal(G[2].add(G[2]), G[4], `2*G + 2*G = 4*G`);
+          equal(G[2].add(G[2]), G[4], '2*G + 2*G = 4*G');
-      equal(G[7].add(G[3].negate()), G[4], `7*G - 3*G = 4*G`);
+          equal(G[7].add(G[3].negate()), G[4], '7*G - 3*G = 4*G');
         });
-    should(`${name}/${pointName}/Basic group laws (addition commutativity)`, () => {
+        should('(addition commutativity)', () => {
-      equal(G[4].add(G[3]), G[3].add(G[4]), `4*G + 3*G = 3*G + 4*G`);
+          equal(G[4].add(G[3]), G[3].add(G[4]), '4*G + 3*G = 3*G + 4*G');
-      equal(G[4].add(G[3]), G[3].add(G[2]).add(G[2]), `4*G + 3*G = 3*G + 2*G + 2*G`);
+          equal(G[4].add(G[3]), G[3].add(G[2]).add(G[2]), '4*G + 3*G = 3*G + 2*G + 2*G');
         });
-    should(`${name}/${pointName}/Basic group laws (double)`, () => {
+        should('(double)', () => {
           equal(G[3].double(), G[6], '(3*G).double() = 6*G');
         });
-    should(`${name}/${pointName}/Basic group laws (multiply)`, () => {
+        should('(multiply)', () => {
           equal(G[2].multiply(3), G[6], '(2*G).multiply(3) = 6*G');
         });
-    should(`${name}/${pointName}/Basic group laws (same point addition)`, () => {
+        should('(same point addition)', () => {
-      equal(G[3].add(G[3]), G[6], `3*G + 3*G = 6*G`);
+          equal(G[3].add(G[3]), G[6], '3*G + 3*G = 6*G');
         });
-    should(`${name}/${pointName}/Basic group laws (same point (negative) addition)`, () => {
+        should('(same point (negative) addition)', () => {
           equal(G[3].add(G[3].negate()), G[0], '3*G + (- 3*G) = 0*G');
           equal(G[3].subtract(G[3]), G[0], '3*G - 3*G = 0*G');
         });
-    should(`${name}/${pointName}/Basic group laws (curve order)`, () => {
+        should('(curve order)', () => {
           equal(G[1].multiply(CURVE_ORDER - 1n).add(G[1]), G[0], '(N-1)*G + G = 0');
           equal(G[1].multiply(CURVE_ORDER - 1n).add(G[2]), G[1], '(N-1)*G + 2*G = 1*G');
           equal(G[1].multiply(CURVE_ORDER - 2n).add(G[2]), G[0], '(N-2)*G + 2*G = 0');
@@ -355,7 +393,7 @@ for (const name in CURVES) {
           const carry = CURVE_ORDER % 2n === 1n ? G[1] : G[0];
           equal(G[1].multiply(half).double().add(carry), G[0], '((N/2) * G).double() = 0');
         });
-    should(`${name}/${pointName}/Basic group laws (inversion)`, () => {
+        should('(inversion)', () => {
           const a = 1234n;
           const b = 5678n;
           const c = a * b;
@@ -363,7 +401,7 @@ for (const name in CURVES) {
           const inv = mod.invert(b, CURVE_ORDER);
           equal(G[1].multiply(c).multiply(inv), G[1].multiply(a), 'c*G * (1/b)*G = a*G');
         });
-    should(`${name}/${pointName}/Basic group laws (multiply, rand)`, () =>
+        should('(multiply, rand)', () =>
           fc.assert(
             fc.property(FC_BIGINT, FC_BIGINT, (a, b) => {
               const c = mod.mod(a + b, CURVE_ORDER);
@@ -371,27 +409,29 @@ for (const name in CURVES) {
               const pA = G[1].multiply(a);
               const pB = G[1].multiply(b);
               const pC = G[1].multiply(c);
-          equal(pA.add(pB), pB.add(pA), `pA + pB = pB + pA`);
+              equal(pA.add(pB), pB.add(pA), 'pA + pB = pB + pA');
-          equal(pA.add(pB), pC, `pA + pB = pC`);
+              equal(pA.add(pB), pC, 'pA + pB = pC');
             }),
             { numRuns: NUM_RUNS }
           )
         );
-    should(`${name}/${pointName}/Basic group laws (multiply2, rand)`, () =>
+        should('(multiply2, rand)', () =>
           fc.assert(
             fc.property(FC_BIGINT, FC_BIGINT, (a, b) => {
               const c = mod.mod(a * b, CURVE_ORDER);
               const pA = G[1].multiply(a);
               const pB = G[1].multiply(b);
-          equal(pA.multiply(b), pB.multiply(a), `b*pA = a*pB`);
+              equal(pA.multiply(b), pB.multiply(a), 'b*pA = a*pB');
-          equal(pA.multiply(b), G[1].multiply(c), `b*pA = c*G`);
+              equal(pA.multiply(b), G[1].multiply(c), 'b*pA = c*G');
             }),
             { numRuns: NUM_RUNS }
           )
         );
+      });
 
       for (const op of ['add', 'subtract']) {
-      should(`${name}/${pointName}/${op} type check`, () => {
+        describe(op, () => {
+          should('type check', () => {
             throws(() => G[1][op](0), '0');
             throws(() => G[1][op](0n), '0n');
             G[1][op](G[2]);
@@ -399,17 +439,21 @@ for (const name in CURVES) {
             throws(() => G[1][op](123.456), '123.456');
             throws(() => G[1][op](true), 'true');
             throws(() => G[1][op]('1'), "'1'");
-        throws(() => G[1][op]({ x: 1n, y: 1n, z: 1n, t: 1n }), '{ x: 1n, y: 1n, z: 1n, t: 1n }');
+            throws(
+              () => G[1][op]({ x: 1n, y: 1n, z: 1n, t: 1n }),
+              '{ x: 1n, y: 1n, z: 1n, t: 1n }'
+            );
             throws(() => G[1][op](new Uint8Array([])), 'ui8a([])');
             throws(() => G[1][op](new Uint8Array([0])), 'ui8a([0])');
             throws(() => G[1][op](new Uint8Array([1])), 'ui8a([1])');
             throws(() => G[1][op](new Uint8Array(4096).fill(1)), 'ui8a(4096*[1])');
-        if (G[1].toAffine) throws(() => G[1][op](C.Point.BASE), `Point ${op} ${pointName}`);
+            if (G[1].toAffine) throws(() => G[1][op](C.Point.BASE), 'Point ${op} ${pointName}');
-        throws(() => G[1][op](o.BASE), `${op}/other curve point`);
+            throws(() => G[1][op](o.BASE), '${op}/other curve point');
+          });
         });
       }
 
-    should(`${name}/${pointName}/equals type check`, () => {
+      should('equals type check', () => {
         throws(() => G[1].equals(0), '0');
         throws(() => G[1].equals(0n), '0n');
         deepStrictEqual(G[1].equals(G[2]), false, '1*G != 2*G');
@@ -424,13 +468,14 @@ for (const name in CURVES) {
         throws(() => G[1].equals(new Uint8Array([0])), 'ui8a([0])');
         throws(() => G[1].equals(new Uint8Array([1])), 'ui8a([1])');
         throws(() => G[1].equals(new Uint8Array(4096).fill(1)), 'ui8a(4096*[1])');
-      if (G[1].toAffine) throws(() => G[1].equals(C.Point.BASE), `Point.equals(${pointName})`);
+        if (G[1].toAffine) throws(() => G[1].equals(C.Point.BASE), 'Point.equals(${pointName})');
         throws(() => G[1].equals(o.BASE), 'other curve point');
       });
 
       for (const op of ['multiply', 'multiplyUnsafe']) {
         if (!p.BASE[op]) continue;
-      should(`${name}/${pointName}/${op} type check`, () => {
+        describe(op, () => {
+          should('type check', () => {
             if (op !== 'multiplyUnsafe') {
               throws(() => G[1][op](0), '0');
               throws(() => G[1][op](0n), '0n');
@@ -449,23 +494,24 @@ for (const name in CURVES) {
             throws(() => G[1][op](new Uint8Array(4096).fill(1)), 'ui8a(4096*[1])');
             throws(() => G[1][op](o.BASE), 'other curve point');
           });
+        });
       }
       // Complex point (Extended/Jacobian/Projective?)
       if (p.BASE.toAffine) {
-      should(`${name}/${pointName}/toAffine()`, () => {
+        should('toAffine()', () => {
-        equal(p.ZERO.toAffine(), C.Point.ZERO, `0 = 0`);
+          equal(p.ZERO.toAffine(), C.Point.ZERO, '0 = 0');
-        equal(p.BASE.toAffine(), C.Point.BASE, `1 = 1`);
+          equal(p.BASE.toAffine(), C.Point.BASE, '1 = 1');
         });
       }
       if (p.fromAffine) {
-      should(`${name}/${pointName}/fromAffine()`, () => {
+        should('fromAffine()', () => {
-        equal(p.ZERO, p.fromAffine(C.Point.ZERO), `0 = 0`);
+          equal(p.ZERO, p.fromAffine(C.Point.ZERO), '0 = 0');
-        equal(p.BASE, p.fromAffine(C.Point.BASE), `1 = 1`);
+          equal(p.BASE, p.fromAffine(C.Point.BASE), '1 = 1');
         });
       }
       // toHex/fromHex (if available)
       if (p.fromHex && p.BASE.toHex) {
-      should(`${name}/${pointName}/fromHex(toHex()) roundtrip`, () => {
+        should('fromHex(toHex()) roundtrip', () => {
           fc.assert(
             fc.property(FC_BIGINT, (x) => {
               const hex = p.BASE.multiply(x).toHex();
@@ -474,9 +520,11 @@ for (const name in CURVES) {
           );
         });
       }
+    });
   }
+  describe(name, () => {
     // Generic complex things (getPublicKey/sign/verify/getSharedSecret)
-  should(`${name}/getPublicKey type check`, () => {
+    should('getPublicKey type check', () => {
       throws(() => C.getPublicKey(0), '0');
       throws(() => C.getPublicKey(0n), '0n');
       throws(() => C.getPublicKey(false), 'false');
@@ -491,23 +539,27 @@ for (const name in CURVES) {
       throws(() => C.getPublicKey(new Uint8Array([1])));
       throws(() => C.getPublicKey(new Uint8Array(4096).fill(1)));
     });
-  should(`${name}.verify()/should verify random signatures`, () =>
+    should('.verify() should verify random signatures', () =>
       fc.assert(
         fc.property(fc.hexaString({ minLength: 64, maxLength: 64 }), (msg) => {
           const priv = C.utils.randomPrivateKey();
           const pub = C.getPublicKey(priv);
           const sig = C.sign(msg, priv);
-        deepStrictEqual(C.verify(sig, msg, pub), true);
+          deepStrictEqual(
+            C.verify(sig, msg, pub),
+            true,
+            'priv=${toHex(priv)},pub=${toHex(pub)},msg=${msg}'
+          );
         }),
         { numRuns: NUM_RUNS }
       )
     );
-  should(`${name}.sign()/edge cases`, () => {
+    should('.sign() edge cases', () => {
       throws(() => C.sign());
       throws(() => C.sign(''));
     });
 
