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Просмотр файла: ed448.ts
/**
* Edwards448 (not Ed448-Goldilocks) curve with following addons:
* - X448 ECDH
* - Decaf cofactor elimination
* - Elligator hash-to-group / point indistinguishability
* Conforms to RFC 8032 https://www.rfc-editor.org/rfc/rfc8032.html#section-5.2
* @module
*/
/*! noble-curves - MIT License (c) 2022 Paul Miller (paulmillr.com) */
import { shake256 } from '@noble/hashes/sha3.js';
import { abytes, concatBytes, createHasher as wrapConstructor } from '@noble/hashes/utils.js';
import type { AffinePoint } from './abstract/curve.ts';
import { pippenger } from './abstract/curve.ts';
import {
edwards,
PrimeEdwardsPoint,
twistedEdwards,
type CurveFn,
type EdwardsOpts,
type EdwardsPoint,
type EdwardsPointCons,
} from './abstract/edwards.ts';
import {
_DST_scalar,
createHasher,
expand_message_xof,
type H2CHasher,
type H2CHasherBase,
type H2CMethod,
type htfBasicOpts,
} from './abstract/hash-to-curve.ts';
import { Field, FpInvertBatch, isNegativeLE, mod, pow2, type IField } from './abstract/modular.ts';
import { montgomery, type MontgomeryECDH as XCurveFn } from './abstract/montgomery.ts';
import { asciiToBytes, bytesToNumberLE, ensureBytes, equalBytes, type Hex } from './utils.ts';
// edwards448 curve
// a = 1n
// d = Fp.neg(39081n)
// Finite field 2n**448n - 2n**224n - 1n
// Subgroup order
// 2n**446n - 13818066809895115352007386748515426880336692474882178609894547503885n
const ed448_CURVE: EdwardsOpts = {
p: BigInt(
'0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffff'
),
n: BigInt(
'0x3fffffffffffffffffffffffffffffffffffffffffffffffffffffff7cca23e9c44edb49aed63690216cc2728dc58f552378c292ab5844f3'
),
h: BigInt(4),
a: BigInt(1),
d: BigInt(
'0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffffffffffffffffffffffffffffffffffffffffffffffff6756'
),
Gx: BigInt(
'0x4f1970c66bed0ded221d15a622bf36da9e146570470f1767ea6de324a3d3a46412ae1af72ab66511433b80e18b00938e2626a82bc70cc05e'
),
Gy: BigInt(
'0x693f46716eb6bc248876203756c9c7624bea73736ca3984087789c1e05a0c2d73ad3ff1ce67c39c4fdbd132c4ed7c8ad9808795bf230fa14'
),
};
// E448 NIST curve is identical to edwards448, except for:
// d = 39082/39081
// Gx = 3/2
const E448_CURVE: EdwardsOpts = Object.assign({}, ed448_CURVE, {
d: BigInt(
'0xd78b4bdc7f0daf19f24f38c29373a2ccad46157242a50f37809b1da3412a12e79ccc9c81264cfe9ad080997058fb61c4243cc32dbaa156b9'
),
Gx: BigInt(
'0x79a70b2b70400553ae7c9df416c792c61128751ac92969240c25a07d728bdc93e21f7787ed6972249de732f38496cd11698713093e9c04fc'
),
Gy: BigInt(
'0x7fffffffffffffffffffffffffffffffffffffffffffffffffffffff80000000000000000000000000000000000000000000000000000001'
),
});
const shake256_114 = /* @__PURE__ */ wrapConstructor(() => shake256.create({ dkLen: 114 }));
const shake256_64 = /* @__PURE__ */ wrapConstructor(() => shake256.create({ dkLen: 64 }));
// prettier-ignore
const _1n = BigInt(1), _2n = BigInt(2), _3n = BigInt(3), _4n = BigInt(4), _11n = BigInt(11);
// prettier-ignore
const _22n = BigInt(22), _44n = BigInt(44), _88n = BigInt(88), _223n = BigInt(223);
// powPminus3div4 calculates z = x^k mod p, where k = (p-3)/4.
// Used for efficient square root calculation.
