@noble/curves/src/abstract/weierstrass.ts

Audited & minimal JS implementation of elliptic curve cryptography

/**
 * Short Weierstrass curve methods. The formula is: y² = x³ + ax + b.
 *
 * ### Design rationale for types
 *
 * * Interaction between classes from different curves should fail:
 *   `k256.Point.BASE.add(p256.Point.BASE)`
 * * For this purpose we want to use `instanceof` operator, which is fast and works during runtime
 * * Different calls of `curve()` would return different classes -
 *   `curve(params) !== curve(params)`: if somebody decided to monkey-patch their curve,
 *   it won't affect others
 *
 * TypeScript can't infer types for classes created inside a function. Classes is one instance
 * of nominative types in TypeScript and interfaces only check for shape, so it's hard to create
 * unique type for every function call.
 *
 * We can use generic types via some param, like curve opts, but that would:
 *     1. Enable interaction between `curve(params)` and `curve(params)` (curves of same params)
 *     which is hard to debug.
 *     2. Params can be generic and we can't enforce them to be constant value:
 *     if somebody creates curve from non-constant params,
 *     it would be allowed to interact with other curves with non-constant params
 *
 * @todo https://www.typescriptlang.org/docs/handbook/release-notes/typescript-2-7.html#unique-symbol
 * @module
 */
/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { hmac as nobleHmac } from '@noble/hashes/hmac.js';
import { ahash } from '@noble/hashes/utils.js';
import {
  abool,
  abytes,
  aInRange,
  bitLen,
  bitMask,
  bytesToHex,
  bytesToNumberBE,
  concatBytes,
  createHmacDrbg,
  hexToBytes,
  isBytes,
  memoized,
  numberToHexUnpadded,
  validateObject,
  randomBytes as wcRandomBytes,
  type CHash,
  type Signer,
} from '../utils.ts';
import {
  createCurveFields,
  createKeygen,
  mulEndoUnsafe,
  negateCt,
  normalizeZ,
  wNAF,
  type AffinePoint,
  type CurveLengths,
  type CurvePoint,
  type CurvePointCons,
} from './curve.ts';
import {
  FpInvertBatch,
  getMinHashLength,
  mapHashToField,
  validateField,
  type IField,
} from './modular.ts';

export type { AffinePoint };

type EndoBasis = [[bigint, bigint], [bigint, bigint]];
/**
 * When Weierstrass curve has `a=0`, it becomes Koblitz curve.
 * Koblitz curves allow using **efficiently-computable GLV endomorphism ψ**.
 * Endomorphism uses 2x less RAM, speeds up precomputation by 2x and ECDH / key recovery by 20%.
 * For precomputed wNAF it trades off 1/2 init time & 1/3 ram for 20% perf hit.
 *
 * Endomorphism consists of beta, lambda and splitScalar:
 *
 * 1. GLV endomorphism ψ transforms a point: `P = (x, y) ↦ ψ(P) = (β·x mod p, y)`
 * 2. GLV scalar decomposition transforms a scalar: `k ≡ k₁ + k₂·λ (mod n)`
 * 3. Then these are combined: `k·P = k₁·P + k₂·ψ(P)`
 * 4. Two 128-bit point-by-scalar multiplications + one point addition is faster than
 *    one 256-bit multiplication.
 *
 * where
 * * beta: β ∈ Fₚ with β³ = 1, β ≠ 1
 * * lambda: λ ∈ Fₙ with λ³ = 1, λ ≠ 1
 * * splitScalar decomposes k ↦ k₁, k₂, by using reduced basis vectors.
 *   Gauss lattice reduction calculates them from initial basis vectors `(n, 0), (-λ, 0)`
 *
 * Check out `test/misc/endomorphism.js` and
 * [gist](https://gist.github.com/paulmillr/eb670806793e84df628a7c434a873066).
 */
export type EndomorphismOpts = {
  beta: bigint;
  basises?: EndoBasis;
  splitScalar?: (k: bigint) => { k1neg: boolean; k1: bigint; k2neg: boolean; k2: bigint };
};
// We construct basis in such way that den is always positive and equals n, but num sign depends on basis (not on secret value)
const divNearest = (num: bigint, den: bigint) => (num + (num >= 0 ? den : -den) / _2n) / den;

export type ScalarEndoParts = { k1neg: boolean; k1: bigint; k2neg: boolean; k2: bigint };

/**
 * Splits scalar for GLV endomorphism.
 */
export function _splitEndoScalar(k: bigint, basis: EndoBasis, n: bigint): ScalarEndoParts {
  // Split scalar into two such that part is ~half bits: `abs(part) < sqrt(N)`
  // Since part can be negative, we need to do this on point.
  // TODO: verifyScalar function which consumes lambda
  const [[a1, b1], [a2, b2]] = basis;
  const c1 = divNearest(b2 * k, n);
  const c2 = divNearest(-b1 * k, n);
  // |k1|/|k2| is < sqrt(N), but can be negative.
  // If we do `k1 mod N`, we'll get big scalar (`> sqrt(N)`): so, we do cheaper negation instead.
  let k1 = k - c1 * a1 - c2 * a2;
  let k2 = -c1 * b1 - c2 * b2;
  const k1neg = k1 < _0n;
  const k2neg = k2 < _0n;
  if (k1neg) k1 = -k1;
  if (k2neg) k2 = -k2;
  // Double check that resulting scalar less than half bits of N: otherwise wNAF will fail.
  // This should only happen on wrong basises. Also, math inside is too complex and I don't trust it.
  const MAX_NUM = bitMask(Math.ceil(bitLen(n) / 2)) + _1n; // Half bits of N
  if (k1 < _0n || k1 >= MAX_NUM || k2 < _0n || k2 >= MAX_NUM) {
    throw new Error('splitScalar (endomorphism): failed, k=' + k);
  }
  return { k1neg, k1, k2neg, k2 };
}

/**
 * Option to enable hedged signatures with improved security.
 *
 * * Randomly generated k is bad, because broken CSPRNG would leak private keys.
 * * Deterministic k (RFC6979) is better; but is suspectible to fault attacks.
 *
 * We allow using technique described in RFC6979 3.6: additional k', a.k.a. adding randomness
 * to deterministic sig. If CSPRNG is broken & randomness is weak, it would STILL be as secure
 * as ordinary sig without ExtraEntropy.
 *
 * * `true` means "fetch data, from CSPRNG, incorporate it into k generation"
 * * `false` means "disable extra entropy, use purely deterministic k"
 * * `Uint8Array` passed means "incorporate following data into k generation"
 *
 * https://paulmillr.com/posts/deterministic-signatures/
 */
export type ECDSAExtraEntropy = boolean | Uint8Array;
/**
 * - `compact` is the default format
 * - `recovered` is the same as compact, but with an extra byte indicating recovery byte
 * - `der` is ASN.1 DER encoding
 */
export type ECDSASignatureFormat = 'compact' | 'recovered' | 'der';
/**
 * - `prehash`: (default: true) indicates whether to do sha256(message).
 *   When a custom hash is used, it must be set to `false`.
 */
export type ECDSARecoverOpts = {
  prehash?: boolean;
};
/**
 * - `prehash`: (default: true) indicates whether to do sha256(message).
 *   When a custom hash is used, it must be set to `false`.
 * - `lowS`: (default: true) prohibits signatures which have (sig.s >= CURVE.n/2n).
 *   Compatible with BTC/ETH. Setting `lowS: false` allows to create malleable signatures,
 *   which is default openssl behavior.
 *   Non-malleable signatures can still be successfully verified in openssl.
 * - `format`: (default: 'compact') 'compact' or 'recovered' with recovery byte
 */
export type ECDSAVerifyOpts = {
  prehash?: boolean;
  lowS?: boolean;
  format?: ECDSASignatureFormat;
};
/**
 * - `prehash`: (default: true) indicates whether to do sha256(message).
 *   When a custom hash is used, it must be set to `false`.
 * - `lowS`: (default: true) prohibits signatures which have (sig.s >= CURVE.n/2n).
 *   Compatible with BTC/ETH. Setting `lowS: false` allows to create malleable signatures,
 *   which is default openssl behavior.
 *   Non-malleable signatures can still be successfully verified in openssl.
 * - `format`: (default: 'compact') 'compact' or 'recovered' with recovery byte
 * - `extraEntropy`: (default: false) creates sigs with increased security, see {@link ECDSAExtraEntropy}
 */
export type ECDSASignOpts = {
  prehash?: boolean;
  lowS?: boolean;
  format?: ECDSASignatureFormat;
  extraEntropy?: ECDSAExtraEntropy;
};