-  should(`${name}.verify()/should not verify signature with wrong hash`, () => {
+    should('.verify() should not verify signature with wrong hash', () => {
       const MSG = '01'.repeat(32);
       const PRIV_KEY = 0x2n;
       const WRONG_MSG = '11'.repeat(32);
@@ -517,7 +569,7 @@ for (const name in CURVES) {
     });
     // NOTE: fails for ed, because of empty message. Since we convert it to scalar,
     // need to check what other implementations do. Empty message != new Uint8Array([0]), but what scalar should be in that case?
-  // should(`${name}/should not verify signature with wrong message`, () => {
+    // should('should not verify signature with wrong message', () => {
     //   fc.assert(
     //     fc.property(
     //       fc.array(fc.integer({ min: 0x00, max: 0xff })),
@@ -539,7 +591,7 @@ for (const name in CURVES) {
     // });
 
     if (C.getSharedSecret) {
-    should(`${name}/getSharedSecret() should be commutative`, () => {
+      should('getSharedSecret() should be commutative', () => {
         for (let i = 0; i < NUM_RUNS; i++) {
           const asec = C.utils.randomPrivateKey();
           const apub = C.getPublicKey(asec);
@@ -554,7 +606,24 @@ for (const name in CURVES) {
         }
       });
     }
+  });
 }
+
+should('secp224k1 sqrt bug', () => {
+  const { Fp } = secp224r1.CURVE;
+  const sqrtMinus1 = Fp.sqrt(-1n);
+  // Verified against sage
+  deepStrictEqual(
+    sqrtMinus1,
+    23621584063597419797792593680131996961517196803742576047493035507225n
+  );
+  deepStrictEqual(
+    Fp.negate(sqrtMinus1),
+    3338362603553219996874421406887633712040719456283732096017030791656n
+  );
+  deepStrictEqual(Fp.square(sqrtMinus1), Fp.create(-1n));
+});
+
 // ESM is broken.
 import url from 'url';
 if (import.meta.url === url.pathToFileURL(process.argv[1]).href) {

test/bls12-381.test.js

@@ -1,5 +1,5 @@
 import { bls12_381 } from '../lib/esm/bls12-381.js';
-import { should } from 'micro-should';
+import { describe, should } from 'micro-should';
 import { deepStrictEqual, notDeepStrictEqual, throws } from 'assert';
 import { sha512 } from '@noble/hashes/sha512';
 import * as fc from 'fast-check';
@@ -38,11 +38,11 @@ const B_384_40 = '40'.padEnd(384, '0'); // [0x40, 0, 0...]
 const getPubKey = (priv) => bls.getPublicKey(priv);
 
 // Fp
-{
+describe('bls12-381 Fp', () => {
   const Fp = bls.Fp;
   const FC_BIGINT = fc.bigInt(1n, Fp.ORDER - 1n);
 
-  should('bls12-381/Fp/multiply/sqrt', () => {
+  should('multiply/sqrt', () => {
     let sqr1 = Fp.sqrt(Fp.create(300855555557n));
     deepStrictEqual(
       sqr1 && sqr1.toString(),
@@ -50,16 +50,16 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
     );
     throws(() => Fp.sqrt(Fp.create(72057594037927816n)));
   });
-}
+});
 
 // Fp2
-{
+describe('bls12-381 Fp2', () => {
   const Fp = bls.Fp;
   const Fp2 = bls.Fp2;
   const FC_BIGINT = fc.bigInt(1n, Fp.ORDER - 1n);
   const FC_BIGINT_2 = fc.array(FC_BIGINT, { minLength: 2, maxLength: 2 });
 
-  should('bls12-381 Fp2/non-equality', () => {
+  should('non-equality', () => {
     fc.assert(
       fc.property(FC_BIGINT_2, FC_BIGINT_2, (num1, num2) => {
         const a = Fp2.fromBigTuple([num1[0], num1[1]]);
@@ -70,7 +70,7 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
     );
   });
 
-  should('bls12-381 Fp2/div/x/1=x', () => {
+  should('div/x/1=x', () => {
     fc.assert(
       fc.property(FC_BIGINT_2, (num) => {
         const a = Fp2.fromBigTuple([num[0], num[1]]);
@@ -81,7 +81,7 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
     );
   });
 
-  should('bls12-381 Fp2/frobenius', () => {
+  should('frobenius', () => {
     // expect(Fp2.FROBENIUS_COEFFICIENTS[0].equals(Fp.ONE)).toBe(true);
     // expect(
     //   Fp2.FROBENIUS_COEFFICIENTS[1].equals(
@@ -139,16 +139,17 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
       true
     );
   });
-}
+});
 
 // Point
-{
+describe('bls12-381 Point', () => {
   const Fp = bls.Fp;
   const FC_BIGINT = fc.bigInt(1n, Fp.ORDER - 1n);
   const PointG1 = bls.G1.Point;
   const PointG2 = bls.G2.Point;
 
-  should('bls12-381 Point/Point with Fp coordinates/Point equality', () => {
+  describe('with Fp coordinates', () => {
+    should('Point equality', () => {
       fc.assert(
         fc.property(
           fc.array(FC_BIGINT, { minLength: 3, maxLength: 3 }),
@@ -164,16 +165,16 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
         )
       );
     });
-  should('bls12-381 Point/Point with Fp coordinates/should be placed on curve vector 1', () => {
+    should('be placed on curve vector 1', () => {
       const a = new PointG1(Fp.create(0n), Fp.create(0n));
       a.assertValidity();
     });
-  should('bls12-381 Point/Point with Fp coordinates/should not be placed on curve vector 1', () => {
+    should('not be placed on curve vector 1', () => {
       const a = new PointG1(Fp.create(0n), Fp.create(1n));
       throws(() => a.assertValidity());
     });
 
-  should('bls12-381 Point/Point with Fp coordinates/should be placed on curve vector 2', () => {
+    should('be placed on curve vector 2', () => {
       const a = new PointG1(
         Fp.create(
           0x17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bbn
@@ -184,7 +185,7 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
       );
       a.assertValidity();
     });
-  should('bls12-381 Point/Point with Fp coordinates/should be placed on curve vector 3', () => {
+    should('be placed on curve vector 3', () => {
       const a = new PointG1(
         Fp.create(
           3971675556538908004130084773503021351583407620890695272226385332452194486153316625183061567093226342405194446632851n
@@ -196,7 +197,7 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
 
       a.assertValidity();
     });
-  should('bls12-381 Point/Point with Fp coordinates/should not be placed on curve vector 3', () => {
+    should('not be placed on curve vector 3', () => {
       const a = new PointG1(
         Fp.create(
           622186380008502900120948444810967255157373993223369845903602988014033704418470621816206856882891545628885272576827n
@@ -207,7 +208,7 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
       );
       throws(() => a.assertValidity());
     });
-  should('bls12-381 Point/Point with Fp coordinates/should not be placed on curve vector 2', () => {
+    should('not be placed on curve vector 2', () => {
       const a = new PointG1(
         Fp.create(
           0x17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6ban
@@ -219,9 +220,7 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
       throws(() => a.assertValidity());
     });
 
-  should(
+    should('be doubled and placed on curve vector 1', () => {
-    'bls12-381 Point/Point with Fp coordinates/should be doubled and placed on curve vector 1',
-    () => {
       const a = new PointG1(
         Fp.create(
           0x17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac586c55e83ff97a1aeffb3af00adb22c6bbn
@@ -245,11 +244,8 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
       );
       deepStrictEqual(double, a.multiply(2n));
       deepStrictEqual(double, a.add(a));
-    }
+    });
-  );
+    should('be pdoubled and placed on curve vector 2', () => {
-  should(
-    'bls12-381 Point/Point with Fp coordinates/should be pdoubled and placed on curve vector 2',
-    () => {
       const a = new PointG1(
         Fp.create(
           3971675556538908004130084773503021351583407620890695272226385332452194486153316625183061567093226342405194446632851n
@@ -273,9 +269,8 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
       );
       deepStrictEqual(double, a.multiply(2n));
       deepStrictEqual(double, a.add(a));
-    }
+    });
-  );
+    should('not validate incorrect point', () => {
-  should('bls12-381 Point/Point with Fp coordinates/should not validate incorrect point', () => {
       const x =
         499001545268060011619089734015590154568173930614466321429631711131511181286230338880376679848890024401335766847607n;
       const y =
@@ -284,8 +279,10 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
       const p = new PointG1(Fp.create(x), Fp.create(y));
       throws(() => p.assertValidity());
     });
+  });
 