// ((P-3)/4).toString(2) would produce bits [223x 1, 0, 222x 1]
function ed448_pow_Pminus3div4(x: bigint): bigint {
const P = ed448_CURVE.p;
const b2 = (x * x * x) % P;
const b3 = (b2 * b2 * x) % P;
const b6 = (pow2(b3, _3n, P) * b3) % P;
const b9 = (pow2(b6, _3n, P) * b3) % P;
const b11 = (pow2(b9, _2n, P) * b2) % P;
const b22 = (pow2(b11, _11n, P) * b11) % P;
const b44 = (pow2(b22, _22n, P) * b22) % P;
const b88 = (pow2(b44, _44n, P) * b44) % P;
const b176 = (pow2(b88, _88n, P) * b88) % P;
const b220 = (pow2(b176, _44n, P) * b44) % P;
const b222 = (pow2(b220, _2n, P) * b2) % P;
const b223 = (pow2(b222, _1n, P) * x) % P;
return (pow2(b223, _223n, P) * b222) % P;
}
function adjustScalarBytes(bytes: Uint8Array): Uint8Array {
// Section 5: Likewise, for X448, set the two least significant bits of the first byte to 0,
bytes[0] &= 252; // 0b11111100
// and the most significant bit of the last byte to 1.
bytes[55] |= 128; // 0b10000000
// NOTE: is NOOP for 56 bytes scalars (X25519/X448)
bytes[56] = 0; // Byte outside of group (456 buts vs 448 bits)
return bytes;
}
// Constant-time ratio of u to v. Allows to combine inversion and square root u/√v.
// Uses algo from RFC8032 5.1.3.
function uvRatio(u: bigint, v: bigint): { isValid: boolean; value: bigint } {
const P = ed448_CURVE.p;
// https://www.rfc-editor.org/rfc/rfc8032#section-5.2.3
// To compute the square root of (u/v), the first step is to compute the
// candidate root x = (u/v)^((p+1)/4). This can be done using the
// following trick, to use a single modular powering for both the
// inversion of v and the square root:
// x = (u/v)^((p+1)/4) = u³v(u⁵v³)^((p-3)/4) (mod p)
const u2v = mod(u * u * v, P); // u²v
const u3v = mod(u2v * u, P); // u³v
const u5v3 = mod(u3v * u2v * v, P); // u⁵v³
const root = ed448_pow_Pminus3div4(u5v3);
const x = mod(u3v * root, P);
// Verify that root is exists
const x2 = mod(x * x, P); // x²
// If vx² = u, the recovered x-coordinate is x. Otherwise, no
// square root exists, and the decoding fails.
return { isValid: mod(x2 * v, P) === u, value: x };
}
// Finite field 2n**448n - 2n**224n - 1n
// The value fits in 448 bits, but we use 456-bit (57-byte) elements because of bitflags.
// - ed25519 fits in 255 bits, allowing using last 1 byte for specifying bit flag of point negation.
// - ed448 fits in 448 bits. We can't use last 1 byte: we can only use a bit 224 in the middle.
const Fp = /* @__PURE__ */ (() => Field(ed448_CURVE.p, { BITS: 456, isLE: true }))();
const Fn = /* @__PURE__ */ (() => Field(ed448_CURVE.n, { BITS: 456, isLE: true }))();
// decaf448 uses 448-bit (56-byte) keys
const Fp448 = /* @__PURE__ */ (() => Field(ed448_CURVE.p, { BITS: 448, isLE: true }))();
const Fn448 = /* @__PURE__ */ (() => Field(ed448_CURVE.n, { BITS: 448, isLE: true }))();
// SHAKE256(dom4(phflag,context)||x, 114)
function dom4(data: Uint8Array, ctx: Uint8Array, phflag: boolean) {
if (ctx.length > 255) throw new Error('context must be smaller than 255, got: ' + ctx.length);
return concatBytes(
asciiToBytes('SigEd448'),
new Uint8Array([phflag ? 1 : 0, ctx.length]),
ctx,
data
);
}
// const ed448_eddsa_opts = { adjustScalarBytes, domain: dom4 };
// const ed448_Point = edwards(ed448_CURVE, { Fp, Fn, uvRatio });
const ED448_DEF = /* @__PURE__ */ (() => ({
...ed448_CURVE,
Fp,
Fn,
nBitLength: Fn.BITS,
hash: shake256_114,
adjustScalarBytes,
domain: dom4,
uvRatio,
}))();
/**
* ed448 EdDSA curve and methods.