function validateSigFormat(format: string): ECDSASignatureFormat {
  if (!['compact', 'recovered', 'der'].includes(format))
    throw new Error('Signature format must be "compact", "recovered", or "der"');
  return format as ECDSASignatureFormat;
}

function validateSigOpts<T extends ECDSASignOpts, D extends Required<ECDSASignOpts>>(
  opts: T,
  def: D
): Required<ECDSASignOpts> {
  const optsn: ECDSASignOpts = {};
  for (let optName of Object.keys(def)) {
    // @ts-ignore
    optsn[optName] = opts[optName] === undefined ? def[optName] : opts[optName];
  }
  abool(optsn.lowS!, 'lowS');
  abool(optsn.prehash!, 'prehash');
  if (optsn.format !== undefined) validateSigFormat(optsn.format);
  return optsn as Required<ECDSASignOpts>;
}

/** Instance methods for 3D XYZ projective points. */
export interface WeierstrassPoint<T> extends CurvePoint<T, WeierstrassPoint<T>> {
  /** projective X coordinate. Different from affine x. */
  readonly X: T;
  /** projective Y coordinate. Different from affine y. */
  readonly Y: T;
  /** projective z coordinate */
  readonly Z: T;
  /** affine x coordinate. Different from projective X. */
  get x(): T;
  /** affine y coordinate. Different from projective Y. */
  get y(): T;
  /** Encodes point using IEEE P1363 (DER) encoding. First byte is 2/3/4. Default = isCompressed. */
  toBytes(isCompressed?: boolean): Uint8Array;
  toHex(isCompressed?: boolean): string;
}

/** Static methods for 3D XYZ projective points. */
export interface WeierstrassPointCons<T> extends CurvePointCons<WeierstrassPoint<T>> {
  /** Does NOT validate if the point is valid. Use `.assertValidity()`. */
  new (X: T, Y: T, Z: T): WeierstrassPoint<T>;
  CURVE(): WeierstrassOpts<T>;
}

/**
 * Weierstrass curve options.
 *
 * * p: prime characteristic (order) of finite field, in which arithmetics is done
 * * n: order of prime subgroup a.k.a total amount of valid curve points
 * * h: cofactor, usually 1. h*n is group order; n is subgroup order
 * * a: formula param, must be in field of p
 * * b: formula param, must be in field of p
 * * Gx: x coordinate of generator point a.k.a. base point
 * * Gy: y coordinate of generator point
 */
export type WeierstrassOpts<T> = Readonly<{
  p: bigint;
  n: bigint;
  h: bigint;
  a: T;
  b: T;
  Gx: T;
  Gy: T;
}>;

// When a cofactor != 1, there can be an effective methods to:
// 1. Determine whether a point is torsion-free
// 2. Clear torsion component
export type WeierstrassExtraOpts<T> = Partial<{
  Fp: IField<T>;
  Fn: IField<bigint>;
  allowInfinityPoint: boolean;
  endo: EndomorphismOpts;
  isTorsionFree: (c: WeierstrassPointCons<T>, point: WeierstrassPoint<T>) => boolean;
  clearCofactor: (c: WeierstrassPointCons<T>, point: WeierstrassPoint<T>) => WeierstrassPoint<T>;
  fromBytes: (bytes: Uint8Array) => AffinePoint<T>;
  toBytes: (
    c: WeierstrassPointCons<T>,
    point: WeierstrassPoint<T>,
    isCompressed: boolean
  ) => Uint8Array;
}>;

/**
 * Options for ECDSA signatures over a Weierstrass curve.
 *
 * * lowS: (default: true) whether produced / verified signatures occupy low half of ecdsaOpts.p. Prevents malleability.
 * * hmac: (default: noble-hashes hmac) function, would be used to init hmac-drbg for k generation.
 * * randomBytes: (default: webcrypto os-level CSPRNG) custom method for fetching secure randomness.
 * * bits2int, bits2int_modN: used in sigs, sometimes overridden by curves
 */
export type ECDSAOpts = Partial<{
  lowS: boolean;
  hmac: (key: Uint8Array, message: Uint8Array) => Uint8Array;
  randomBytes: (bytesLength?: number) => Uint8Array;
  bits2int: (bytes: Uint8Array) => bigint;
  bits2int_modN: (bytes: Uint8Array) => bigint;
}>;

/**
 * Elliptic Curve Diffie-Hellman interface.
 * Provides keygen, secret-to-public conversion, calculating shared secrets.
 */
export interface ECDH {
  keygen: (seed?: Uint8Array) => { secretKey: Uint8Array; publicKey: Uint8Array };
  getPublicKey: (secretKey: Uint8Array, isCompressed?: boolean) => Uint8Array;
  getSharedSecret: (
    secretKeyA: Uint8Array,
    publicKeyB: Uint8Array,
    isCompressed?: boolean
  ) => Uint8Array;
  Point: WeierstrassPointCons<bigint>;
  utils: {
    isValidSecretKey: (secretKey: Uint8Array) => boolean;
    isValidPublicKey: (publicKey: Uint8Array, isCompressed?: boolean) => boolean;
    randomSecretKey: (seed?: Uint8Array) => Uint8Array;
  };
  lengths: CurveLengths;
}