-  should('bls12-381 Point/Point with Fp2 coordinates/Point equality', () => {
+  describe('with Fp2 coordinates', () => {
+    should('Point equality', () => {
       fc.assert(
         fc.property(
           fc.array(fc.array(FC_BIGINT, { minLength: 2, maxLength: 2 }), {
@@ -297,8 +294,16 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
             maxLength: 3,
           }),
           ([x1, y1, z1], [x2, y2, z2]) => {
-          const p1 = new PointG2(Fp2.fromBigTuple(x1), Fp2.fromBigTuple(y1), Fp2.fromBigTuple(z1));
+            const p1 = new PointG2(
-          const p2 = new PointG2(Fp2.fromBigTuple(x2), Fp2.fromBigTuple(y2), Fp2.fromBigTuple(z2));
+              Fp2.fromBigTuple(x1),
+              Fp2.fromBigTuple(y1),
+              Fp2.fromBigTuple(z1)
+            );
+            const p2 = new PointG2(
+              Fp2.fromBigTuple(x2),
+              Fp2.fromBigTuple(y2),
+              Fp2.fromBigTuple(z2)
+            );
             deepStrictEqual(p1.equals(p1), true);
             deepStrictEqual(p2.equals(p2), true);
             deepStrictEqual(p1.equals(p2), false);
@@ -307,11 +312,11 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
         )
       );
     });
-  // should('bls12-381 Point/Point with Fp2 coordinates/should be placed on curve vector 1', () => {
+    // should('be placed on curve vector 1', () => {
     //   const a = new PointG2(Fp2.fromBigTuple([0n, 0n]), Fp2.fromBigTuple([0n, 0n]));
     //   a.assertValidity();
     // });
-  should('bls12-381 Point/Point with Fp2 coordinates/should be placed on curve vector 2', () => {
+    should('be placed on curve vector 2', () => {
       const a = new PointG2(
         Fp2.fromBigTuple([
           0x024aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4510b647ae3d1770bac0326a805bbefd48056c8c121bdb8n,
@@ -325,7 +330,7 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
       );
       a.assertValidity();
     });
-  should('bls12-381 Point/Point with Fp2 coordinates/should be placed on curve vector 3', () => {
+    should('be placed on curve vector 3', () => {
       const a = new PointG2(
         Fp2.fromBigTuple([
           233289878585407360737561818812172281900488265436962145913969074168503452745466655442125797664134009339799716079103n,
@@ -338,20 +343,15 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
       );
       a.assertValidity();
     });
-  should(
+    should('not be placed on curve vector 1', () => {
-    'bls12-381 Point/Point with Fp2 coordinates/should not be placed on curve vector 1',
-    () => {
       const a = new PointG2(
         Fp2.fromBigTuple([0n, 0n]),
         Fp2.fromBigTuple([1n, 0n]),
         Fp2.fromBigTuple([1n, 0n])
       );
       throws(() => a.assertValidity());
-    }
+    });
-  );
+    should('not be placed on curve vector 2', () => {
-  should(
-    'bls12-381 Point/Point with Fp2 coordinates/should not be placed on curve vector 2',
-    () => {
       const a = new PointG2(
         Fp2.fromBigTuple([
           0x024aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4410b647ae3d1770bac0326a805bbefd48056c8c121bdb8n,
@@ -364,11 +364,8 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
         Fp2.fromBigTuple([1n, 0n])
       );
       throws(() => a.assertValidity());
-    }
+    });
-  );
+    should('not be placed on curve vector 3', () => {
-  should(
-    'bls12-381 Point/Point with Fp2 coordinates/should not be placed on curve vector 3',
-    () => {
       const a = new PointG2(
         Fp2.fromBigTuple([
           0x877d52dd65245f8908a03288adcd396f489ef87ae23fe110c5aa48bc208fbd1a0ed403df5b1ac137922b915f1f38ec37n,
@@ -384,10 +381,10 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
         ])
       );
       throws(() => a.assertValidity());
-    }
+    });
-  );
+  });
 
-  should('bls12-381 Point/should be doubled and placed on curve vector 1', () => {
+  should('be doubled and placed on curve vector 1', () => {
     const a = new PointG2(
       Fp2.fromBigTuple([
         0x024aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4510b647ae3d1770bac0326a805bbefd48056c8c121bdb8n,
@@ -417,7 +414,7 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
     deepStrictEqual(double, a.multiply(2n));
     deepStrictEqual(double, a.add(a));
   });
-  should('bls12-381 Point/should be doubled and placed on curve vector 2', () => {
+  should('be doubled and placed on curve vector 2', () => {
     const a = new PointG2(
       Fp2.fromBigTuple([
         233289878585407360737561818812172281900488265436962145913969074168503452745466655442125797664134009339799716079103n,
@@ -455,49 +452,51 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
     0x53eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001n,
     0x63eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000000n,
   ];
-  should('bls12-381 Point/wNAF multiplication same as unsafe (G1, W=1)', () => {
+  describe('wNAF multiplication same as unsafe', () => {
+    should('(G1, W=1)', () => {
       let G = PointG1.BASE.negate().negate(); // create new point
       G._setWindowSize(1);
       for (let k of wNAF_VECTORS) {
         deepStrictEqual(G.multiply(k).equals(G.multiplyUnsafe(k)), true);
       }
     });
-  should('bls12-381 Point/wNAF multiplication same as unsafe (G1, W=4)', () => {
+    should('(G1, W=4)', () => {
       let G = PointG1.BASE.negate().negate();
       G._setWindowSize(4);
       for (let k of wNAF_VECTORS) {
         deepStrictEqual(G.multiply(k).equals(G.multiplyUnsafe(k)), true);
       }
     });
-  should('bls12-381 Point/wNAF multiplication same as unsafe (G1, W=5)', () => {
+    should('(G1, W=5)', () => {
       let G = PointG1.BASE.negate().negate();
       G._setWindowSize(5);
       for (let k of wNAF_VECTORS) {
         deepStrictEqual(G.multiply(k).equals(G.multiplyUnsafe(k)), true);
       }
     });
-  should('bls12-381 Point/wNAF multiplication same as unsafe (G2, W=1)', () => {
+    should('(G2, W=1)', () => {
       let G = PointG2.BASE.negate().negate();
       G._setWindowSize(1);
       for (let k of wNAF_VECTORS) {
         deepStrictEqual(G.multiply(k).equals(G.multiplyUnsafe(k)), true);
       }
     });
-  should('bls12-381 Point/wNAF multiplication same as unsafe (G2, W=4)', () => {
+    should('(G2, W=4)', () => {
       let G = PointG2.BASE.negate().negate();
       G._setWindowSize(4);
       for (let k of wNAF_VECTORS) {
         deepStrictEqual(G.multiply(k).equals(G.multiplyUnsafe(k)), true);
       }
     });
-  should('bls12-381 Point/wNAF multiplication same as unsafe (G2, W=5)', () => {
+    should('(G2, W=5)', () => {
       let G = PointG2.BASE.negate().negate();
       G._setWindowSize(5);
       for (let k of wNAF_VECTORS) {
         deepStrictEqual(G.multiply(k).equals(G.multiplyUnsafe(k)), true);
       }
     });
-  should('bls12-381 Point/PSI cofactor cleaning same as multiplication', () => {
+  });
+  should('PSI cofactor cleaning same as multiplication', () => {
     const points = [
       new PointG2(
         Fp2.fromBigTuple([
@@ -558,14 +557,15 @@ const getPubKey = (priv) => bls.getPublicKey(priv);
       deepStrictEqual(ours.equals(shouldBe), true, 'clearLast');
     }
   });
-}
+});
 
 // index.ts
 
 // bls.PointG1.BASE.clearMultiplyPrecomputes();
 // bls.PointG1.BASE.calcMultiplyPrecomputes(4);
 
-should('bls12-381/basic/should construct point G1 from its uncompressed form (Raw Bytes)', () => {
+describe('bls12-381/basic', () => {
+  should('construct point G1 from its uncompressed form (Raw Bytes)', () => {
     // Test Zero
     const g1 = bls.G1.Point.fromHex(B_192_40);
     deepStrictEqual(g1.x, bls.G1.Point.ZERO.x);
@@ -590,7 +590,7 @@ should('bls12-381/basic/should construct point G1 from its uncompressed form (Ra
     deepStrictEqual(g1_.y, y);
   });
 
-should('bls12-381/basic/should construct point G1 from its uncompressed form (Hex)', () => {
+  should('construct point G1 from its uncompressed form (Hex)', () => {
     // Test Zero
     const g1 = bls.G1.Point.fromHex(B_192_40);
 
@@ -616,7 +616,7 @@ should('bls12-381/basic/should construct point G1 from its uncompressed form (He
     deepStrictEqual(g1_.y, y);
   });
 
-should('bls12-381/basic/should construct point G2 from its uncompressed form (Raw Bytes)', () => {
+  should('construct point G2 from its uncompressed form (Raw Bytes)', () => {
     // Test Zero
     const g2 = bls.G2.Point.fromHex(B_384_40);
     deepStrictEqual(g2.x, bls.G2.Point.ZERO.x, 'zero(x)');
@@ -647,7 +647,7 @@ should('bls12-381/basic/should construct point G2 from its uncompressed form (Ra
     deepStrictEqual(g2_.y, y);
   });
 
-should('bls12-381/basic/should construct point G2 from its uncompressed form (Hex)', () => {
+  should('construct point G2 from its uncompressed form (Hex)', () => {
     // Test Zero
     const g2 = bls.G2.Point.fromHex(B_384_40);
 
@@ -679,7 +679,7 @@ should('bls12-381/basic/should construct point G2 from its uncompressed form (He
     deepStrictEqual(g2_.y, y);
   });
 
