@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 {
  abignumber,
  abool,
  abytes,
  aInRange,
  asafenumber,
  bitLen,
  bitMask,
  bytesToHex,
  bytesToNumberBE,
  concatBytes,
  createHmacDrbg,
  hexToBytes,
  isBytes,
  numberToHexUnpadded,
  validateObject,
  randomBytes as wcRandomBytes,
  type CHash,
  type HmacFn,
  type Signer,
  type TArg,
  type TRet,
} from '../utils.ts';
import {
  createCurveFields,
  createKeygen,
  mulEndoUnsafe,
  negateCt,
  normalizeZ,
  wNAF,
  type AffinePoint,
  type CurveLengths,
  type CurvePoint,
  type CurvePointCons,
} from './curve.ts';
import {
  FpInvertBatch,
  FpIsSquare,
  getMinHashLength,
  mapHashToField,
  validateField,
  type IField,
} from './modular.ts';

/** Shared affine point shape used by Weierstrass helpers. */
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
 * {@link https://gist.github.com/paulmillr/eb670806793e84df628a7c434a873066 | this endomorphism gist}.
 */
export type EndomorphismOpts = {
  /** Cube root of unity used by the GLV endomorphism. */
  beta: bigint;
  /** Reduced lattice basis used for scalar splitting. */
  basises?: EndoBasis;
  /**
   * Optional custom scalar-splitting helper.
   * Receives one scalar and returns two half-sized scalar components.
   */
  splitScalar?: (k: bigint) => { k1neg: boolean; k1: bigint; k2neg: boolean; k2: bigint };
};
// We construct the basis so `den` is always positive and equals `n`,
// but the `num` sign depends on the basis, not on the secret value.
// Exact half-way cases round away from zero, which keeps the split symmetric
// around the reduced-basis boundaries used by endomorphism decomposition.
const divNearest = (num: bigint, den: bigint) => (num + (num >= 0 ? den : -den) / _2n) / den;

/** Two half-sized scalar components returned by endomorphism splitting. */
export type ScalarEndoParts = {
  /** Whether the first split scalar should be negated. */
  k1neg: boolean;
  /** Absolute value of the first split scalar. */
  k1: bigint;
  /** Whether the second split scalar should be negated. */
  k2neg: boolean;
  /** Absolute value of the second split scalar. */
  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.
  // Callers must provide a reduced GLV basis whose vectors satisfy
  // `a + b * lambda ≡ 0 (mod n)`; this helper only sees the basis and `n`.
  // Reject unreduced scalars instead of silently treating them mod n.
  aInRange('scalar', k, _0n, n);
  // 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 bases.
  // Also, the math inside is complex enough that this guard is worth keeping.
  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 for 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"
 *
 * See {@link https://paulmillr.com/posts/deterministic-signatures/ | 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 = {
  /** Whether to hash the message before signature recovery. */
  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 with `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 = {
  /** Whether to hash the message before verification. */
  prehash?: boolean;
  /** Whether to reject high-S signatures. */
  lowS?: boolean;
  /** Signature encoding to accept. */
  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 with `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 signatures with increased
 *   security, see {@link ECDSAExtraEntropy}
 */
export type ECDSASignOpts = {
  /** Whether to hash the message before signing. */
  prehash?: boolean;
  /** Whether to normalize signatures into the low-S half-order. */
  lowS?: boolean;
  /** Signature encoding to produce. */
  format?: ECDSASignatureFormat;
  /** Optional hedging input for deterministic k generation. */
  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
): D {
  validateObject(opts);
  const optsn = {} as D;
  // Normalize only the declared option subset from `def`; unknown keys are
  // intentionally ignored so shared / superset option bags stay valid here too.
  // `extraEntropy` stays an opaque payload until the signing path consumes it.
  for (let optName of Object.keys(def) as (keyof D)[]) {
    // @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;
}

/** Projective XYZ point used by short Weierstrass curves. */
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;
  /**
   * Encode the point into compressed or uncompressed SEC1 bytes.