/**
 * ECDSA interface.
 * Only supported for prime fields, not Fp2 (extension fields).
 */
export interface ECDSA extends ECDH {
  sign: (message: Uint8Array, secretKey: Uint8Array, opts?: ECDSASignOpts) => Uint8Array;
  verify: (
    signature: Uint8Array,
    message: Uint8Array,
    publicKey: Uint8Array,
    opts?: ECDSAVerifyOpts
  ) => boolean;
  recoverPublicKey(signature: Uint8Array, message: Uint8Array, opts?: ECDSARecoverOpts): Uint8Array;
  Signature: ECDSASignatureCons;
}
export class DERErr extends Error {
  constructor(m = '') {
    super(m);
  }
}
export type IDER = {
  // asn.1 DER encoding utils
  Err: typeof DERErr;
  // Basic building block is TLV (Tag-Length-Value)
  _tlv: {
    encode: (tag: number, data: string) => string;
    // v - value, l - left bytes (unparsed)
    decode(tag: number, data: Uint8Array): { v: Uint8Array; l: Uint8Array };
  };
  // https://crypto.stackexchange.com/a/57734 Leftmost bit of first byte is 'negative' flag,
  // since we always use positive integers here. It must always be empty:
  // - add zero byte if exists
  // - if next byte doesn't have a flag, leading zero is not allowed (minimal encoding)
  _int: {
    encode(num: bigint): string;
    decode(data: Uint8Array): bigint;
  };
  toSig(hex: string | Uint8Array): { r: bigint; s: bigint };
  hexFromSig(sig: { r: bigint; s: bigint }): string;
};
/**
 * ASN.1 DER encoding utilities. ASN is very complex & fragile. Format:
 *
 *     [0x30 (SEQUENCE), bytelength, 0x02 (INTEGER), intLength, R, 0x02 (INTEGER), intLength, S]
 *
 * Docs: https://letsencrypt.org/docs/a-warm-welcome-to-asn1-and-der/, https://luca.ntop.org/Teaching/Appunti/asn1.html
 */
export const DER: IDER = {
  // asn.1 DER encoding utils
  Err: DERErr,
  // Basic building block is TLV (Tag-Length-Value)
  _tlv: {
    encode: (tag: number, data: string): string => {
      const { Err: E } = DER;
      if (tag < 0 || tag > 256) throw new E('tlv.encode: wrong tag');
      if (data.length & 1) throw new E('tlv.encode: unpadded data');
      const dataLen = data.length / 2;
      const len = numberToHexUnpadded(dataLen);
      if ((len.length / 2) & 0b1000_0000) throw new E('tlv.encode: long form length too big');
      // length of length with long form flag
      const lenLen = dataLen > 127 ? numberToHexUnpadded((len.length / 2) | 0b1000_0000) : '';
      const t = numberToHexUnpadded(tag);
      return t + lenLen + len + data;
    },
    // v - value, l - left bytes (unparsed)
    decode(tag: number, data: Uint8Array): { v: Uint8Array; l: Uint8Array } {
      const { Err: E } = DER;
      let pos = 0;
      if (tag < 0 || tag > 256) throw new E('tlv.encode: wrong tag');
      if (data.length < 2 || data[pos++] !== tag) throw new E('tlv.decode: wrong tlv');
      const first = data[pos++];
      const isLong = !!(first & 0b1000_0000); // First bit of first length byte is flag for short/long form
      let length = 0;
      if (!isLong) length = first;
      else {
        // Long form: [longFlag(1bit), lengthLength(7bit), length (BE)]
        const lenLen = first & 0b0111_1111;
        if (!lenLen) throw new E('tlv.decode(long): indefinite length not supported');
        if (lenLen > 4) throw new E('tlv.decode(long): byte length is too big'); // this will overflow u32 in js
        const lengthBytes = data.subarray(pos, pos + lenLen);
        if (lengthBytes.length !== lenLen) throw new E('tlv.decode: length bytes not complete');
        if (lengthBytes[0] === 0) throw new E('tlv.decode(long): zero leftmost byte');
        for (const b of lengthBytes) length = (length << 8) | b;
        pos += lenLen;
        if (length < 128) throw new E('tlv.decode(long): not minimal encoding');
      }
      const v = data.subarray(pos, pos + length);
      if (v.length !== length) throw new E('tlv.decode: wrong value length');
      return { v, l: data.subarray(pos + length) };
    },
  },
  // https://crypto.stackexchange.com/a/57734 Leftmost bit of first byte is 'negative' flag,
  // since we always use positive integers here. It must always be empty:
  // - add zero byte if exists
  // - if next byte doesn't have a flag, leading zero is not allowed (minimal encoding)
  _int: {
    encode(num: bigint): string {
      const { Err: E } = DER;
      if (num < _0n) throw new E('integer: negative integers are not allowed');
      let hex = numberToHexUnpadded(num);
      // Pad with zero byte if negative flag is present
      if (Number.parseInt(hex[0], 16) & 0b1000) hex = '00' + hex;
      if (hex.length & 1) throw new E('unexpected DER parsing assertion: unpadded hex');
      return hex;
    },
    decode(data: Uint8Array): bigint {
      const { Err: E } = DER;
      if (data[0] & 0b1000_0000) throw new E('invalid signature integer: negative');
      if (data[0] === 0x00 && !(data[1] & 0b1000_0000))
        throw new E('invalid signature integer: unnecessary leading zero');
      return bytesToNumberBE(data);
    },
  },
  toSig(bytes: Uint8Array): { r: bigint; s: bigint } {
    // parse DER signature
    const { Err: E, _int: int, _tlv: tlv } = DER;
    const data = abytes(bytes, undefined, 'signature');
    const { v: seqBytes, l: seqLeftBytes } = tlv.decode(0x30, data);
    if (seqLeftBytes.length) throw new E('invalid signature: left bytes after parsing');
    const { v: rBytes, l: rLeftBytes } = tlv.decode(0x02, seqBytes);
    const { v: sBytes, l: sLeftBytes } = tlv.decode(0x02, rLeftBytes);
    if (sLeftBytes.length) throw new E('invalid signature: left bytes after parsing');
    return { r: int.decode(rBytes), s: int.decode(sBytes) };
  },
  hexFromSig(sig: { r: bigint; s: bigint }): string {
    const { _tlv: tlv, _int: int } = DER;
    const rs = tlv.encode(0x02, int.encode(sig.r));
    const ss = tlv.encode(0x02, int.encode(sig.s));
    const seq = rs + ss;
    return tlv.encode(0x30, seq);
  },
};

// Be friendly to bad ECMAScript parsers by not using bigint literals
// prettier-ignore
const _0n = BigInt(0), _1n = BigInt(1), _2n = BigInt(2), _3n = BigInt(3), _4n = BigInt(4);

/**
 * Creates weierstrass Point constructor, based on specified curve options.
 *
 * See {@link WeierstrassOpts}.
 *
 * @example
```js
const opts = {
  p: 0xfffffffffffffffffffffffffffffffeffffac73n,
  n: 0x100000000000000000001b8fa16dfab9aca16b6b3n,
  h: 1n,
  a: 0n,
  b: 7n,
  Gx: 0x3b4c382ce37aa192a4019e763036f4f5dd4d7ebbn,
  Gy: 0x938cf935318fdced6bc28286531733c3f03c4feen,
};
const secp160k1_Point = weierstrass(opts);
```
 */
export function weierstrass<T>(
  params: WeierstrassOpts<T>,
  extraOpts: WeierstrassExtraOpts<T> = {}
): WeierstrassPointCons<T> {
  const validated = createCurveFields('weierstrass', params, extraOpts);
  const { Fp, Fn } = validated;
  let CURVE = validated.CURVE as WeierstrassOpts<T>;
  const { h: cofactor, n: CURVE_ORDER } = CURVE;
  validateObject(
    extraOpts,
    {},
    {
      allowInfinityPoint: 'boolean',
      clearCofactor: 'function',
      isTorsionFree: 'function',
      fromBytes: 'function',
      toBytes: 'function',
      endo: 'object',
    }
  );

  const { endo } = extraOpts;
  if (endo) {
    // validateObject(endo, { beta: 'bigint', splitScalar: 'function' });
    if (!Fp.is0(CURVE.a) || typeof endo.beta !== 'bigint' || !Array.isArray(endo.basises)) {
      throw new Error('invalid endo: expected "beta": bigint and "basises": array');
    }
  }

  const lengths = getWLengths(Fp, Fn);

  function assertCompressionIsSupported() {
    if (!Fp.isOdd) throw new Error('compression is not supported: Field does not have .isOdd()');
  }