-should('bls12-381/basic/should get uncompressed form of point G1 (Raw Bytes)', () => {
+  should('get uncompressed form of point G1 (Raw Bytes)', () => {
     // Test Zero
     deepStrictEqual(bls.G1.Point.ZERO.toHex(false), B_192_40);
     // Test Non-Zero
@@ -700,7 +700,7 @@ should('bls12-381/basic/should get uncompressed form of point G1 (Raw Bytes)', (
     );
   });
 
-should('bls12-381/basic/should get uncompressed form of point G1 (Hex)', () => {
+  should('get uncompressed form of point G1 (Hex)', () => {
     // Test Zero
     deepStrictEqual(bls.G1.Point.ZERO.toHex(false), B_192_40);
     // Test Non-Zero
@@ -721,7 +721,7 @@ should('bls12-381/basic/should get uncompressed form of point G1 (Hex)', () => {
     );
   });
 
-should('bls12-381/basic/should get uncompressed form of point G2 (Raw Bytes)', () => {
+  should('get uncompressed form of point G2 (Raw Bytes)', () => {
     // Test Zero
     deepStrictEqual(bls.G2.Point.ZERO.toHex(false), B_384_40);
     // Test Non-Zero
@@ -748,7 +748,7 @@ should('bls12-381/basic/should get uncompressed form of point G2 (Raw Bytes)', (
     );
   });
 
-should('bls12-381/basic/should get uncompressed form of point G2 (Hex)', () => {
+  should('get uncompressed form of point G2 (Hex)', () => {
     // Test Zero
     deepStrictEqual(bls.G2.Point.ZERO.toHex(false), B_384_40);
 
@@ -776,22 +776,22 @@ should('bls12-381/basic/should get uncompressed form of point G2 (Hex)', () => {
     );
   });
 
-should('bls12-381/basic/should compress and decompress G1 points', async () => {
+  should('compress and decompress G1 points', async () => {
     const priv = bls.G1.Point.fromPrivateKey(42n);
     const publicKey = priv.toHex(true);
     const decomp = bls.G1.Point.fromHex(publicKey);
     deepStrictEqual(publicKey, decomp.toHex(true));
   });
-should('bls12-381/basic/should not compress and decompress zero G1 point', () => {
+  should('not compress and decompress zero G1 point', () => {
     throws(() => bls.G1.Point.fromPrivateKey(0n));
   });
-should('bls12-381/basic/should compress and decompress G2 points', () => {
+  should('compress and decompress G2 points', () => {
     const priv = bls.G2.Point.fromPrivateKey(42n);
     const publicKey = priv.toHex(true);
     const decomp = bls.G2.Point.fromHex(publicKey);
     deepStrictEqual(publicKey, decomp.toHex(true));
   });
-should('bls12-381/basic/should not compress and decompress zero G2 point', () => {
+  should('not compress and decompress zero G2 point', () => {
     throws(() => bls.G2.Point.fromPrivateKey(0n));
   });
   const VALID_G1 = new bls.G1.Point(
@@ -820,7 +820,7 @@ const INVALID_G1 = new bls.G1.Point(
     )
   );
 
-should('bls12-381/basic/should aggregate pubkeys', () => {
+  should('aggregate pubkeys', () => {
     const agg = bls.aggregatePublicKeys([VALID_G1, VALID_G1_2]);
     deepStrictEqual(
       agg.x,
@@ -832,19 +832,19 @@ should('bls12-381/basic/should aggregate pubkeys', () => {
     );
   });
 
-should('bls12-381/basic/should not aggregate invalid pubkeys', () => {
+  should('not aggregate invalid pubkeys', () => {
     throws(() => bls.aggregatePublicKeys([VALID_G1, INVALID_G1]));
   });
   // should aggregate signatures
 
-should(`should produce correct signatures (${G2_VECTORS.length} vectors)`, async () => {
+  should(`produce correct signatures (${G2_VECTORS.length} vectors)`, async () => {
     for (let vector of G2_VECTORS) {
       const [priv, msg, expected] = vector;
       const sig = await bls.sign(msg, priv);
       deepStrictEqual(bls.utils.bytesToHex(sig), expected);
     }
   });
-should(`should produce correct scalars (${SCALAR_VECTORS.length} vectors)`, async () => {
+  should(`produce correct scalars (${SCALAR_VECTORS.length} vectors)`, async () => {
     const options = {
       p: bls.CURVE.r,
       m: 1,
@@ -858,7 +858,7 @@ should(`should produce correct scalars (${SCALAR_VECTORS.length} vectors)`, asyn
       deepStrictEqual(scalars[0][0], expected);
     }
   });
-should('bls12-381/basic/should verify signed message', async () => {
+  should('verify signed message', async () => {
     for (let i = 0; i < NUM_RUNS; i++) {
       const [priv, msg] = G2_VECTORS[i];
       const sig = await bls.sign(msg, priv);
@@ -867,7 +867,7 @@ should('bls12-381/basic/should verify signed message', async () => {
       deepStrictEqual(res, true);
     }
   });
-should('bls12-381/basic/should not verify signature with wrong message', async () => {
+  should('not verify signature with wrong message', async () => {
     for (let i = 0; i < NUM_RUNS; i++) {
       const [priv, msg] = G2_VECTORS[i];
       const invMsg = G2_VECTORS[i + 1][1];
@@ -877,7 +877,7 @@ should('bls12-381/basic/should not verify signature with wrong message', async (
       deepStrictEqual(res, false);
     }
   });
-should('bls12-381/basic/should not verify signature with wrong key', async () => {
+  should('not verify signature with wrong key', async () => {
     for (let i = 0; i < NUM_RUNS; i++) {
       const [priv, msg] = G2_VECTORS[i];
       const sig = await bls.sign(msg, priv);
@@ -887,7 +887,7 @@ should('bls12-381/basic/should not verify signature with wrong key', async () =>
       deepStrictEqual(res, false);
     }
   });
-should('bls12-381/basic/should verify multi-signature', async () => {
+  should('verify multi-signature', async () => {
     await fc.assert(
       fc.asyncProperty(FC_MSG_5, FC_BIGINT_5, async (messages, privateKeys) => {
         privateKeys = privateKeys.slice(0, messages.length);
@@ -901,7 +901,7 @@ should('bls12-381/basic/should verify multi-signature', async () => {
       })
     );
   });
-should('bls12-381/basic/should batch verify multi-signatures', async () => {
+  should('batch verify multi-signatures', async () => {
     await fc.assert(
       fc.asyncProperty(
         FC_MSG_5,
@@ -926,7 +926,7 @@ should('bls12-381/basic/should batch verify multi-signatures', async () => {
       )
     );
   });
-should('bls12-381/basic/should not verify multi-signature with wrong public keys', async () => {
+  should('not verify multi-signature with wrong public keys', async () => {
     await fc.assert(
       fc.asyncProperty(
         FC_MSG_5,
@@ -951,7 +951,7 @@ should('bls12-381/basic/should not verify multi-signature with wrong public keys
       )
     );
   });
-should('bls12-381/basic/should verify multi-signature as simple signature', async () => {
+  should('verify multi-signature as simple signature', async () => {
     await fc.assert(
       fc.asyncProperty(FC_MSG, FC_BIGINT_5, async (message, privateKeys) => {
         const publicKey = await Promise.all(privateKeys.map(getPubKey));
@@ -964,7 +964,7 @@ should('bls12-381/basic/should verify multi-signature as simple signature', asyn
       })
     );
   });
-should('bls12-381/basic/should not verify wrong multi-signature as simple signature', async () => {
+  should('not verify wrong multi-signature as simple signature', async () => {
     await fc.assert(
       fc.asyncProperty(FC_MSG, FC_MSG, FC_BIGINT_5, async (message, wrongMessage, privateKeys) => {
         const publicKey = await Promise.all(privateKeys.map(getPubKey));
@@ -980,34 +980,35 @@ should('bls12-381/basic/should not verify wrong multi-signature as simple signat
       })
     );
   });
+});
 
 // Pairing
-{
+describe('pairing', () => {
   const { pairing, Fp12 } = bls;
   const G1 = bls.G1.Point.BASE;
   const G2 = bls.G2.Point.BASE;
 