   * @param isCompressed - Whether to use the compressed form.
   * @returns Encoded point bytes.
   */
  toBytes(isCompressed?: boolean): TRet<Uint8Array>;
  /**
   * Encode the point into compressed or uncompressed SEC1 hex.
   * @param isCompressed - Whether to use the compressed form.
   * @returns Encoded point hex.
   */
  toHex(isCompressed?: boolean): string;
}

/** Constructor and metadata helpers for Weierstrass 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>;
  /**
   * Return the curve parameters captured by this point constructor.
   * @returns Curve parameters.
   */
  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<{
  /** Base-field modulus. */
  p: bigint;
  /** Prime subgroup order. */
  n: bigint;
  /** Curve cofactor. */
  h: bigint;
  /** Weierstrass curve parameter `a`. */
  a: T;
  /** Weierstrass curve parameter `b`. */
  b: T;
  /** Generator x coordinate. */
  Gx: T;
  /** Generator y coordinate. */
  Gy: T;
}>;

/**
 * Optional helpers and overrides for a Weierstrass point constructor.
 *
 * When a cofactor != 1, there can be effective methods to:
 * 1. Determine whether a point is torsion-free
 * 2. Clear torsion component
 */
export type WeierstrassExtraOpts<T> = Partial<{
  /** Optional base-field override. */
  Fp: IField<T>;
  /** Optional scalar-field override. */
  Fn: IField<bigint>;
  /** Whether the point constructor accepts infinity points. */
  allowInfinityPoint: boolean;
  /** Optional GLV endomorphism data. */
  endo: EndomorphismOpts;
  /** Optional torsion-check override. */
  isTorsionFree: (c: WeierstrassPointCons<T>, point: WeierstrassPoint<T>) => boolean;
  /** Optional cofactor-clearing override. */
  clearCofactor: (c: WeierstrassPointCons<T>, point: WeierstrassPoint<T>) => WeierstrassPoint<T>;
  /** Optional custom point decoder. */
  fromBytes: (bytes: TArg<Uint8Array>) => AffinePoint<T>;
  /** Optional custom point encoder. */
  toBytes: (
    c: WeierstrassPointCons<T>,
    point: WeierstrassPoint<T>,
    isCompressed: boolean
  ) => TRet<Uint8Array>;
}>;

/**
 * Options for ECDSA signatures over a Weierstrass curve.
 *
 * * lowS: (default: true) whether produced or verified signatures occupy the
 *   low half of `ecdsaOpts.n`. 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. Custom hooks are
 *   treated as pure functions over validated bytes and MUST NOT mutate caller-owned buffers or
 *   closure-captured option bags. `bits2int_modN` must also return a canonical scalar in
 *   `[0..Point.Fn.ORDER-1]`.
 */
export type ECDSAOpts = Partial<{
  /** Default low-S policy for this ECDSA instance. */
  lowS: boolean;
  /** HMAC implementation used by RFC6979 DRBG. */
  hmac: HmacFn;
  /** RNG override used by helper constructors. */
  randomBytes: (bytesLength?: number) => TRet<Uint8Array>;
  /** Hash-to-integer conversion override. */
  bits2int: (bytes: TArg<Uint8Array>) => bigint;
  /** Hash-to-integer-mod-n conversion override. Returns a canonical scalar in `[0..Fn.ORDER-1]`. */
  bits2int_modN: (bytes: TArg<Uint8Array>) => bigint;
}>;

/** Elliptic Curve Diffie-Hellman helper namespace. */
export interface ECDH {
  /**
   * Generate a secret/public key pair.
   * @param seed - Optional seed material.
   * @returns Secret/public key pair.
   */
  keygen: (seed?: TArg<Uint8Array>) => { secretKey: TRet<Uint8Array>; publicKey: TRet<Uint8Array> };
  /**
   * Derive the public key from a secret key.
   * @param secretKey - Secret key bytes.
   * @param isCompressed - Whether to emit compressed SEC1 bytes.
   * @returns Encoded public key.