  // Implements IEEE P1363 point encoding
  function pointToBytes(
    _c: WeierstrassPointCons<T>,
    point: WeierstrassPoint<T>,
    isCompressed: boolean
  ): Uint8Array {
    const { x, y } = point.toAffine();
    const bx = Fp.toBytes(x);
    abool(isCompressed, 'isCompressed');
    if (isCompressed) {
      assertCompressionIsSupported();
      const hasEvenY = !Fp.isOdd!(y);
      return concatBytes(pprefix(hasEvenY), bx);
    } else {
      return concatBytes(Uint8Array.of(0x04), bx, Fp.toBytes(y));
    }
  }
  function pointFromBytes(bytes: Uint8Array) {
    abytes(bytes, undefined, 'Point');
    const { publicKey: comp, publicKeyUncompressed: uncomp } = lengths; // e.g. for 32-byte: 33, 65
    const length = bytes.length;
    const head = bytes[0];
    const tail = bytes.subarray(1);
    // No actual validation is done here: use .assertValidity()
    if (length === comp && (head === 0x02 || head === 0x03)) {
      const x = Fp.fromBytes(tail);
      if (!Fp.isValid(x)) throw new Error('bad point: is not on curve, wrong x');
      const y2 = weierstrassEquation(x); // y² = x³ + ax + b
      let y: T;
      try {
        y = Fp.sqrt(y2); // y = y² ^ (p+1)/4
      } catch (sqrtError) {
        const err = sqrtError instanceof Error ? ': ' + sqrtError.message : '';
        throw new Error('bad point: is not on curve, sqrt error' + err);
      }
      assertCompressionIsSupported();
      const evenY = Fp.isOdd!(y);
      const evenH = (head & 1) === 1; // ECDSA-specific
      if (evenH !== evenY) y = Fp.neg(y);
      return { x, y };
    } else if (length === uncomp && head === 0x04) {
      // TODO: more checks
      const L = Fp.BYTES;
      const x = Fp.fromBytes(tail.subarray(0, L));
      const y = Fp.fromBytes(tail.subarray(L, L * 2));
      if (!isValidXY(x, y)) throw new Error('bad point: is not on curve');
      return { x, y };
    } else {
      throw new Error(
        `bad point: got length ${length}, expected compressed=${comp} or uncompressed=${uncomp}`
      );
    }
  }

  const encodePoint = extraOpts.toBytes || pointToBytes;
  const decodePoint = extraOpts.fromBytes || pointFromBytes;
  function weierstrassEquation(x: T): T {
    const x2 = Fp.sqr(x); // x * x
    const x3 = Fp.mul(x2, x); // x² * x
    return Fp.add(Fp.add(x3, Fp.mul(x, CURVE.a)), CURVE.b); // x³ + a * x + b
  }

  // TODO: move top-level
  /** Checks whether equation holds for given x, y: y² == x³ + ax + b */
  function isValidXY(x: T, y: T): boolean {
    const left = Fp.sqr(y); // y²
    const right = weierstrassEquation(x); // x³ + ax + b
    return Fp.eql(left, right);
  }

  // Validate whether the passed curve params are valid.
  // Test 1: equation y² = x³ + ax + b should work for generator point.
  if (!isValidXY(CURVE.Gx, CURVE.Gy)) throw new Error('bad curve params: generator point');

  // Test 2: discriminant Δ part should be non-zero: 4a³ + 27b² != 0.
  // Guarantees curve is genus-1, smooth (non-singular).
  const _4a3 = Fp.mul(Fp.pow(CURVE.a, _3n), _4n);
  const _27b2 = Fp.mul(Fp.sqr(CURVE.b), BigInt(27));
  if (Fp.is0(Fp.add(_4a3, _27b2))) throw new Error('bad curve params: a or b');

  /** Asserts coordinate is valid: 0 <= n < Fp.ORDER. */
  function acoord(title: string, n: T, banZero = false) {
    if (!Fp.isValid(n) || (banZero && Fp.is0(n))) throw new Error(`bad point coordinate ${title}`);
    return n;
  }

  function aprjpoint(other: unknown) {
    if (!(other instanceof Point)) throw new Error('Weierstrass Point expected');
  }

  function splitEndoScalarN(k: bigint) {
    if (!endo || !endo.basises) throw new Error('no endo');
    return _splitEndoScalar(k, endo.basises, Fn.ORDER);
  }

  // Memoized toAffine / validity check. They are heavy. Points are immutable.

  // Converts Projective point to affine (x, y) coordinates.
  // Can accept precomputed Z^-1 - for example, from invertBatch.
  // (X, Y, Z) ∋ (x=X/Z, y=Y/Z)
  const toAffineMemo = memoized((p: Point, iz?: T): AffinePoint<T> => {
    const { X, Y, Z } = p;
    // Fast-path for normalized points
    if (Fp.eql(Z, Fp.ONE)) return { x: X, y: Y };
    const is0 = p.is0();
    // If invZ was 0, we return zero point. However we still want to execute
    // all operations, so we replace invZ with a random number, 1.
    if (iz == null) iz = is0 ? Fp.ONE : Fp.inv(Z);
    const x = Fp.mul(X, iz);
    const y = Fp.mul(Y, iz);
    const zz = Fp.mul(Z, iz);
    if (is0) return { x: Fp.ZERO, y: Fp.ZERO };
    if (!Fp.eql(zz, Fp.ONE)) throw new Error('invZ was invalid');
    return { x, y };
  });
  // NOTE: on exception this will crash 'cached' and no value will be set.
  // Otherwise true will be return
  const assertValidMemo = memoized((p: Point) => {
    if (p.is0()) {
      // (0, 1, 0) aka ZERO is invalid in most contexts.
      // In BLS, ZERO can be serialized, so we allow it.
      // (0, 0, 0) is invalid representation of ZERO.
      if (extraOpts.allowInfinityPoint && !Fp.is0(p.Y)) return;
      throw new Error('bad point: ZERO');
    }
    // Some 3rd-party test vectors require different wording between here & `fromCompressedHex`
    const { x, y } = p.toAffine();
    if (!Fp.isValid(x) || !Fp.isValid(y)) throw new Error('bad point: x or y not field elements');
    if (!isValidXY(x, y)) throw new Error('bad point: equation left != right');
    if (!p.isTorsionFree()) throw new Error('bad point: not in prime-order subgroup');
    return true;
  });

  function finishEndo(
    endoBeta: EndomorphismOpts['beta'],
    k1p: Point,
    k2p: Point,
    k1neg: boolean,
    k2neg: boolean
  ) {
    k2p = new Point(Fp.mul(k2p.X, endoBeta), k2p.Y, k2p.Z);
    k1p = negateCt(k1neg, k1p);
    k2p = negateCt(k2neg, k2p);
    return k1p.add(k2p);
  }

  /**
   * Projective Point works in 3d / projective (homogeneous) coordinates:(X, Y, Z) ∋ (x=X/Z, y=Y/Z).
   * Default Point works in 2d / affine coordinates: (x, y).
   * We're doing calculations in projective, because its operations don't require costly inversion.
   */
  class Point implements WeierstrassPoint<T> {
    // base / generator point
    static readonly BASE = new Point(CURVE.Gx, CURVE.Gy, Fp.ONE);
    // zero / infinity / identity point
    static readonly ZERO = new Point(Fp.ZERO, Fp.ONE, Fp.ZERO); // 0, 1, 0
    // math field
    static readonly Fp = Fp;
    // scalar field
    static readonly Fn = Fn;

    readonly X: T;
    readonly Y: T;
    readonly Z: T;