-  should('pairing/creates negative G1 pairing', () => {
+  should('creates negative G1 pairing', () => {
     const p1 = pairing(G1, G2);
     const p2 = pairing(G1.negate(), G2);
     deepStrictEqual(Fp12.mul(p1, p2), Fp12.ONE);
   });
-  should('pairing/creates negative G2 pairing', () => {
+  should('creates negative G2 pairing', () => {
     const p2 = pairing(G1.negate(), G2);
     const p3 = pairing(G1, G2.negate());
     deepStrictEqual(p2, p3);
   });
-  should('pairing/creates proper pairing output order', () => {
+  should('creates proper pairing output order', () => {
     const p1 = pairing(G1, G2);
     const p2 = Fp12.pow(p1, CURVE_ORDER);
     deepStrictEqual(p2, Fp12.ONE);
   });
-  should('pairing/G1 billinearity', () => {
+  should('G1 billinearity', () => {
     const p1 = pairing(G1, G2);
     const p2 = pairing(G1.multiply(2n), G2);
     deepStrictEqual(Fp12.mul(p1, p1), p2);
   });
-  should('pairing/should not degenerate', () => {
+  should('should not degenerate', () => {
     const p1 = pairing(G1, G2);
     const p2 = pairing(G1.multiply(2n), G2);
     const p3 = pairing(G1, G2.negate());
@@ -1015,17 +1016,17 @@ should('bls12-381/basic/should not verify wrong multi-signature as simple signat
     notDeepStrictEqual(p1, p3);
     notDeepStrictEqual(p2, p3);
   });
-  should('pairing/G2 billinearity', () => {
+  should('G2 billinearity', () => {
     const p1 = pairing(G1, G2);
     const p2 = pairing(G1, G2.multiply(2n));
     deepStrictEqual(Fp12.mul(p1, p1), p2);
   });
-  should('pairing/proper pairing composite check', () => {
+  should('proper pairing composite check', () => {
     const p1 = pairing(G1.multiply(37n), G2.multiply(27n));
     const p2 = pairing(G1.multiply(999n), G2);
     deepStrictEqual(p1, p2);
   });
-  should('pairing/vectors from https://github.com/zkcrypto/pairing', () => {
+  should('vectors from https://github.com/zkcrypto/pairing', () => {
     const p1 = pairing(G1, G2);
     deepStrictEqual(
       p1,
@@ -1045,7 +1046,7 @@ should('bls12-381/basic/should not verify wrong multi-signature as simple signat
       ])
     );
   });
-  should('pairing/finalExponentiate is correct', () => {
+  should('finalExponentiate is correct', () => {
     const p1 = Fp12.fromBigTwelve([
       690392658038414015999440694435086329841032295415825549843130960252222448232974816207293269712691075396080336239827n,
       1673244384695948045466836192250093912021245353707563547917201356526057153141766171738038843400145227470982267854187n,
@@ -1078,9 +1079,9 @@ should('bls12-381/basic/should not verify wrong multi-signature as simple signat
       ])
     );
   });
-}
+});
 // hashToCurve
-{
+describe('hash-to-curve', () => {
   const DST = 'QUUX-V01-CS02-with-expander-SHA256-128';
   const VECTORS = [
     {
@@ -1824,9 +1825,9 @@ should('bls12-381/basic/should not verify wrong multi-signature as simple signat
       deepStrictEqual(p.toHex(), t.expected);
     });
   }
-}
+});
 // Deterministic
-{
+describe('bls12-381 deterministic', () => {
   // NOTE: Killic returns all items in reversed order, which looks strange:
   // instead of `Fp2(${this.c0} + ${this.c1}×i)`; it returns `Fp2(${this.c0}×i + ${this.c1})`;
   const killicHex = (lst) =>
@@ -1934,7 +1935,7 @@ should('bls12-381/basic/should not verify wrong multi-signature as simple signat
       }
     }
   });
-}
+});
 
 // ESM is broken.
 import url from 'url';

test/ed25519.test.js

@@ -1,7 +1,14 @@
-import { deepStrictEqual, throws } from 'assert';
+import { deepEqual, deepStrictEqual, strictEqual, throws } from 'assert';
-import { should } from 'micro-should';
+import { describe, should } from 'micro-should';
 import * as fc from 'fast-check';
-import { ed25519, ed25519ctx, ed25519ph, x25519, RistrettoPoint } from '../lib/esm/ed25519.js';
+import {
+  ed25519,
+  ed25519ctx,
+  ed25519ph,
+  x25519,
+  RistrettoPoint,
+  ED25519_TORSION_SUBGROUP,
+} from '../lib/esm/ed25519.js';
 import { readFileSync } from 'fs';
 import { default as zip215 } from './ed25519/zip215.json' assert { type: 'json' };
 import { hexToBytes, bytesToHex, randomBytes } from '@noble/hashes/utils';
@@ -10,6 +17,7 @@ import { sha512 } from '@noble/hashes/sha512';
 import { default as ed25519vectors } from './wycheproof/eddsa_test.json' assert { type: 'json' };
 import { default as x25519vectors } from './wycheproof/x25519_test.json' assert { type: 'json' };
 
+describe('ed25519', () => {
   const ed = ed25519;
   const hex = bytesToHex;
 
@@ -27,12 +35,12 @@ function utf8ToBytes(str) {
 
   ed.utils.precompute(8);
 
-should('ed25519/should not accept >32byte private keys', () => {
+  should('not accept >32byte private keys', () => {
     const invalidPriv =
       100000000000000000000000000000000000009000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000090000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000800073278156000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000n;
     throws(() => ed.getPublicKey(invalidPriv));
   });
-should('ed25519/should verify recent signature', () => {
+  should('verify recent signature', () => {
     fc.assert(
       fc.property(
         fc.hexaString({ minLength: 2, maxLength: 32 }),
@@ -48,7 +56,7 @@ should('ed25519/should verify recent signature', () => {
       { numRuns: 5 }
     );
   });
-should('ed25519/should not verify signature with wrong message', () => {
+  should('not verify signature with wrong message', () => {
     fc.assert(
       fc.property(
         fc.array(fc.integer({ min: 0x00, max: 0xff })),
@@ -72,38 +80,40 @@ should('ed25519/should not verify signature with wrong message', () => {
   const privKey = to32Bytes('a665a45920422f9d417e4867ef');
   const msg = hexToBytes('874f9960c5d2b7a9b5fad383e1ba44719ebb743a');
   const wrongMsg = hexToBytes('589d8c7f1da0a24bc07b7381ad48b1cfc211af1c');
-should('ed25519/basic methods/should sign and verify', () => {
+  describe('basic methods', () => {
+    should('sign and verify', () => {
       const publicKey = ed.getPublicKey(privKey);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, msg, publicKey), true);
     });
-should('ed25519/basic methods/should not verify signature with wrong public key', () => {
+    should('not verify signature with wrong public key', () => {
       const publicKey = ed.getPublicKey(12);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, msg, publicKey), false);
     });
-should('ed25519/basic methods/should not verify signature with wrong hash', () => {
+    should('not verify signature with wrong hash', () => {
       const publicKey = ed.getPublicKey(privKey);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, wrongMsg, publicKey), false);
     });
-
+  });
-should('ed25519/sync methods/should sign and verify', () => {
+  describe('sync methods', () => {
+    should('sign and verify', () => {
       const publicKey = ed.getPublicKey(privKey);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, msg, publicKey), true);
     });
-should('ed25519/sync methods/should not verify signature with wrong public key', () => {
+    should('not verify signature with wrong public key', () => {
       const publicKey = ed.getPublicKey(12);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, msg, publicKey), false);
     });
-should('ed25519/sync methods/should not verify signature with wrong hash', () => {
+    should('not verify signature with wrong hash', () => {
       const publicKey = ed.getPublicKey(privKey);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, wrongMsg, publicKey), false);
     });
-
+  });
   // https://xmr.llcoins.net/addresstests.html
   should(
     'ed25519/BASE_POINT.multiply()/should create right publicKey without SHA-512 hashing TEST 1',
@@ -646,10 +656,23 @@ for (let i = 0; i < VECTORS_RFC8032_PH.length; i++) {
 
   should('X25519 base point', () => {
     const { y } = ed25519.Point.BASE;
-  const u = ed25519.utils.mod((y + 1n) * ed25519.utils.invert(1n - y, ed25519.CURVE.P));
+    const { Fp } = ed25519.CURVE;
+    const u = Fp.create((y + 1n) * Fp.invert(1n - y));
     deepStrictEqual(hex(numberToBytesLE(u, 32)), x25519.Gu);
   });
 
+  should('isTorsionFree()', () => {
+    const orig = ed.utils.getExtendedPublicKey(ed.utils.randomPrivateKey()).point;
+    for (const hex of ED25519_TORSION_SUBGROUP.slice(1)) {
+      const dirty = orig.add(ed.Point.fromHex(hex));
+      const cleared = dirty.clearCofactor();
+      strictEqual(orig.isTorsionFree(), true, `orig must be torsionFree: ${hex}`);
+      strictEqual(dirty.isTorsionFree(), false, `dirty must not be torsionFree: ${hex}`);
+      strictEqual(cleared.isTorsionFree(), true, `cleared must be torsionFree: ${hex}`);
+    }
+  });
+});
+
 // ESM is broken.
 import url from 'url';
 if (import.meta.url === url.pathToFileURL(process.argv[1]).href) {

test/ed448.test.js

@@ -1,5 +1,5 @@
 import { deepStrictEqual, throws } from 'assert';
-import { should } from 'micro-should';
+import { describe, should } from 'micro-should';
 import * as fc from 'fast-check';
 import { ed448, ed448ph, x448 } from '../lib/esm/ed448.js';
 import { hexToBytes, bytesToHex, randomBytes } from '@noble/hashes/utils';
@@ -7,6 +7,7 @@ import { numberToBytesLE } from '../lib/esm/abstract/utils.js';
 import { default as ed448vectors } from './wycheproof/ed448_test.json' assert { type: 'json' };
 import { default as x448vectors } from './wycheproof/x448_test.json' assert { type: 'json' };
 
+describe('ed448', () => {
   const ed = ed448;
   const hex = bytesToHex;
   ed.utils.precompute(4);
@@ -323,7 +324,7 @@ for (let i = 0; i < VECTORS_RFC8032.length; i++) {
     });
   }
 
-should('ed448/should not accept >57byte private keys', async () => {
+  should('not accept >57byte private keys', async () => {
     const invalidPriv =
       100000000000000000000000000000000000009000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000090000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000800073278156000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000n;
     throws(() => ed.getPublicKey(invalidPriv));
@@ -334,7 +335,7 @@ function to57Bytes(numOrStr) {
     return hexToBytes(hex.padStart(114, '0'));
   }
 
-should('ed448/should verify recent signature', () => {
+  should('verify recent signature', () => {
     fc.assert(
       fc.property(
         fc.hexaString({ minLength: 2, maxLength: 57 }),
@@ -350,7 +351,7 @@ should('ed448/should verify recent signature', () => {
       { numRuns: 5 }
     );
   });
-should('ed448/should not verify signature with wrong message', () => {
+  should('not verify signature with wrong message', () => {
     fc.assert(
       fc.property(
         fc.array(fc.integer({ min: 0x00, max: 0xff })),
@@ -374,39 +375,43 @@ should('ed448/should not verify signature with wrong message', () => {
   const privKey = to57Bytes('a665a45920422f9d417e4867ef');
   const msg = hexToBytes('874f9960c5d2b7a9b5fad383e1ba44719ebb743a');
   const wrongMsg = hexToBytes('589d8c7f1da0a24bc07b7381ad48b1cfc211af1c');
-should('ed25519/basic methods/should sign and verify', () => {
+  describe('basic methods', () => {
+    should('sign and verify', () => {
       const publicKey = ed.getPublicKey(privKey);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, msg, publicKey), true);
     });
-should('ed25519/basic methods/should not verify signature with wrong public key', () => {
+    should('not verify signature with wrong public key', () => {
       const publicKey = ed.getPublicKey(12);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, msg, publicKey), false);
     });
-should('ed25519/basic methods/should not verify signature with wrong hash', () => {
+    should('not verify signature with wrong hash', () => {
       const publicKey = ed.getPublicKey(privKey);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, wrongMsg, publicKey), false);
     });
+  });
 