   */
  getPublicKey: (secretKey: TArg<Uint8Array>, isCompressed?: boolean) => TRet<Uint8Array>;
  /**
   * Compute the shared secret point from a secret key and peer public key.
   * @param secretKeyA - Local secret key bytes.
   * @param publicKeyB - Peer public key bytes.
   * @param isCompressed - Whether to emit compressed SEC1 bytes.
   * @returns Encoded shared point.
   */
  getSharedSecret: (
    secretKeyA: TArg<Uint8Array>,
    publicKeyB: TArg<Uint8Array>,
    isCompressed?: boolean
  ) => TRet<Uint8Array>;
  /** Point constructor used by this ECDH instance. */
  Point: WeierstrassPointCons<bigint>;
  /** Validation and random-key helpers. */
  utils: {
    /** Check whether a secret key has the expected encoding. */
    isValidSecretKey: (secretKey: TArg<Uint8Array>) => boolean;
    /** Check whether a public key decodes to a valid point. */
    isValidPublicKey: (publicKey: TArg<Uint8Array>, isCompressed?: boolean) => boolean;
    /** Generate a valid random secret key. */
    randomSecretKey: (seed?: TArg<Uint8Array>) => TRet<Uint8Array>;
  };
  /** Byte lengths for keys and signatures exposed by this curve. */
  lengths: CurveLengths;
}

/**
 * ECDSA interface.
 * Only supported for prime fields, not Fp2 (extension fields).
 */
export interface ECDSA extends ECDH {
  /**
   * Sign a message with the given secret key.
   * @param message - Message bytes.
   * @param secretKey - Secret key bytes.
   * @param opts - Optional signing tweaks. See {@link ECDSASignOpts}.
   * @returns Encoded signature bytes.
   */
  sign: (
    message: TArg<Uint8Array>,
    secretKey: TArg<Uint8Array>,
    opts?: TArg<ECDSASignOpts>
  ) => TRet<Uint8Array>;
  /**
   * Verify a signature against a message and public key.
   * @param signature - Encoded signature bytes.
   * @param message - Message bytes.
   * @param publicKey - Encoded public key.
   * @param opts - Optional verification tweaks. See {@link ECDSAVerifyOpts}.
   * @returns Whether the signature is valid.
   */
  verify: (
    signature: TArg<Uint8Array>,
    message: TArg<Uint8Array>,
    publicKey: TArg<Uint8Array>,
    opts?: TArg<ECDSAVerifyOpts>
  ) => boolean;
  /**
   * Recover the public key encoded into a recoverable signature.
   * @param signature - Recoverable signature bytes.
   * @param message - Message bytes.
   * @param opts - Optional recovery tweaks. See {@link ECDSARecoverOpts}.
   * @returns Encoded recovered public key.
   */
  recoverPublicKey(
    signature: TArg<Uint8Array>,
    message: TArg<Uint8Array>,
    opts?: TArg<ECDSARecoverOpts>
  ): TRet<Uint8Array>;
  /** Signature constructor and parser helpers. */
  Signature: ECDSASignatureCons;
}
/**
 * @param m - Error message.
 * @example
 * Throw a DER-specific error when signature parsing encounters invalid bytes.
 *
 * ```ts
 * new DERErr('bad der');
 * ```
 */
export class DERErr extends Error {
  constructor(m = '') {
    super(m);
  }
}
/** DER helper namespace used by ECDSA signature parsing and encoding. */
export type IDER = {
  // asn.1 DER encoding utils
  /**
   * DER-specific error constructor.
   * @param m - Error message.
   * @returns DER-specific error instance.
   */
  Err: typeof DERErr;
  // Basic building block is TLV (Tag-Length-Value)
  /** Low-level tag-length-value helpers used by DER encoders. */
  _tlv: {
    /**
     * Encode one TLV record.
     * @param tag - ASN.1 tag byte.
     * @param data - Hex-encoded value payload.
     * @returns Encoded TLV string.