    /** Does NOT validate if the point is valid. Use `.assertValidity()`. */
    constructor(X: T, Y: T, Z: T) {
      this.X = acoord('x', X);
      this.Y = acoord('y', Y, true);
      this.Z = acoord('z', Z);
      Object.freeze(this);
    }

    static CURVE(): WeierstrassOpts<T> {
      return CURVE;
    }

    /** Does NOT validate if the point is valid. Use `.assertValidity()`. */
    static fromAffine(p: AffinePoint<T>): Point {
      const { x, y } = p || {};
      if (!p || !Fp.isValid(x) || !Fp.isValid(y)) throw new Error('invalid affine point');
      if (p instanceof Point) throw new Error('projective point not allowed');
      // (0, 0) would've produced (0, 0, 1) - instead, we need (0, 1, 0)
      if (Fp.is0(x) && Fp.is0(y)) return Point.ZERO;
      return new Point(x, y, Fp.ONE);
    }

    static fromBytes(bytes: Uint8Array): Point {
      const P = Point.fromAffine(decodePoint(abytes(bytes, undefined, 'point')));
      P.assertValidity();
      return P;
    }

    static fromHex(hex: string): Point {
      return Point.fromBytes(hexToBytes(hex));
    }

    get x(): T {
      return this.toAffine().x;
    }
    get y(): T {
      return this.toAffine().y;
    }

    /**
     *
     * @param windowSize
     * @param isLazy true will defer table computation until the first multiplication
     * @returns
     */
    precompute(windowSize: number = 8, isLazy = true): Point {
      wnaf.createCache(this, windowSize);
      if (!isLazy) this.multiply(_3n); // random number
      return this;
    }

    // TODO: return `this`
    /** A point on curve is valid if it conforms to equation. */
    assertValidity(): void {
      assertValidMemo(this);
    }

    hasEvenY(): boolean {
      const { y } = this.toAffine();
      if (!Fp.isOdd) throw new Error("Field doesn't support isOdd");
      return !Fp.isOdd(y);
    }

    /** Compare one point to another. */
    equals(other: Point): boolean {
      aprjpoint(other);
      const { X: X1, Y: Y1, Z: Z1 } = this;
      const { X: X2, Y: Y2, Z: Z2 } = other;
      const U1 = Fp.eql(Fp.mul(X1, Z2), Fp.mul(X2, Z1));
      const U2 = Fp.eql(Fp.mul(Y1, Z2), Fp.mul(Y2, Z1));
      return U1 && U2;
    }

    /** Flips point to one corresponding to (x, -y) in Affine coordinates. */
    negate(): Point {
      return new Point(this.X, Fp.neg(this.Y), this.Z);
    }

    // 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
    // Cost: 8M + 3S + 3*a + 2*b3 + 15add.
    double() {
      const { a, b } = CURVE;
      const b3 = Fp.mul(b, _3n);
      const { X: X1, Y: Y1, Z: Z1 } = this;
      let X3 = Fp.ZERO, Y3 = Fp.ZERO, Z3 = Fp.ZERO; // prettier-ignore
      let t0 = Fp.mul(X1, X1); // step 1
      let t1 = Fp.mul(Y1, Y1);
      let t2 = Fp.mul(Z1, Z1);
      let t3 = Fp.mul(X1, Y1);
      t3 = Fp.add(t3, t3); // step 5
      Z3 = Fp.mul(X1, Z1);
      Z3 = Fp.add(Z3, Z3);
      X3 = Fp.mul(a, Z3);
      Y3 = Fp.mul(b3, t2);
      Y3 = Fp.add(X3, Y3); // step 10
      X3 = Fp.sub(t1, Y3);
      Y3 = Fp.add(t1, Y3);
      Y3 = Fp.mul(X3, Y3);
      X3 = Fp.mul(t3, X3);
      Z3 = Fp.mul(b3, Z3); // step 15
      t2 = Fp.mul(a, t2);
      t3 = Fp.sub(t0, t2);
      t3 = Fp.mul(a, t3);
      t3 = Fp.add(t3, Z3);
      Z3 = Fp.add(t0, t0); // step 20
      t0 = Fp.add(Z3, t0);
      t0 = Fp.add(t0, t2);
      t0 = Fp.mul(t0, t3);
      Y3 = Fp.add(Y3, t0);
      t2 = Fp.mul(Y1, Z1); // step 25
      t2 = Fp.add(t2, t2);
      t0 = Fp.mul(t2, t3);
      X3 = Fp.sub(X3, t0);
      Z3 = Fp.mul(t2, t1);
      Z3 = Fp.add(Z3, Z3); // step 30
      Z3 = Fp.add(Z3, Z3);
      return new Point(X3, Y3, Z3);
    }

    // Renes-Costello-Batina exception-free addition formula.
    // There is 30% faster Jacobian formula, but it is not complete.
    // https://eprint.iacr.org/2015/1060, algorithm 1
    // Cost: 12M + 0S + 3*a + 3*b3 + 23add.
    add(other: Point): Point {
      aprjpoint(other);
      const { X: X1, Y: Y1, Z: Z1 } = this;
      const { X: X2, Y: Y2, Z: Z2 } = other;
      let X3 = Fp.ZERO, Y3 = Fp.ZERO, Z3 = Fp.ZERO; // prettier-ignore
      const a = CURVE.a;
      const b3 = Fp.mul(CURVE.b, _3n);
      let t0 = Fp.mul(X1, X2); // step 1
      let t1 = Fp.mul(Y1, Y2);
      let t2 = Fp.mul(Z1, Z2);
      let t3 = Fp.add(X1, Y1);
      let t4 = Fp.add(X2, Y2); // step 5
      t3 = Fp.mul(t3, t4);
      t4 = Fp.add(t0, t1);
      t3 = Fp.sub(t3, t4);
      t4 = Fp.add(X1, Z1);
      let t5 = Fp.add(X2, Z2); // step 10
      t4 = Fp.mul(t4, t5);
      t5 = Fp.add(t0, t2);
      t4 = Fp.sub(t4, t5);
      t5 = Fp.add(Y1, Z1);
      X3 = Fp.add(Y2, Z2); // step 15
      t5 = Fp.mul(t5, X3);
      X3 = Fp.add(t1, t2);
      t5 = Fp.sub(t5, X3);
      Z3 = Fp.mul(a, t4);
      X3 = Fp.mul(b3, t2); // step 20
      Z3 = Fp.add(X3, Z3);
      X3 = Fp.sub(t1, Z3);
      Z3 = Fp.add(t1, Z3);
      Y3 = Fp.mul(X3, Z3);
      t1 = Fp.add(t0, t0); // step 25
      t1 = Fp.add(t1, t0);
      t2 = Fp.mul(a, t2);
      t4 = Fp.mul(b3, t4);
      t1 = Fp.add(t1, t2);
      t2 = Fp.sub(t0, t2); // step 30
      t2 = Fp.mul(a, t2);
      t4 = Fp.add(t4, t2);
      t0 = Fp.mul(t1, t4);
      Y3 = Fp.add(Y3, t0);
      t0 = Fp.mul(t5, t4); // step 35
      X3 = Fp.mul(t3, X3);
      X3 = Fp.sub(X3, t0);
      t0 = Fp.mul(t3, t1);
      Z3 = Fp.mul(t5, Z3);
      Z3 = Fp.add(Z3, t0); // step 40
      return new Point(X3, Y3, Z3);
    }

    subtract(other: Point) {
      return this.add(other.negate());
    }

    is0(): boolean {
      return this.equals(Point.ZERO);
    }

    /**
     * Constant time multiplication.
     * Uses wNAF method. Windowed method may be 10% faster,
     * but takes 2x longer to generate and consumes 2x memory.
     * Uses precomputes when available.
     * Uses endomorphism for Koblitz curves.
     * @param scalar by which the point would be multiplied
     * @returns New point
     */
    multiply(scalar: bigint): Point {
      const { endo } = extraOpts;
      if (!Fn.isValidNot0(scalar)) throw new Error('invalid scalar: out of range'); // 0 is invalid
      let point: Point, fake: Point; // Fake point is used to const-time mult
      const mul = (n: bigint) => wnaf.cached(this, n, (p) => normalizeZ(Point, p));
      /** See docs for {@link EndomorphismOpts} */
      if (endo) {
        const { k1neg, k1, k2neg, k2 } = splitEndoScalarN(scalar);
        const { p: k1p, f: k1f } = mul(k1);
        const { p: k2p, f: k2f } = mul(k2);
        fake = k1f.add(k2f);
        point = finishEndo(endo.beta, k1p, k2p, k1neg, k2neg);
      } else {
        const { p, f } = mul(scalar);
        point = p;
        fake = f;
      }
      // Normalize `z` for both points, but return only real one
      return normalizeZ(Point, [point, fake])[0];
    }