* @example
* import { ed448 } from '@noble/curves/ed448';
* const { secretKey, publicKey } = ed448.keygen();
* const msg = new TextEncoder().encode('hello');
* const sig = ed448.sign(msg, secretKey);
* const isValid = ed448.verify(sig, msg, publicKey);
*/
export const ed448: CurveFn = twistedEdwards(ED448_DEF);
// There is no ed448ctx, since ed448 supports ctx by default
/** Prehashed version of ed448. Accepts already-hashed messages in sign() and verify(). */
export const ed448ph: CurveFn = /* @__PURE__ */ (() =>
twistedEdwards({
...ED448_DEF,
prehash: shake256_64,
}))();
/**
* E448 curve, defined by NIST.
* E448 != edwards448 used in ed448.
* E448 is birationally equivalent to edwards448.
*/
export const E448: EdwardsPointCons = edwards(E448_CURVE);
/**
* ECDH using curve448 aka x448.
* x448 has 56-byte keys as per RFC 7748, while
* ed448 has 57-byte keys as per RFC 8032.
*/
export const x448: XCurveFn = /* @__PURE__ */ (() => {
const P = ed448_CURVE.p;
return montgomery({
P,
type: 'x448',
powPminus2: (x: bigint): bigint => {
const Pminus3div4 = ed448_pow_Pminus3div4(x);
const Pminus3 = pow2(Pminus3div4, _2n, P);
return mod(Pminus3 * x, P); // Pminus3 * x = Pminus2
},
adjustScalarBytes,
});
})();
// Hash To Curve Elligator2 Map
const ELL2_C1 = /* @__PURE__ */ (() => (Fp.ORDER - BigInt(3)) / BigInt(4))(); // 1. c1 = (q - 3) / 4 # Integer arithmetic
const ELL2_J = /* @__PURE__ */ BigInt(156326);
function map_to_curve_elligator2_curve448(u: bigint) {
let tv1 = Fp.sqr(u); // 1. tv1 = u^2
let e1 = Fp.eql(tv1, Fp.ONE); // 2. e1 = tv1 == 1
tv1 = Fp.cmov(tv1, Fp.ZERO, e1); // 3. tv1 = CMOV(tv1, 0, e1) # If Z * u^2 == -1, set tv1 = 0
let xd = Fp.sub(Fp.ONE, tv1); // 4. xd = 1 - tv1
let x1n = Fp.neg(ELL2_J); // 5. x1n = -J
let tv2 = Fp.sqr(xd); // 6. tv2 = xd^2
let gxd = Fp.mul(tv2, xd); // 7. gxd = tv2 * xd # gxd = xd^3
let gx1 = Fp.mul(tv1, Fp.neg(ELL2_J)); // 8. gx1 = -J * tv1 # x1n + J * xd
gx1 = Fp.mul(gx1, x1n); // 9. gx1 = gx1 * x1n # x1n^2 + J * x1n * xd
gx1 = Fp.add(gx1, tv2); // 10. gx1 = gx1 + tv2 # x1n^2 + J * x1n * xd + xd^2
gx1 = Fp.mul(gx1, x1n); // 11. gx1 = gx1 * x1n # x1n^3 + J * x1n^2 * xd + x1n * xd^2
let tv3 = Fp.sqr(gxd); // 12. tv3 = gxd^2
tv2 = Fp.mul(gx1, gxd); // 13. tv2 = gx1 * gxd # gx1 * gxd
tv3 = Fp.mul(tv3, tv2); // 14. tv3 = tv3 * tv2 # gx1 * gxd^3
let y1 = Fp.pow(tv3, ELL2_C1); // 15. y1 = tv3^c1 # (gx1 * gxd^3)^((p - 3) / 4)
y1 = Fp.mul(y1, tv2); // 16. y1 = y1 * tv2 # gx1 * gxd * (gx1 * gxd^3)^((p - 3) / 4)
let x2n = Fp.mul(x1n, Fp.neg(tv1)); // 17. x2n = -tv1 * x1n # x2 = x2n / xd = -1 * u^2 * x1n / xd
let y2 = Fp.mul(y1, u); // 18. y2 = y1 * u
y2 = Fp.cmov(y2, Fp.ZERO, e1); // 19. y2 = CMOV(y2, 0, e1)
tv2 = Fp.sqr(y1); // 20. tv2 = y1^2
tv2 = Fp.mul(tv2, gxd); // 21. tv2 = tv2 * gxd
let e2 = Fp.eql(tv2, gx1); // 22. e2 = tv2 == gx1
let xn = Fp.cmov(x2n, x1n, e2); // 23. xn = CMOV(x2n, x1n, e2) # If e2, x = x1, else x = x2
let y = Fp.cmov(y2, y1, e2); // 24. y = CMOV(y2, y1, e2) # If e2, y = y1, else y = y2
let e3 = Fp.isOdd(y); // 25. e3 = sgn0(y) == 1 # Fix sign of y
y = Fp.cmov(y, Fp.neg(y), e2 !== e3); // 26. y = CMOV(y, -y, e2 XOR e3)
return { xn, xd, yn: y, yd: Fp.ONE }; // 27. return (xn, xd, y, 1)