     */
    encode: (tag: number, data: string) => string;
    // v - value, l - left bytes (unparsed)
    /**
     * Decode one TLV record and return the value plus leftover bytes.
     * @param tag - Expected ASN.1 tag byte.
     * @param data - Remaining DER bytes.
     * @returns Parsed value plus leftover bytes.
     */
    decode(tag: number, data: TArg<Uint8Array>): TRet<{ 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)
  /** Positive-integer DER helpers used by ECDSA signature encoding. */
  _int: {
    /**
     * Encode one positive bigint as a DER INTEGER.
     * @param num - Positive integer to encode.
     * @returns Encoded DER INTEGER.
     */
    encode(num: bigint): string;
    /**
     * Decode one DER INTEGER into a bigint.
     * @param data - DER INTEGER bytes.
     * @returns Decoded bigint.
     */
    decode(data: TArg<Uint8Array>): bigint;
  };
  /**
   * Parse a DER signature into `{ r, s }`.
   * @param bytes - DER signature bytes.
   * @returns Parsed signature components.
   */
  toSig(bytes: TArg<Uint8Array>): { r: bigint; s: bigint };
  /**
   * Encode `{ r, s }` as a DER signature.
   * @param sig - Signature components.
   * @returns DER-encoded signature hex.
   */
  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: {@link https://letsencrypt.org/docs/a-warm-welcome-to-asn1-and-der/ | Let's Encrypt ASN.1 guide} and
 * {@link https://luca.ntop.org/Teaching/Appunti/asn1.html | Luca Deri's ASN.1 notes}.
 * @example
 * ASN.1 DER encoding utilities.
 *
 * ```ts
 * const der = DER.hexFromSig({ r: 1n, s: 2n });
 * ```
 */
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;
      asafenumber(tag, 'tag');
      if (tag < 0 || tag > 255) throw new E('tlv.encode: wrong tag');
      if (typeof data !== 'string')
        throw new TypeError('"data" expected string, got type=' + typeof data);
      // Internal helper: callers hand this already-validated hex payload, so we only enforce
      // byte alignment here instead of re-validating every nibble.
      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: TArg<Uint8Array>): TRet<{ v: Uint8Array; l: Uint8Array }> {
      const { Err: E } = DER;
      data = abytes(data, undefined, 'DER data');
      let pos = 0;
      if (tag < 0 || tag > 255) 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++];
      // First bit of first length byte is the short/long form flag.
      const isLong = !!(first & 0b1000_0000);
      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');
        // This would overflow u32 in JS.
        if (lenLen > 4) throw new E('tlv.decode(long): byte length is too big');
        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) } as TRet<{ 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 {
      const { Err: E } = DER;
      abignumber(num);
      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: TArg<Uint8Array>): bigint {
      const { Err: E } = DER;
      if (data.length < 1) throw new E('invalid signature integer: empty');
      if (data[0] & 0b1000_0000) throw new E('invalid signature integer: negative');
      // Single-byte zero `00` is the canonical DER INTEGER encoding for zero.
      if (data.length > 1 && data[0] === 0x00 && !(data[1] & 0b1000_0000))
        throw new E('invalid signature integer: unnecessary leading zero');
      return bytesToNumberBE(data);
    },
  },
  toSig(bytes: TArg<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);
  },
};
Object.freeze(DER._tlv);
Object.freeze(DER._int);
Object.freeze(DER);

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

/**
 * Creates weierstrass Point constructor, based on specified curve options.
 *
 * See {@link WeierstrassOpts}.
 * @param params - Curve parameters. See {@link WeierstrassOpts}.
 * @param extraOpts - Optional helpers and overrides. See {@link WeierstrassExtraOpts}.
 * @returns Weierstrass point constructor.
 * @throws If the curve parameters, overrides, or point codecs are invalid. {@link Error}
 *
 * @example
 * Construct a point type from explicit Weierstrass curve parameters.