-should('ed25519/sync methods/should sign and verify', () => {
+  describe('sync methods', () => {
+    should('sign and verify', () => {
       const publicKey = ed.getPublicKey(privKey);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, msg, publicKey), true);
     });
-should('ed25519/sync methods/should not verify signature with wrong public key', async () => {
+    should('not verify signature with wrong public key', async () => {
       const publicKey = ed.getPublicKey(12);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, msg, publicKey), false);
     });
-should('ed25519/sync methods/should not verify signature with wrong hash', async () => {
+    should('not verify signature with wrong hash', async () => {
       const publicKey = ed.getPublicKey(privKey);
       const signature = ed.sign(msg, privKey);
       deepStrictEqual(ed.verify(signature, wrongMsg, publicKey), false);
     });
+  });
 
-should('ed25519/BASE_POINT.multiply()/should throw Point#multiply on TEST 5', () => {
+  should('BASE_POINT.multiply() throws in Point#multiply on TEST 5', () => {
     for (const num of [0n, 0, -1n, -1, 1.1]) {
       throws(() => ed.Point.BASE.multiply(num));
     }
@@ -651,11 +656,13 @@ for (let i = 0; i < VECTORS_RFC8032_PH.length; i++) {
 
   should('X448 base point', () => {
     const { x, y } = ed448.Point.BASE;
-  const { P } = ed448.CURVE;
+    const { Fp } = ed448.CURVE;
-  const invX = ed448.utils.invert(x * x, P); // x²
+    // const invX = Fp.invert(x * x); // x²
-  const u = ed448.utils.mod(y * y * invX, P); // (y²/x²)
+    const u = Fp.div(Fp.create(y * y), Fp.create(x * x)); // (y²/x²)
+    // const u = Fp.create(y * y * invX);
     deepStrictEqual(hex(numberToBytesLE(u, 56)), x448.Gu);
   });
+});
 
 // ESM is broken.
 import url from 'url';

test/hash-to-curve.test.js

@@ -1,5 +1,5 @@
 import { deepStrictEqual } from 'assert';
-import { should } from 'micro-should';
+import { describe, should } from 'micro-should';
 import { bytesToHex } from '@noble/hashes/utils';
 // Generic tests for all curves in package
 import { sha256 } from '@noble/hashes/sha256';
@@ -51,9 +51,10 @@ import { default as ed448_ro } from './hash-to-curve/edwards448_XOF:SHAKE256_ELL
 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}`, () => {
     for (let i = 0; i < vectors.tests.length; i++) {
       const t = vectors.tests[i];
-    should(`expand_message_xmd/${vectors.hash}/${vectors.DST.length}/${i}`, () => {
+      should(`${vectors.hash}/${vectors.DST.length}/${i}`, () => {
         const p = expand_message_xmd(
           stringToBytes(t.msg),
           stringToBytes(vectors.DST),
@@ -63,16 +64,20 @@ function testExpandXMD(hash, vectors) {
         deepStrictEqual(bytesToHex(p), t.uniform_bytes);
       });
     }
+  });
 }
 
+describe('expand_message_xmd', () => {
   testExpandXMD(sha256, xmd_sha256_38);
   testExpandXMD(sha256, xmd_sha256_256);
   testExpandXMD(sha512, xmd_sha512_38);
+});
 
 function testExpandXOF(hash, vectors) {
+  describe(`${vectors.hash}/${vectors.DST.length}`, () => {
     for (let i = 0; i < vectors.tests.length; i++) {
       const t = vectors.tests[i];
-    should(`expand_message_xof/${vectors.hash}/${vectors.DST.length}/${i}`, () => {
+      should(`${i}`, () => {
         const p = expand_message_xof(
           stringToBytes(t.msg),
           stringToBytes(vectors.DST),
@@ -83,11 +88,14 @@ function testExpandXOF(hash, vectors) {
         deepStrictEqual(bytesToHex(p), t.uniform_bytes);
       });
     }
+  });
 }
 
+describe('expand_message_xof', () => {
   testExpandXOF(shake128, xof_shake128_36);
   testExpandXOF(shake128, xof_shake128_256);
   testExpandXOF(shake256, xof_shake256_36);
+});
 
 function stringToFp(s) {
   // bls-G2 support
@@ -99,9 +107,10 @@ function stringToFp(s) {
 }
 
 function testCurve(curve, ro, nu) {
+  describe(`${ro.curve}/${ro.ciphersuite}`, () => {
     for (let i = 0; i < ro.vectors.length; i++) {
       const t = ro.vectors[i];
-    should(`${ro.curve}/${ro.ciphersuite}(${i})`, () => {
+      should(`(${i})`, () => {
         const p = curve.Point.hashToCurve(stringToBytes(t.msg), {
           DST: ro.dst,
         });
@@ -109,9 +118,11 @@ function testCurve(curve, ro, nu) {
         deepStrictEqual(p.y, stringToFp(t.P.y), 'Py');
       });
     }
+  });
+  describe(`${nu.curve}/${nu.ciphersuite}`, () => {
     for (let i = 0; i < nu.vectors.length; i++) {
       const t = nu.vectors[i];
-    should(`${nu.curve}/${nu.ciphersuite}(${i})`, () => {
+      should(`(${i})`, () => {
         const p = curve.Point.encodeToCurve(stringToBytes(t.msg), {
           DST: nu.dst,
         });
@@ -119,6 +130,7 @@ function testCurve(curve, ro, nu) {
         deepStrictEqual(p.y, stringToFp(t.P.y), 'Py');
       });
     }
+  });
 }
 
 testCurve(secp256r1, p256_ro, p256_nu);

test/jubjub.test.js

@@ -1,5 +1,5 @@
 import { jubjub, findGroupHash } from '../lib/esm/jubjub.js';
-import { should } from 'micro-should';
+import { describe, should } from 'micro-should';
 import { deepStrictEqual, throws } from 'assert';
 import { hexToBytes, bytesToHex } from '@noble/hashes/utils';
 
@@ -18,6 +18,7 @@ const G_PROOF = new jubjub.ExtendedPoint(
 
 const getXY = (p) => ({ x: p.x, y: p.y });
 
+describe('jubjub', () => {
   should('toHex/fromHex', () => {
     // More than field
     throws(() =>
@@ -32,8 +33,8 @@ should('toHex/fromHex', () => {
     throws(() =>
       jubjub.Point.fromHex(
         new Uint8Array([
-        7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
+          7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
-        0,
+          0, 0,
         ])
       )
     );
@@ -61,11 +62,18 @@ should('toHex/fromHex', () => {
   });
 
   should('Find generators', () => {
-  const spend = findGroupHash(new Uint8Array(), new Uint8Array([90, 99, 97, 115, 104, 95, 71, 95]));
+    const spend = findGroupHash(
-  const proof = findGroupHash(new Uint8Array(), new Uint8Array([90, 99, 97, 115, 104, 95, 72, 95]));
+      new Uint8Array(),
+      new Uint8Array([90, 99, 97, 115, 104, 95, 71, 95])
+    );
+    const proof = findGroupHash(
+      new Uint8Array(),
+      new Uint8Array([90, 99, 97, 115, 104, 95, 72, 95])
+    );
     deepStrictEqual(getXY(spend.toAffine()), getXY(G_SPEND.toAffine()));
     deepStrictEqual(getXY(proof.toAffine()), getXY(G_PROOF.toAffine()));
   });
+});
 
 // ESM is broken.
 import url from 'url';

test/nist.test.js

@@ -1,5 +1,5 @@
 import { deepStrictEqual, throws } from 'assert';
-import { should } from 'micro-should';
+import { describe, should } from 'micro-should';
 import { secp192r1, P192 } from '../lib/esm/p192.js';
 import { secp224r1, P224 } from '../lib/esm/p224.js';
 import { secp256r1, P256 } from '../lib/esm/p256.js';
@@ -344,10 +344,11 @@ function runWycheproof(name, CURVE, group, index) {
 
 for (const name in WYCHEPROOF_ECDSA) {
   const { curve, hashes } = WYCHEPROOF_ECDSA[name];
+  describe('Wycheproof/WYCHEPROOF_ECDSA', () => {
     for (const hName in hashes) {
       const { hash, tests } = hashes[hName];
       const CURVE = curve.create(hash);
-    should(`Wycheproof/WYCHEPROOF_ECDSA ${name}/${hName}`, () => {
+      should(`${name}/${hName}`, () => {
         for (let i = 0; i < tests.length; i++) {
           const groups = tests[i].testGroups;
           for (let j = 0; j < groups.length; j++) {
@@ -357,6 +358,7 @@ for (const name in WYCHEPROOF_ECDSA) {
         }
       });
     }
+  });
 }
 
 const hexToBigint = (hex) => BigInt(`0x${hex}`);

test/secp256k1.test.js

@@ -1,12 +1,13 @@
 import * as fc from 'fast-check';
 import { secp256k1, schnorr } from '../lib/esm/secp256k1.js';
+import { Fp } from '../lib/esm/abstract/modular.js';
 import { readFileSync } from 'fs';
 import { default as ecdsa } from './vectors/ecdsa.json' assert { type: 'json' };
 import { default as ecdh } from './vectors/ecdh.json' assert { type: 'json' };
 import { default as privates } from './vectors/privates.json' assert { type: 'json' };
 import { default as points } from './vectors/points.json' assert { type: 'json' };
 import { default as wp } from './vectors/wychenproof.json' assert { type: 'json' };
-import { should } from 'micro-should';
+import { should, describe } from 'micro-should';
 import { deepStrictEqual, throws } from 'assert';
 import { hexToBytes, bytesToHex } from '@noble/hashes/utils';
 