    /**
     * 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 secret key e.g. sig verification, which works over *public* keys.
     */
    multiplyUnsafe(sc: bigint): Point {
      const { endo } = extraOpts;
      const p = this as Point;
      if (!Fn.isValid(sc)) throw new Error('invalid scalar: out of range'); // 0 is valid
      if (sc === _0n || p.is0()) return Point.ZERO; // 0
      if (sc === _1n) return p; // 1
      if (wnaf.hasCache(this)) return this.multiply(sc); // precomputes
      // We don't have method for double scalar multiplication (aP + bQ):
      // Even with using Strauss-Shamir trick, it's 35% slower than naïve mul+add.
      if (endo) {
        const { k1neg, k1, k2neg, k2 } = splitEndoScalarN(sc);
        const { p1, p2 } = mulEndoUnsafe(Point, p, k1, k2); // 30% faster vs wnaf.unsafe
        return finishEndo(endo.beta, p1, p2, k1neg, k2neg);
      } else {
        return wnaf.unsafe(p, sc);
      }
    }

    /**
     * Converts Projective point to affine (x, y) coordinates.
     * @param invertedZ Z^-1 (inverted zero) - optional, precomputation is useful for invertBatch
     */
    toAffine(invertedZ?: T): AffinePoint<T> {
      return toAffineMemo(this, invertedZ);
    }

    /**
     * Checks whether Point is free of torsion elements (is in prime subgroup).
     * Always torsion-free for cofactor=1 curves.
     */
    isTorsionFree(): boolean {
      const { isTorsionFree } = extraOpts;
      if (cofactor === _1n) return true;
      if (isTorsionFree) return isTorsionFree(Point, this);
      return wnaf.unsafe(this, CURVE_ORDER).is0();
    }

    clearCofactor(): Point {
      const { clearCofactor } = extraOpts;
      if (cofactor === _1n) return this; // Fast-path
      if (clearCofactor) return clearCofactor(Point, this) as Point;
      return this.multiplyUnsafe(cofactor);
    }

    isSmallOrder(): boolean {
      // can we use this.clearCofactor()?
      return this.multiplyUnsafe(cofactor).is0();
    }

    toBytes(isCompressed = true): Uint8Array {
      abool(isCompressed, 'isCompressed');
      this.assertValidity();
      return encodePoint(Point, this, isCompressed);
    }

    toHex(isCompressed = true): string {
      return bytesToHex(this.toBytes(isCompressed));
    }

    toString() {
      return `<Point ${this.is0() ? 'ZERO' : this.toHex()}>`;
    }
  }
  const bits = Fn.BITS;
  const wnaf = new wNAF(Point, extraOpts.endo ? Math.ceil(bits / 2) : bits);
  Point.BASE.precompute(8); // Enable precomputes. Slows down first publicKey computation by 20ms.
  return Point;
}

/** Methods of ECDSA signature instance. */
export interface ECDSASignature {
  readonly r: bigint;
  readonly s: bigint;
  readonly recovery?: number;
  addRecoveryBit(recovery: number): ECDSASignature & { readonly recovery: number };
  hasHighS(): boolean;
  recoverPublicKey(messageHash: Uint8Array): WeierstrassPoint<bigint>;
  toBytes(format?: string): Uint8Array;
  toHex(format?: string): string;
}
/** Methods of ECDSA signature constructor. */
export type ECDSASignatureCons = {
  new (r: bigint, s: bigint, recovery?: number): ECDSASignature;
  fromBytes(bytes: Uint8Array, format?: ECDSASignatureFormat): ECDSASignature;
  fromHex(hex: string, format?: ECDSASignatureFormat): ECDSASignature;
};

// Points start with byte 0x02 when y is even; otherwise 0x03
function pprefix(hasEvenY: boolean): Uint8Array {
  return Uint8Array.of(hasEvenY ? 0x02 : 0x03);
}