 *
 * ```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 = validated.Fp as IField<T>;
  const Fn = validated.Fn as IField<bigint>;
  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',
    }
  );

  // Snapshot constructor-time flags whose later mutation would otherwise change
  // validity semantics of an already-built point type.
  const { endo, allowInfinityPoint } = 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 as TArg<IField<T>>, 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
  ): TRet<Uint8Array> {
    // SEC 1 v2.0 §2.3.3 encodes infinity as the single octet 0x00. Only curves
    // that opt into infinity as a public point value should expose that byte form.
    if (allowInfinityPoint && point.is0()) return Uint8Array.of(0) as TRet<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) as TRet<Uint8Array>;
    } else {
      return concatBytes(Uint8Array.of(0x04), bx, Fp.toBytes(y)) as TRet<Uint8Array>;
    }
  }
  function pointFromBytes(bytes: TArg<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);
    if (allowInfinityPoint && length === 1 && head === 0x00) return { x: Fp.ZERO, y: Fp.ZERO };
    // SEC 1 v2.0 §2.3.4 decodes 0x00 as infinity, but §3.2.2 public-key validation
    // rejects infinity. We therefore keep 0x00 rejected by default because callers
    // reuse this parser as the strict public-key boundary, and only admit it when
    // the curve explicitly opts into infinity as a public point value. secp256k1
    // crosstests show OpenSSL raw point codecs accept 0x00 too.
    // 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 === undefined ? pointToBytes : extraOpts.toBytes;
  const decodePoint = extraOpts.fromBytes === undefined ? pointFromBytes : extraOpts.fromBytes;
  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);
  }

  // Keep constructor-time generator validation cheap: callers are responsible for supplying the
  // correct prime-order base point, while eager subgroup checks here would slow heavy module imports.
  // 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): asserts other is Point {
    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);
  }

  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 is not just about ZERO / infinity: ambient curves can have real
      // finite points with y=0. Those points are 2-torsion, so they cannot lie
      // in the odd prime-order subgroups this point type is meant to represent.
      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: TArg<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 {
      const p = this;
      if (p.is0()) {
        // (0, 1, 0) aka ZERO is invalid in most contexts.
        // In BLS, ZERO can be serialized, so we allow it.
        // Keep the accepted infinity encoding canonical: projective-equivalent (X, Y, 0) points
        // like (1, 1, 0) compare equal to ZERO, but only (0, 1, 0) should pass this guard.
        if (extraOpts.allowInfinityPoint && Fp.is0(p.X) && Fp.eql(p.Y, Fp.ONE) && Fp.is0(p.Z))
          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');
    }

    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: WeierstrassPoint<T>): 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: WeierstrassPoint<T>): 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: WeierstrassPoint<T>) {
      // Validate before calling `negate()` so wrong inputs fail with the point guard
      // instead of leaking a foreign `negate()` error.
      aprjpoint(other);
      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;
      // Keep the subgroup-scalar contract strict instead of reducing 0 / n to ZERO.
      // In key/signature-style callers, those values usually mean broken hash/scalar plumbing,
      // and failing closed is safer than silently producing the identity point.
      if (!Fn.isValidNot0(scalar)) throw new RangeError('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(scalar: bigint): Point {
      const { endo } = extraOpts;
      const p = this as Point;
      const sc = scalar;
      // Public-scalar callers may need 0, but n and larger values stay rejected here too.
      // Reducing them mod n would turn bad caller input into an accidental identity point.
      if (!Fn.isValid(sc)) throw new RangeError('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.
     * (X, Y, Z) ∋ (x=X/Z, y=Y/Z).
     * @param invertedZ - Z^-1 (inverted zero) - optional, precomputation is useful for invertBatch
     */
    toAffine(invertedZ?: T): AffinePoint<T> {
      const p = this;
      let iz = invertedZ;
      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 };
    }

    /**
     * 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;
      // Default fallback assumes the cofactor fits the usual subgroup-scalar
      // multiplyUnsafe() contract. Curves with larger / structured cofactors
      // should define a clearCofactor override anyway (e.g. psi/Frobenius maps).
      return this.multiplyUnsafe(cofactor);
    }

    isSmallOrder(): boolean {
      if (cofactor === _1n) return this.is0(); // Fast-path
      return this.clearCofactor().is0();
    }

    toBytes(isCompressed = true): TRet<Uint8Array> {
      abool(isCompressed, 'isCompressed');
      // Same policy as pointFromBytes(): keep ZERO out of the default byte surface because
      // callers use these encodings as public keys, where SEC 1 validation rejects infinity.