@@ -16,7 +17,6 @@ const privatesTxt = readFileSync('./test/vectors/privates-2.txt', 'utf-8');
 const schCsv = readFileSync('./test/vectors/schnorr.csv', 'utf-8');
 
 const FC_BIGINT = fc.bigInt(1n + 1n, secp.CURVE.n - 1n);
-const P = secp.CURVE.Fp.ORDER;
 // prettier-ignore
 const INVALID_ITEMS = ['deadbeef', Math.pow(2, 53), [1], 'xyzxyzxyxyzxyzxyxyzxyzxyxyzxyzxyxyzxyzxyxyzxyzxyxyzxyzxyxyzxyzxy', secp.CURVE.n + 2n];
 
@@ -30,7 +30,8 @@ function hexToNumber(hex) {
   return BigInt(`0x${hex}`);
 }
 
-should('secp256k1.getPublicKey()', () => {
+describe('secp256k1', () => {
+  should('getPublicKey()', () => {
     const data = privatesTxt
       .split('\n')
       .filter((line) => line)
@@ -49,12 +50,12 @@ should('secp256k1.getPublicKey()', () => {
       deepStrictEqual(toBEHex(point3.y), y);
     }
   });
-should('secp256k1.getPublicKey() rejects invalid keys', () => {
+  should('getPublicKey() rejects invalid keys', () => {
-  // for (const item of INVALID_ITEMS) {
+    for (const item of INVALID_ITEMS) {
-  //   throws(() => secp.getPublicKey(item));
+      throws(() => secp.getPublicKey(item));
-  // }
+    }
   });
-should('secp256k1.precompute', () => {
+  should('precompute', () => {
     secp.utils.precompute(4);
     const data = privatesTxt
       .split('\n')
@@ -75,7 +76,7 @@ should('secp256k1.precompute', () => {
     }
   });
 
-should('secp256k1.Point.isValidPoint()', () => {
+  should('Point.isValidPoint()', () => {
     for (const vector of points.valid.isPoint) {
       const { P, expected } = vector;
       if (expected) {
@@ -86,7 +87,7 @@ should('secp256k1.Point.isValidPoint()', () => {
     }
   });
 
-should('secp256k1.Point.fromPrivateKey()', () => {
+  should('Point.fromPrivateKey()', () => {
     for (const vector of points.valid.pointFromScalar) {
       const { d, expected } = vector;
       let p = secp.Point.fromPrivateKey(d);
@@ -94,7 +95,7 @@ should('secp256k1.Point.fromPrivateKey()', () => {
     }
   });
 
-should('secp256k1.Point#toHex(compressed)', () => {
+  should('Point#toHex(compressed)', () => {
     for (const vector of points.valid.pointCompress) {
       const { P, compress, expected } = vector;
       let p = secp.Point.fromHex(P);
@@ -102,7 +103,7 @@ should('secp256k1.Point#toHex(compressed)', () => {
     }
   });
 
-should('secp256k1.Point#toHex() roundtrip (failed case)', () => {
+  should('Point#toHex() roundtrip (failed case)', () => {
     const point1 =
       secp.Point.fromPrivateKey(
         88572218780422190464634044548753414301110513745532121983949500266768436236425n
@@ -111,7 +112,7 @@ should('secp256k1.Point#toHex() roundtrip (failed case)', () => {
     // deepStrictEqual(secp.Point.fromHex(hex).toHex(true), hex);
   });
 
-should('secp256k1.Point#toHex() roundtrip', () => {
+  should('Point#toHex() roundtrip', () => {
     fc.assert(
       fc.property(FC_BIGINT, (x) => {
         const point1 = secp.Point.fromPrivateKey(x);
@@ -121,7 +122,7 @@ should('secp256k1.Point#toHex() roundtrip', () => {
     );
   });
 
-should('secp256k1.Point#add(other)', () => {
+  should('Point#add(other)', () => {
     for (const vector of points.valid.pointAdd) {
       const { P, Q, expected } = vector;
       let p = secp.Point.fromHex(P);
@@ -136,7 +137,7 @@ should('secp256k1.Point#add(other)', () => {
     }
   });
 
-should('secp256k1.Point#multiply(privateKey)', () => {
+  should('Point#multiply(privateKey)', () => {
     for (const vector of points.valid.pointMultiply) {
       const { P, d, expected } = vector;
       const p = secp.Point.fromHex(P);
@@ -175,7 +176,7 @@ should('secp256k1.Point#multiply(privateKey)', () => {
   //   console.log(secp.ProjectivePoint.normalizeZ([p0.multiplyUnsafe(z)])[0])
   // });
 
-should('secp256k1.Signature.fromCompactHex() roundtrip', () => {
+  should('Signature.fromCompactHex() roundtrip', () => {
     fc.assert(
       fc.property(FC_BIGINT, FC_BIGINT, (r, s) => {
         const sig = new secp.Signature(r, s);
@@ -184,7 +185,7 @@ should('secp256k1.Signature.fromCompactHex() roundtrip', () => {
     );
   });
 
-should('secp256k1.Signature.fromDERHex() roundtrip', () => {
+  should('Signature.fromDERHex() roundtrip', () => {
     fc.assert(
       fc.property(FC_BIGINT, FC_BIGINT, (r, s) => {
         const sig = new secp.Signature(r, s);
@@ -193,7 +194,7 @@ should('secp256k1.Signature.fromDERHex() roundtrip', () => {
     );
   });
 
-should('secp256k1.sign()/should create deterministic signatures with RFC 6979', () => {
+  should('sign()/should create deterministic signatures with RFC 6979', () => {
     for (const vector of ecdsa.valid) {
       let usig = secp.sign(vector.m, vector.d);
       let sig = usig.toCompactHex();
@@ -203,18 +204,21 @@ should('secp256k1.sign()/should create deterministic signatures with RFC 6979',
     }
   });
 
-should('secp256k1.sign()/should not create invalid deterministic signatures with RFC 6979', () => {
+  should(
+    'secp256k1.sign()/should not create invalid deterministic signatures with RFC 6979',
+    () => {
       for (const vector of ecdsa.invalid.sign) {
         throws(() => secp.sign(vector.m, vector.d));
       }
-});
+    }
+  );
 
-should('secp256k1.sign()/edge cases', () => {
+  should('sign()/edge cases', () => {
     throws(() => secp.sign());
     throws(() => secp.sign(''));
   });
 
-should('secp256k1.sign()/should create correct DER encoding against libsecp256k1', () => {
+  should('sign()/should create correct DER encoding against libsecp256k1', () => {
     const CASES = [
       [
         'd1a9dc8ed4e46a6a3e5e594615ca351d7d7ef44df1e4c94c1802f3592183794b',
@@ -237,7 +241,7 @@ should('secp256k1.sign()/should create correct DER encoding against libsecp256k1
       deepStrictEqual(secp.Signature.fromCompact(rs).toDERHex(), exp);
     }
   });
-should('secp256k1.sign()/sign ecdsa extraData', () => {
+  should('sign()/sign ecdsa extraData', () => {
     const ent1 = '0000000000000000000000000000000000000000000000000000000000000000';
     const ent2 = '0000000000000000000000000000000000000000000000000000000000000001';
     const ent3 = '6e723d3fd94ed5d2b6bdd4f123364b0f3ca52af829988a63f8afe91d29db1c33';
@@ -258,7 +262,7 @@ should('secp256k1.sign()/sign ecdsa extraData', () => {
     }
   });
 