/**
 * Implementation of the Shallue and van de Woestijne method for any weierstrass curve.
 * TODO: check if there is a way to merge this with uvRatio in Edwards; move to modular.
 * b = True and y = sqrt(u / v) if (u / v) is square in F, and
 * b = False and y = sqrt(Z * (u / v)) otherwise.
 * @param Fp
 * @param Z
 * @returns
 */
export function SWUFpSqrtRatio<T>(
  Fp: IField<T>,
  Z: T
): (u: T, v: T) => { isValid: boolean; value: T } {
  // Generic implementation
  const q = Fp.ORDER;
  let l = _0n;
  for (let o = q - _1n; o % _2n === _0n; o /= _2n) l += _1n;
  const c1 = l; // 1. c1, the largest integer such that 2^c1 divides q - 1.
  // We need 2n ** c1 and 2n ** (c1-1). We can't use **; but we can use <<.
  // 2n ** c1 == 2n << (c1-1)
  const _2n_pow_c1_1 = _2n << (c1 - _1n - _1n);
  const _2n_pow_c1 = _2n_pow_c1_1 * _2n;
  const c2 = (q - _1n) / _2n_pow_c1; // 2. c2 = (q - 1) / (2^c1)  # Integer arithmetic
  const c3 = (c2 - _1n) / _2n; // 3. c3 = (c2 - 1) / 2            # Integer arithmetic
  const c4 = _2n_pow_c1 - _1n; // 4. c4 = 2^c1 - 1                # Integer arithmetic
  const c5 = _2n_pow_c1_1; // 5. c5 = 2^(c1 - 1)                  # Integer arithmetic
  const c6 = Fp.pow(Z, c2); // 6. c6 = Z^c2
  const c7 = Fp.pow(Z, (c2 + _1n) / _2n); // 7. c7 = Z^((c2 + 1) / 2)
  let sqrtRatio = (u: T, v: T): { isValid: boolean; value: T } => {
    let tv1 = c6; // 1. tv1 = c6
    let tv2 = Fp.pow(v, c4); // 2. tv2 = v^c4
    let tv3 = Fp.sqr(tv2); // 3. tv3 = tv2^2
    tv3 = Fp.mul(tv3, v); // 4. tv3 = tv3 * v
    let tv5 = Fp.mul(u, tv3); // 5. tv5 = u * tv3
    tv5 = Fp.pow(tv5, c3); // 6. tv5 = tv5^c3
    tv5 = Fp.mul(tv5, tv2); // 7. tv5 = tv5 * tv2
    tv2 = Fp.mul(tv5, v); // 8. tv2 = tv5 * v
    tv3 = Fp.mul(tv5, u); // 9. tv3 = tv5 * u
    let tv4 = Fp.mul(tv3, tv2); // 10. tv4 = tv3 * tv2
    tv5 = Fp.pow(tv4, c5); // 11. tv5 = tv4^c5
    let isQR = Fp.eql(tv5, Fp.ONE); // 12. isQR = tv5 == 1
    tv2 = Fp.mul(tv3, c7); // 13. tv2 = tv3 * c7
    tv5 = Fp.mul(tv4, tv1); // 14. tv5 = tv4 * tv1
    tv3 = Fp.cmov(tv2, tv3, isQR); // 15. tv3 = CMOV(tv2, tv3, isQR)
    tv4 = Fp.cmov(tv5, tv4, isQR); // 16. tv4 = CMOV(tv5, tv4, isQR)
    // 17. for i in (c1, c1 - 1, ..., 2):
    for (let i = c1; i > _1n; i--) {
      let tv5 = i - _2n; // 18.    tv5 = i - 2
      tv5 = _2n << (tv5 - _1n); // 19.    tv5 = 2^tv5
      let tvv5 = Fp.pow(tv4, tv5); // 20.    tv5 = tv4^tv5
      const e1 = Fp.eql(tvv5, Fp.ONE); // 21.    e1 = tv5 == 1
      tv2 = Fp.mul(tv3, tv1); // 22.    tv2 = tv3 * tv1
      tv1 = Fp.mul(tv1, tv1); // 23.    tv1 = tv1 * tv1
      tvv5 = Fp.mul(tv4, tv1); // 24.    tv5 = tv4 * tv1
      tv3 = Fp.cmov(tv2, tv3, e1); // 25.    tv3 = CMOV(tv2, tv3, e1)
      tv4 = Fp.cmov(tvv5, tv4, e1); // 26.    tv4 = CMOV(tv5, tv4, e1)
    }
    return { isValid: isQR, value: tv3 };
  };
  if (Fp.ORDER % _4n === _3n) {
    // sqrt_ratio_3mod4(u, v)
    const c1 = (Fp.ORDER - _3n) / _4n; // 1. c1 = (q - 3) / 4     # Integer arithmetic
    const c2 = Fp.sqrt(Fp.neg(Z)); // 2. c2 = sqrt(-Z)
    sqrtRatio = (u: T, v: T) => {
      let tv1 = Fp.sqr(v); // 1. tv1 = v^2
      const tv2 = Fp.mul(u, v); // 2. tv2 = u * v
      tv1 = Fp.mul(tv1, tv2); // 3. tv1 = tv1 * tv2
      let y1 = Fp.pow(tv1, c1); // 4. y1 = tv1^c1
      y1 = Fp.mul(y1, tv2); // 5. y1 = y1 * tv2
      const y2 = Fp.mul(y1, c2); // 6. y2 = y1 * c2
      const tv3 = Fp.mul(Fp.sqr(y1), v); // 7. tv3 = y1^2; 8. tv3 = tv3 * v
      const isQR = Fp.eql(tv3, u); // 9. isQR = tv3 == u
      let y = Fp.cmov(y2, y1, isQR); // 10. y = CMOV(y2, y1, isQR)
      return { isValid: isQR, value: y }; // 11. return (isQR, y) isQR ? y : y*c2
    };
  }
  // No curves uses that
  // if (Fp.ORDER % _8n === _5n) // sqrt_ratio_5mod8
  return sqrtRatio;
}
/**
 * Simplified Shallue-van de Woestijne-Ulas Method
 * https://www.rfc-editor.org/rfc/rfc9380#section-6.6.2
 */
export function mapToCurveSimpleSWU<T>(
  Fp: IField<T>,
  opts: {
    A: T;
    B: T;
    Z: T;
  }
): (u: T) => { x: T; y: T } {
  validateField(Fp);
  const { A, B, Z } = opts;
  if (!Fp.isValid(A) || !Fp.isValid(B) || !Fp.isValid(Z))
    throw new Error('mapToCurveSimpleSWU: invalid opts');
  const sqrtRatio = SWUFpSqrtRatio(Fp, Z);
  if (!Fp.isOdd) throw new Error('Field does not have .isOdd()');
  // Input: u, an element of F.
  // Output: (x, y), a point on E.
  return (u: T): { x: T; y: T } => {
    // prettier-ignore
    let tv1, tv2, tv3, tv4, tv5, tv6, x, y;
    tv1 = Fp.sqr(u); // 1.  tv1 = u^2
    tv1 = Fp.mul(tv1, Z); // 2.  tv1 = Z * tv1
    tv2 = Fp.sqr(tv1); // 3.  tv2 = tv1^2
    tv2 = Fp.add(tv2, tv1); // 4.  tv2 = tv2 + tv1
    tv3 = Fp.add(tv2, Fp.ONE); // 5.  tv3 = tv2 + 1
    tv3 = Fp.mul(tv3, B); // 6.  tv3 = B * tv3
    tv4 = Fp.cmov(Z, Fp.neg(tv2), !Fp.eql(tv2, Fp.ZERO)); // 7.  tv4 = CMOV(Z, -tv2, tv2 != 0)
    tv4 = Fp.mul(tv4, A); // 8.  tv4 = A * tv4
    tv2 = Fp.sqr(tv3); // 9.  tv2 = tv3^2
    tv6 = Fp.sqr(tv4); // 10. tv6 = tv4^2
    tv5 = Fp.mul(tv6, A); // 11. tv5 = A * tv6
    tv2 = Fp.add(tv2, tv5); // 12. tv2 = tv2 + tv5
    tv2 = Fp.mul(tv2, tv3); // 13. tv2 = tv2 * tv3
    tv6 = Fp.mul(tv6, tv4); // 14. tv6 = tv6 * tv4
    tv5 = Fp.mul(tv6, B); // 15. tv5 = B * tv6
    tv2 = Fp.add(tv2, tv5); // 16. tv2 = tv2 + tv5
    x = Fp.mul(tv1, tv3); // 17.   x = tv1 * tv3
    const { isValid, value } = sqrtRatio(tv2, tv6); // 18. (is_gx1_square, y1) = sqrt_ratio(tv2, tv6)
    y = Fp.mul(tv1, u); // 19.   y = tv1 * u  -> Z * u^3 * y1
    y = Fp.mul(y, value); // 20.   y = y * y1
    x = Fp.cmov(x, tv3, isValid); // 21.   x = CMOV(x, tv3, is_gx1_square)
    y = Fp.cmov(y, value, isValid); // 22.   y = CMOV(y, y1, is_gx1_square)
    const e1 = Fp.isOdd!(u) === Fp.isOdd!(y); // 23.  e1 = sgn0(u) == sgn0(y)
    y = Fp.cmov(Fp.neg(y), y, e1); // 24.   y = CMOV(-y, y, e1)
    const tv4_inv = FpInvertBatch(Fp, [tv4], true)[0];
    x = Fp.mul(x, tv4_inv); // 25.   x = x / tv4
    return { x, y };
  };
}

function getWLengths<T>(Fp: IField<T>, Fn: IField<bigint>) {
  return {
    secretKey: Fn.BYTES,
    publicKey: 1 + Fp.BYTES,
    publicKeyUncompressed: 1 + 2 * Fp.BYTES,
    publicKeyHasPrefix: true,
    signature: 2 * Fn.BYTES,
  };
}

/**
 * Sometimes users only need getPublicKey, getSharedSecret, and secret key handling.
 * This helper ensures no signature functionality is present. Less code, smaller bundle size.
 */
export function ecdh(
  Point: WeierstrassPointCons<bigint>,
  ecdhOpts: { randomBytes?: (bytesLength?: number) => Uint8Array } = {}
): ECDH {
  const { Fn } = Point;
  const randomBytes_ = ecdhOpts.randomBytes || wcRandomBytes;
  const lengths = Object.assign(getWLengths(Point.Fp, Fn), { seed: getMinHashLength(Fn.ORDER) });

  function isValidSecretKey(secretKey: Uint8Array) {
    try {
      const num = Fn.fromBytes(secretKey);
      return Fn.isValidNot0(num);
    } catch (error) {
      return false;
    }
  }

  function isValidPublicKey(publicKey: Uint8Array, isCompressed?: boolean): boolean {
    const { publicKey: comp, publicKeyUncompressed } = lengths;
    try {
      const l = publicKey.length;
      if (isCompressed === true && l !== comp) return false;
      if (isCompressed === false && l !== publicKeyUncompressed) return false;
      return !!Point.fromBytes(publicKey);
    } catch (error) {
      return false;
    }
  }