      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);
  // Tiny toy curves can have scalar fields narrower than 8 bits. Skip the
  // eager W=8 cache there instead of rejecting an otherwise valid constructor.
  if (bits >= 8) Point.BASE.precompute(8); // Enable precomputes. Slows down first publicKey computation by 20ms.
  Object.freeze(Point.prototype);
  Object.freeze(Point);
  return Point;
}

/** Parsed ECDSA signature with helpers for recovery and re-encoding. */
export interface ECDSASignature {
  /** Signature component `r`. */
  readonly r: bigint;
  /** Signature component `s`. */
  readonly s: bigint;
  /** Optional recovery bit for recoverable signatures. */
  readonly recovery?: number;
  /**
   * Return a copy of the signature with a recovery bit attached.
   * @param recovery - Recovery bit to attach.
   * @returns Signature with an attached recovery bit.
   */
  addRecoveryBit(recovery: number): ECDSASignature & { readonly recovery: number };
  /**
   * Check whether the signature uses the high-S half-order.
   * @returns Whether the signature uses the high-S half-order.
   */
  hasHighS(): boolean;
  /**
   * Recover the public key from the hashed message and recovery bit.
   * @param messageHash - Hashed message bytes.
   * @returns Recovered public-key point.
   */
  recoverPublicKey(messageHash: TArg<Uint8Array>): WeierstrassPoint<bigint>;
  /**
   * Encode the signature into bytes.
   * @param format - Signature encoding to produce.
   * @returns Encoded signature bytes.
   */
  toBytes(format?: string): TRet<Uint8Array>;
  /**
   * Encode the signature into hex.
   * @param format - Signature encoding to produce.
   * @returns Encoded signature hex.
   */
  toHex(format?: string): string;
}
/** Constructor and decoding helpers for ECDSA signatures. */
export type ECDSASignatureCons = {
  /** Create a signature from `r`, `s`, and an optional recovery bit. */
  new (r: bigint, s: bigint, recovery?: number): ECDSASignature;
  /**
   * Decode a signature from bytes.
   * @param bytes - Encoded signature bytes.
   * @param format - Signature encoding to parse.
   * @returns Parsed signature.
   */
  fromBytes(bytes: TArg<Uint8Array>, format?: ECDSASignatureFormat): ECDSASignature;
  /**
   * Decode a signature from hex.
   * @param hex - Encoded signature hex.
   * @param format - Signature encoding to parse.
   * @returns Parsed signature.
   */
  fromHex(hex: string, format?: ECDSASignatureFormat): ECDSASignature;
};

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

/**
 * 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.
 * RFC 9380 expects callers to provide `v != 0`; this helper does not enforce it.
 * @param Fp - Field implementation.
 * @param Z - Simplified SWU map parameter.
 * @returns Square-root ratio helper.
 * @example
 * Build the square-root ratio helper used by SWU map implementations.
 *
 * ```ts
 * import { SWUFpSqrtRatio } from '@noble/curves/abstract/weierstrass.js';
 * import { Field } from '@noble/curves/abstract/modular.js';
 * const Fp = Field(17n);
 * const sqrtRatio = SWUFpSqrtRatio(Fp, 3n);
 * const out = sqrtRatio(4n, 1n);
 * ```
 */
export function SWUFpSqrtRatio<T>(
  Fp: TArg<IField<T>>,
  Z: T
): (u: T, v: T) => { isValid: boolean; value: T } {
  // Fail with the usual field-shape error before touching pow/cmov on malformed field shims.
  const F = validateField(Fp as IField<T>) as IField<T>;
  // Generic implementation
  const q = F.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