-should('secp256k1.verify()/should verify signature', () => {
+  should('verify()/should verify signature', () => {
     const MSG = '01'.repeat(32);
     const PRIV_KEY = 0x2n;
     const signature = secp.sign(MSG, PRIV_KEY);
@@ -266,7 +270,7 @@ should('secp256k1.verify()/should verify signature', () => {
     deepStrictEqual(publicKey.length, 65);
     deepStrictEqual(secp.verify(signature, MSG, publicKey), true);
   });
-should('secp256k1.verify()/should not verify signature with wrong public key', () => {
+  should('verify()/should not verify signature with wrong public key', () => {
     const MSG = '01'.repeat(32);
     const PRIV_KEY = 0x2n;
     const WRONG_PRIV_KEY = 0x22n;
@@ -275,7 +279,7 @@ should('secp256k1.verify()/should not verify signature with wrong public key', (
     deepStrictEqual(publicKey.length, 130);
     deepStrictEqual(secp.verify(signature, MSG, publicKey), false);
   });
-should('secp256k1.verify()/should not verify signature with wrong hash', () => {
+  should('verify()/should not verify signature with wrong hash', () => {
     const MSG = '01'.repeat(32);
     const PRIV_KEY = 0x2n;
     const WRONG_MSG = '11'.repeat(32);
@@ -284,7 +288,7 @@ should('secp256k1.verify()/should not verify signature with wrong hash', () => {
     deepStrictEqual(publicKey.length, 65);
     deepStrictEqual(secp.verify(signature, WRONG_MSG, publicKey), false);
   });
-should('secp256k1.verify()/should verify random signatures', () =>
+  should('verify()/should verify random signatures', () =>
     fc.assert(
       fc.property(FC_BIGINT, fc.hexaString({ minLength: 64, maxLength: 64 }), (privKey, msg) => {
         const pub = secp.getPublicKey(privKey);
@@ -293,10 +297,11 @@ should('secp256k1.verify()/should verify random signatures', () =>
       })
     )
   );
-should('secp256k1.verify()/should not verify signature with invalid r/s', () => {
+  should('verify()/should not verify signature with invalid r/s', () => {
     const msg = new Uint8Array([
-    0xbb, 0x5a, 0x52, 0xf4, 0x2f, 0x9c, 0x92, 0x61, 0xed, 0x43, 0x61, 0xf5, 0x94, 0x22, 0xa1, 0xe3,
+      0xbb, 0x5a, 0x52, 0xf4, 0x2f, 0x9c, 0x92, 0x61, 0xed, 0x43, 0x61, 0xf5, 0x94, 0x22, 0xa1,
-    0x00, 0x36, 0xe7, 0xc3, 0x2b, 0x27, 0x0c, 0x88, 0x07, 0xa4, 0x19, 0xfe, 0xca, 0x60, 0x50, 0x23,
+      0xe3, 0x00, 0x36, 0xe7, 0xc3, 0x2b, 0x27, 0x0c, 0x88, 0x07, 0xa4, 0x19, 0xfe, 0xca, 0x60,
+      0x50, 0x23,
     ]);
     const x = 100260381870027870612475458630405506840396644859280795015145920502443964769584n;
     const y = 41096923727651821103518389640356553930186852801619204169823347832429067794568n;
@@ -312,7 +317,7 @@ should('secp256k1.verify()/should not verify signature with invalid r/s', () =>
     // Verifies, but it shouldn't, because signature S > curve order
     deepStrictEqual(verified, false);
   });
-should('secp256k1.verify()/should not verify msg = curve order', () => {
+  should('verify()/should not verify msg = curve order', () => {
     const msg = 'fffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141';
     const x = 55066263022277343669578718895168534326250603453777594175500187360389116729240n;
     const y = 32670510020758816978083085130507043184471273380659243275938904335757337482424n;
@@ -322,7 +327,7 @@ should('secp256k1.verify()/should not verify msg = curve order', () => {
     const sig = new secp.Signature(r, s);
     deepStrictEqual(secp.verify(sig, msg, pub), false);
   });
-should('secp256k1.verify()/should verify non-strict msg bb5a...', () => {
+  should('verify()/should verify non-strict msg bb5a...', () => {
     const msg = 'bb5a52f42f9c9261ed4361f59422a1e30036e7c32b270c8807a419feca605023';
     const x = 3252872872578928810725465493269682203671229454553002637820453004368632726370n;
     const y = 17482644437196207387910659778872952193236850502325156318830589868678978890912n;
@@ -362,7 +367,7 @@ for (let vec of vectors) {
     });
   }
 
-should('secp256k1.recoverPublicKey()/should recover public key from recovery bit', () => {
+  should('recoverPublicKey()/should recover public key from recovery bit', () => {
     const message = '00000000000000000000000000000000000000000000000000000000deadbeef';
     const privateKey = 123456789n;
     const publicKey = secp.Point.fromHex(secp.getPublicKey(privateKey)).toHex(false);
@@ -370,17 +375,17 @@ should('secp256k1.recoverPublicKey()/should recover public key from recovery bit
     const recoveredPubkey = sig.recoverPublicKey(message);
     // const recoveredPubkey = secp.recoverPublicKey(message, signature, recovery);
     deepStrictEqual(recoveredPubkey !== null, true);
-  deepStrictEqual(recoveredPubkey.toHex(), publicKey);
+    deepStrictEqual(recoveredPubkey.toHex(false), publicKey);
     deepStrictEqual(secp.verify(sig, message, publicKey), true);
   });
-should('secp256k1.recoverPublicKey()/should not recover zero points', () => {
+  should('recoverPublicKey()/should not recover zero points', () => {
     const msgHash = '6b8d2c81b11b2d699528dde488dbdf2f94293d0d33c32e347f255fa4a6c1f0a9';
     const sig =
       '79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f817986b8d2c81b11b2d699528dde488dbdf2f94293d0d33c32e347f255fa4a6c1f0a9';
     const recovery = 0;
     throws(() => secp.recoverPublicKey(msgHash, sig, recovery));
   });
-should('secp256k1.recoverPublicKey()/should handle all-zeros msghash', () => {
+  should('recoverPublicKey()/should handle all-zeros msghash', () => {
     const privKey = secp.utils.randomPrivateKey();
     const pub = secp.getPublicKey(privKey);
     const zeros = '0000000000000000000000000000000000000000000000000000000000000000';
@@ -388,7 +393,7 @@ should('secp256k1.recoverPublicKey()/should handle all-zeros msghash', () => {
     const recoveredKey = sig.recoverPublicKey(zeros);
     deepStrictEqual(recoveredKey.toRawBytes(), pub);
   });
-should('secp256k1.recoverPublicKey()/should handle RFC 6979 vectors', () => {
+  should('recoverPublicKey()/should handle RFC 6979 vectors', () => {
     for (const vector of ecdsa.valid) {
       let usig = secp.sign(vector.m, vector.d);
       let sig = usig.toDERHex();
@@ -402,7 +407,7 @@ should('secp256k1.recoverPublicKey()/should handle RFC 6979 vectors', () => {
   function derToPub(der) {
     return der.slice(46);
   }
-should('secp256k1.getSharedSecret()/should produce correct results', () => {
+  should('getSharedSecret()/should produce correct results', () => {
     // TODO: Once der is there, run all tests.
     for (const vector of ecdh.testGroups[0].tests.slice(0, 230)) {
       if (vector.result === 'invalid' || vector.private.length !== 64) {
@@ -415,7 +420,7 @@ should('secp256k1.getSharedSecret()/should produce correct results', () => {
       }
     }
   });
-should('secp256k1.getSharedSecret()/priv/pub order matters', () => {
+  should('getSharedSecret()/priv/pub order matters', () => {
     for (const vector of ecdh.testGroups[0].tests.slice(0, 100)) {
       if (vector.result === 'valid') {
         let priv = vector.private;
@@ -424,27 +429,36 @@ should('secp256k1.getSharedSecret()/priv/pub order matters', () => {
       }
     }
   });
-should('secp256k1.getSharedSecret()/rejects invalid keys', () => {
+  should('getSharedSecret()/rejects invalid keys', () => {
     throws(() => secp.getSharedSecret('01', '02'));
   });
 
-should('secp256k1.utils.isValidPrivateKey()', () => {
+  should('utils.isValidPrivateKey()', () => {
     for (const vector of privates.valid.isPrivate) {
       const { d, expected } = vector;
       deepStrictEqual(secp.utils.isValidPrivateKey(d), expected);
     }
   });
+  should('have proper curve equation in assertValidity()', () => {
+    throws(() => {
+      const { Fp } = secp.CURVE;
+      let point = new secp.Point(Fp.create(-2n), Fp.create(-1n));
+      point.assertValidity();
+    });
+  });
+
+  const Fn = Fp(secp.CURVE.n);
   const normal = secp.utils._normalizePrivateKey;
   const tweakUtils = {
     privateAdd: (privateKey, tweak) => {
       const p = normal(privateKey);
       const t = normal(tweak);
-    return secp.utils._bigintToBytes(secp.utils.mod(p + t, secp.CURVE.n));
+      return secp.utils._bigintToBytes(Fn.create(p + t));
     },
 
     privateNegate: (privateKey) => {
       const p = normal(privateKey);
-    return secp.utils._bigintToBytes(secp.CURVE.n - p);
+      return secp.utils._bigintToBytes(Fn.negate(p));
     },
 
     pointAddScalar: (p, tweak, isCompressed) => {
@@ -463,45 +477,45 @@ const tweakUtils = {
     },
   };
 
-should('secp256k1.privateAdd()', () => {
+  should('privateAdd()', () => {
     for (const vector of privates.valid.add) {
       const { a, b, expected } = vector;
       deepStrictEqual(bytesToHex(tweakUtils.privateAdd(a, b)), expected);
     }
   });
-should('secp256k1.privateNegate()', () => {
+  should('privateNegate()', () => {
     for (const vector of privates.valid.negate) {
       const { a, expected } = vector;
       deepStrictEqual(bytesToHex(tweakUtils.privateNegate(a)), expected);
     }
   });
-should('secp256k1.pointAddScalar()', () => {
+  should('pointAddScalar()', () => {
     for (const vector of points.valid.pointAddScalar) {
       const { description, P, d, expected } = vector;
       const compressed = !!expected && expected.length === 66; // compressed === 33 bytes
       deepStrictEqual(bytesToHex(tweakUtils.pointAddScalar(P, d, compressed)), expected);
     }
   });
-should('secp256k1.pointAddScalar() invalid', () => {
+  should('pointAddScalar() invalid', () => {
     for (const vector of points.invalid.pointAddScalar) {
       const { P, d, exception } = vector;
       throws(() => tweakUtils.pointAddScalar(P, d));
     }
   });
-should('secp256k1.pointMultiply()', () => {
+  should('pointMultiply()', () => {
     for (const vector of points.valid.pointMultiply) {
       const { P, d, expected } = vector;
       deepStrictEqual(bytesToHex(tweakUtils.pointMultiply(P, d, true)), expected);
     }
   });
-should('secp256k1.pointMultiply() invalid', () => {
+  should('pointMultiply() invalid', () => {
     for (const vector of points.invalid.pointMultiply) {
       const { P, d, exception } = vector;
       throws(() => tweakUtils.pointMultiply(P, d));
     }
   });
 
-should('secp256k1.wychenproof vectors', () => {
+  should('wychenproof vectors', () => {
     for (let group of wp.testGroups) {
       const pubKey = secp.Point.fromHex(group.key.uncompressed);
       for (let test of group.tests) {
@@ -528,6 +542,7 @@ should('secp256k1.wychenproof vectors', () => {
       }
     }
   });
+});
 
 // ESM is broken.
 import url from 'url';