}
function map_to_curve_elligator2_edwards448(u: bigint) {
let { xn, xd, yn, yd } = map_to_curve_elligator2_curve448(u); // 1. (xn, xd, yn, yd) = map_to_curve_elligator2_curve448(u)
let xn2 = Fp.sqr(xn); // 2. xn2 = xn^2
let xd2 = Fp.sqr(xd); // 3. xd2 = xd^2
let xd4 = Fp.sqr(xd2); // 4. xd4 = xd2^2
let yn2 = Fp.sqr(yn); // 5. yn2 = yn^2
let yd2 = Fp.sqr(yd); // 6. yd2 = yd^2
let xEn = Fp.sub(xn2, xd2); // 7. xEn = xn2 - xd2
let tv2 = Fp.sub(xEn, xd2); // 8. tv2 = xEn - xd2
xEn = Fp.mul(xEn, xd2); // 9. xEn = xEn * xd2
xEn = Fp.mul(xEn, yd); // 10. xEn = xEn * yd
xEn = Fp.mul(xEn, yn); // 11. xEn = xEn * yn
xEn = Fp.mul(xEn, _4n); // 12. xEn = xEn * 4
tv2 = Fp.mul(tv2, xn2); // 13. tv2 = tv2 * xn2
tv2 = Fp.mul(tv2, yd2); // 14. tv2 = tv2 * yd2
let tv3 = Fp.mul(yn2, _4n); // 15. tv3 = 4 * yn2
let tv1 = Fp.add(tv3, yd2); // 16. tv1 = tv3 + yd2
tv1 = Fp.mul(tv1, xd4); // 17. tv1 = tv1 * xd4
let xEd = Fp.add(tv1, tv2); // 18. xEd = tv1 + tv2
tv2 = Fp.mul(tv2, xn); // 19. tv2 = tv2 * xn
let tv4 = Fp.mul(xn, xd4); // 20. tv4 = xn * xd4
let yEn = Fp.sub(tv3, yd2); // 21. yEn = tv3 - yd2
yEn = Fp.mul(yEn, tv4); // 22. yEn = yEn * tv4
yEn = Fp.sub(yEn, tv2); // 23. yEn = yEn - tv2
tv1 = Fp.add(xn2, xd2); // 24. tv1 = xn2 + xd2
tv1 = Fp.mul(tv1, xd2); // 25. tv1 = tv1 * xd2
tv1 = Fp.mul(tv1, xd); // 26. tv1 = tv1 * xd
tv1 = Fp.mul(tv1, yn2); // 27. tv1 = tv1 * yn2
tv1 = Fp.mul(tv1, BigInt(-2)); // 28. tv1 = -2 * tv1
let yEd = Fp.add(tv2, tv1); // 29. yEd = tv2 + tv1
tv4 = Fp.mul(tv4, yd2); // 30. tv4 = tv4 * yd2
yEd = Fp.add(yEd, tv4); // 31. yEd = yEd + tv4
tv1 = Fp.mul(xEd, yEd); // 32. tv1 = xEd * yEd
let e = Fp.eql(tv1, Fp.ZERO); // 33. e = tv1 == 0
xEn = Fp.cmov(xEn, Fp.ZERO, e); // 34. xEn = CMOV(xEn, 0, e)
xEd = Fp.cmov(xEd, Fp.ONE, e); // 35. xEd = CMOV(xEd, 1, e)
yEn = Fp.cmov(yEn, Fp.ONE, e); // 36. yEn = CMOV(yEn, 1, e)
yEd = Fp.cmov(yEd, Fp.ONE, e); // 37. yEd = CMOV(yEd, 1, e)
const inv = FpInvertBatch(Fp, [xEd, yEd], true); // batch division
return { x: Fp.mul(xEn, inv[0]), y: Fp.mul(yEn, inv[1]) }; // 38. return (xEn, xEd, yEn, yEd)
}
/** Hashing / encoding to ed448 points / field. RFC 9380 methods. */
export const ed448_hasher: H2CHasher<bigint> = /* @__PURE__ */ (() =>
createHasher(ed448.Point, (scalars: bigint[]) => map_to_curve_elligator2_edwards448(scalars[0]), {
DST: 'edwards448_XOF:SHAKE256_ELL2_RO_',
encodeDST: 'edwards448_XOF:SHAKE256_ELL2_NU_',
p: Fp.ORDER,
m: 1,
k: 224,
expand: 'xof',
hash: shake256,
}))();
// 1-d
const ONE_MINUS_D = /* @__PURE__ */ BigInt('39082');
// 1-2d
const ONE_MINUS_TWO_D = /* @__PURE__ */ BigInt('78163');
// √(-d)
const SQRT_MINUS_D = /* @__PURE__ */ BigInt(
'98944233647732219769177004876929019128417576295529901074099889598043702116001257856802131563896515373927712232092845883226922417596214'
);
// 1 / √(-d)
const INVSQRT_MINUS_D = /* @__PURE__ */ BigInt(
'315019913931389607337177038330951043522456072897266928557328499619017160722351061360252776265186336876723201881398623946864393857820716'
);
// Calculates 1/√(number)
const invertSqrt = (number: bigint) => uvRatio(_1n, number);
/**
* Elligator map for hash-to-curve of decaf448.