  /**
   * Produces cryptographically secure secret key from random of size
   * (groupLen + ceil(groupLen / 2)) with modulo bias being negligible.
   */
  function randomSecretKey(seed = randomBytes_(lengths.seed)): Uint8Array {
    return mapHashToField(abytes(seed, lengths.seed, 'seed'), Fn.ORDER);
  }

  /**
   * Computes public key for a secret key. Checks for validity of the secret key.
   * @param isCompressed whether to return compact (default), or full key
   * @returns Public key, full when isCompressed=false; short when isCompressed=true
   */
  function getPublicKey(secretKey: Uint8Array, isCompressed = true): Uint8Array {
    return Point.BASE.multiply(Fn.fromBytes(secretKey)).toBytes(isCompressed);
  }

  /**
   * Quick and dirty check for item being public key. Does not validate hex, or being on-curve.
   */
  function isProbPub(item: Uint8Array): boolean | undefined {
    const { secretKey, publicKey, publicKeyUncompressed } = lengths;
    if (!isBytes(item)) return undefined;
    if (('_lengths' in Fn && Fn._lengths) || secretKey === publicKey) return undefined;
    const l = abytes(item, undefined, 'key').length;
    return l === publicKey || l === publicKeyUncompressed;
  }

  /**
   * ECDH (Elliptic Curve Diffie Hellman).
   * Computes shared public key from secret key A and public key B.
   * Checks: 1) secret key validity 2) shared key is on-curve.
   * Does NOT hash the result.
   * @param isCompressed whether to return compact (default), or full key
   * @returns shared public key
   */
  function getSharedSecret(
    secretKeyA: Uint8Array,
    publicKeyB: Uint8Array,
    isCompressed = true
  ): Uint8Array {
    if (isProbPub(secretKeyA) === true) throw new Error('first arg must be private key');
    if (isProbPub(publicKeyB) === false) throw new Error('second arg must be public key');
    const s = Fn.fromBytes(secretKeyA);
    const b = Point.fromBytes(publicKeyB); // checks for being on-curve
    return b.multiply(s).toBytes(isCompressed);
  }

  const utils = {
    isValidSecretKey,
    isValidPublicKey,
    randomSecretKey,
  };
  const keygen = createKeygen(randomSecretKey, getPublicKey);

  return Object.freeze({ getPublicKey, getSharedSecret, keygen, Point, utils, lengths });
}

/**
 * Creates ECDSA signing interface for given elliptic curve `Point` and `hash` function.
 *
 * @param Point created using {@link weierstrass} function
 * @param hash used for 1) message prehash-ing 2) k generation in `sign`, using hmac_drbg(hash)
 * @param ecdsaOpts rarely needed, see {@link ECDSAOpts}
 *
 * @example
 * ```js
 * const p256_Point = weierstrass(...);
 * const p256_sha256 = ecdsa(p256_Point, sha256);
 * const p256_sha224 = ecdsa(p256_Point, sha224);
 * const p256_sha224_r = ecdsa(p256_Point, sha224, { randomBytes: (length) => { ... } });
 * ```
 */
export function ecdsa(
  Point: WeierstrassPointCons<bigint>,
  hash: CHash,
  ecdsaOpts: ECDSAOpts = {}
): ECDSA {
  ahash(hash);
  validateObject(
    ecdsaOpts,
    {},
    {
      hmac: 'function',
      lowS: 'boolean',
      randomBytes: 'function',
      bits2int: 'function',
      bits2int_modN: 'function',
    }
  );
  ecdsaOpts = Object.assign({}, ecdsaOpts);
  const randomBytes = ecdsaOpts.randomBytes || wcRandomBytes;
  const hmac = ecdsaOpts.hmac || ((key, msg) => nobleHmac(hash, key, msg));

  const { Fp, Fn } = Point;
  const { ORDER: CURVE_ORDER, BITS: fnBits } = Fn;
  const { keygen, getPublicKey, getSharedSecret, utils, lengths } = ecdh(Point, ecdsaOpts);
  const defaultSigOpts: Required<ECDSASignOpts> = {
    prehash: true,
    lowS: typeof ecdsaOpts.lowS === 'boolean' ? ecdsaOpts.lowS : true,
    format: 'compact' as ECDSASignatureFormat,
    extraEntropy: false,
  };
  const hasLargeCofactor = CURVE_ORDER * _2n < Fp.ORDER; // Won't CURVE().h > 2n be more effective?

  function isBiggerThanHalfOrder(number: bigint) {
    const HALF = CURVE_ORDER >> _1n;
    return number > HALF;
  }
  function validateRS(title: string, num: bigint): bigint {
    if (!Fn.isValidNot0(num))
      throw new Error(`invalid signature ${title}: out of range 1..Point.Fn.ORDER`);
    return num;
  }
  function assertSmallCofactor(): void {
    // ECDSA recovery is hard for cofactor > 1 curves.
    // In sign, `r = q.x mod n`, and here we recover q.x from r.
    // While recovering q.x >= n, we need to add r+n for cofactor=1 curves.
    // However, for cofactor>1, r+n may not get q.x:
    // r+n*i would need to be done instead where i is unknown.
    // To easily get i, we either need to:
    // a. increase amount of valid recid values (4, 5...); OR
    // b. prohibit non-prime-order signatures (recid > 1).
    if (hasLargeCofactor)
      throw new Error('"recovered" sig type is not supported for cofactor >2 curves');
  }
  function validateSigLength(bytes: Uint8Array, format: ECDSASignatureFormat) {
    validateSigFormat(format);
    const size = lengths.signature!;
    const sizer = format === 'compact' ? size : format === 'recovered' ? size + 1 : undefined;
    return abytes(bytes, sizer);
  }

  /**
   * ECDSA signature with its (r, s) properties. Supports compact, recovered & DER representations.
   */
  class Signature implements ECDSASignature {
    readonly r: bigint;
    readonly s: bigint;
    readonly recovery?: number;

    constructor(r: bigint, s: bigint, recovery?: number) {
      this.r = validateRS('r', r); // r in [1..N-1];
      this.s = validateRS('s', s); // s in [1..N-1];
      if (recovery != null) {
        assertSmallCofactor();
        if (![0, 1, 2, 3].includes(recovery)) throw new Error('invalid recovery id');
        this.recovery = recovery;
      }
      Object.freeze(this);
    }

    static fromBytes(
      bytes: Uint8Array,
      format: ECDSASignatureFormat = defaultSigOpts.format
    ): Signature {
      validateSigLength(bytes, format);
      let recid: number | undefined;
      if (format === 'der') {
        const { r, s } = DER.toSig(abytes(bytes));
        return new Signature(r, s);
      }
      if (format === 'recovered') {
        recid = bytes[0];
        format = 'compact';
        bytes = bytes.subarray(1);
      }
      const L = lengths.signature! / 2;
      const r = bytes.subarray(0, L);
      const s = bytes.subarray(L, L * 2);
      return new Signature(Fn.fromBytes(r), Fn.fromBytes(s), recid);
    }

    static fromHex