* Described in [RFC9380](https://www.rfc-editor.org/rfc/rfc9380#appendix-C)
* and [RFC9496](https://www.rfc-editor.org/rfc/rfc9496#name-element-derivation-2).
*/
function calcElligatorDecafMap(r0: bigint): EdwardsPoint {
const { d } = ed448_CURVE;
const P = Fp.ORDER;
const mod = (n: bigint) => Fp.create(n);
const r = mod(-(r0 * r0)); // 1
const u0 = mod(d * (r - _1n)); // 2
const u1 = mod((u0 + _1n) * (u0 - r)); // 3
const { isValid: was_square, value: v } = uvRatio(ONE_MINUS_TWO_D, mod((r + _1n) * u1)); // 4
let v_prime = v; // 5
if (!was_square) v_prime = mod(r0 * v);
let sgn = _1n; // 6
if (!was_square) sgn = mod(-_1n);
const s = mod(v_prime * (r + _1n)); // 7
let s_abs = s;
if (isNegativeLE(s, P)) s_abs = mod(-s);
const s2 = s * s;
const W0 = mod(s_abs * _2n); // 8
const W1 = mod(s2 + _1n); // 9
const W2 = mod(s2 - _1n); // 10
const W3 = mod(v_prime * s * (r - _1n) * ONE_MINUS_TWO_D + sgn); // 11
return new ed448.Point(mod(W0 * W3), mod(W2 * W1), mod(W1 * W3), mod(W0 * W2));
}
function decaf448_map(bytes: Uint8Array): _DecafPoint {
abytes(bytes, 112);
const skipValidation = true;
// Note: Similar to the field element decoding described in
// [RFC7748], and unlike the field element decoding described in
// Section 5.3.1, non-canonical values are accepted.
const r1 = Fp448.create(Fp448.fromBytes(bytes.subarray(0, 56), skipValidation));
const R1 = calcElligatorDecafMap(r1);
const r2 = Fp448.create(Fp448.fromBytes(bytes.subarray(56, 112), skipValidation));
const R2 = calcElligatorDecafMap(r2);
return new _DecafPoint(R1.add(R2));
}
/**
* Each ed448/EdwardsPoint has 4 different equivalent points. This can be
* a source of bugs for protocols like ring signatures. Decaf was created to solve this.
* Decaf point operates in X:Y:Z:T extended coordinates like EdwardsPoint,
* but it should work in its own namespace: do not combine those two.
* See [RFC9496](https://www.rfc-editor.org/rfc/rfc9496).
*/
class _DecafPoint extends PrimeEdwardsPoint<_DecafPoint> {
// The following gymnastics is done because typescript strips comments otherwise
// prettier-ignore
static BASE: _DecafPoint =
/* @__PURE__ */ (() => new _DecafPoint(ed448.Point.BASE).multiplyUnsafe(_2n))();
// prettier-ignore
static ZERO: _DecafPoint =
/* @__PURE__ */ (() => new _DecafPoint(ed448.Point.ZERO))();
// prettier-ignore
static Fp: IField<bigint> =
/* @__PURE__ */ (() => Fp448)();
// prettier-ignore
static Fn: IField<bigint> =
/* @__PURE__ */ (() => Fn448)();
constructor(ep: EdwardsPoint) {
super(ep);
}
static fromAffine(ap: AffinePoint<bigint>): _DecafPoint {
return new _DecafPoint(ed448.Point.fromAffine(ap));
}
protected assertSame(other: _DecafPoint): void {
if (!(other instanceof _DecafPoint)) throw new Error('DecafPoint expected');
}
protected init(ep: EdwardsPoint): _DecafPoint {
return new _DecafPoint(ep);
}
/** @deprecated use `import { decaf448_hasher } from '@noble/curves/ed448.js';` */
static hashToCurve(hex: Hex): _DecafPoint {
return decaf448_map(ensureBytes('decafHash', hex, 112));
}
static fromBytes(bytes: Uint8Array): _DecafPoint {
abytes(bytes, 56);
const { d } = ed448_CURVE;
const P = Fp.ORDER;
const mod = (n: bigint) => Fp448.create(n);
const s = Fp448.fromBytes(bytes);
// 1. Check that s_bytes is the canonical encoding of a field element, or else abort.
// 2. Check that s is non-negative, or else abort
if (!equalBytes(Fn448.toBytes(s), bytes) || isNegativeLE(s, P))
throw new Error('invalid decaf448 encoding 1');
const s2 = mod(s * s); // 1
const u1 = mod(_1n + s2); // 2
const u1sq = mod(u1 * u1);
const u2 = mod(u1sq - _4n * d * s2); // 3
const { isValid, value: invsqrt } = invertSqrt(mod(u2 * u1sq)); // 4
let u3 = mod((s + s) * invsqrt * u1 * SQRT_MINUS_D); // 5
if (isNegativeLE(u3, P)) u3 = mod(-u3);
const x = mod(u3 * invsqrt * u2 * INVSQRT_MINUS_D); // 6
const y = mod((_1n - s2) * invsqrt * u1); // 7
const t = mod(x * y); // 8
if (!isValid) throw new Error('invalid decaf448 encoding 2');
return new _DecafPoint(new ed448.Point(x, y, _1n, t));
}
/**
* Converts decaf-encoded string to decaf point.
* Described in [RFC9496](https://www.rfc-editor.org/rfc/rfc9496#name-decode-2).
* @param hex Decaf-encoded 56 bytes. Not every 56-byte string is valid decaf encoding
*/
static fromHex(hex: Hex): _DecafPoint {
return _DecafPoint.fromBytes(ensureBytes('decafHex', hex, 56));
}
/** @deprecated use `import { pippenger } from '@noble/curves/abstract/curve.js';` */
static msm(points: _DecafPoint[], scalars: bigint[]): _DecafPoint {
return pippenger(_DecafPoint, Fn, points, scalars);
}
/**
* Encodes decaf point to Uint8Array.
* Described in [RFC9496](https://www.rfc-editor.org/rfc/rfc9496#name-encode-2).
*/
toBytes(): Uint8Array {
const { X, Z, T } = this.ep;
const P = Fp.ORDER;
const mod = (n: bigint) => Fp.create(n);
const u1 = mod(mod(X + T) * mod(X - T)); // 1
const x2 = mod(X * X);
const { value: invsqrt } = invertSqrt(mod(u1 * ONE_MINUS_D * x2)); // 2
let ratio = mod(invsqrt * u1 * SQRT_MINUS_D); // 3
if (isNegativeLE(ratio, P)) ratio = mod(-ratio);
const u2 = mod(INVSQRT_MINUS_D * ratio * Z - T); // 4
let s = mod(ONE_MINUS_D * invsqrt * X * u2); // 5
if (isNegativeLE(s, P)) s = mod(-s);
return Fn448.toBytes(s);
}
/**
* Compare one point to another.
* Described in [RFC9496](https://www.rfc-editor.org/rfc/rfc9496#name-equals-2).
*/
equals(other: _DecafPoint): boolean {
this.assertSame(other);
const { X: X1, Y: Y1 } = this.ep;
const { X: X2, Y: Y2 } = other.ep;
// (x1 * y2 == y1 * x2)
return Fp.create(X1 * Y2) === Fp.create(Y1 * X2);
}
is0(): boolean {
return this.equals(_DecafPoint.ZERO);
}
}
export const decaf448: {
Point: typeof _DecafPoint;
} = { Point: _DecafPoint };
/** Hashing to decaf448 points / field. RFC 9380 methods. */
export const decaf448_hasher: H2CHasherBase<bigint> = {
hashToCurve(msg: Uint8Array, options?: htfBasicOpts): _DecafPoint {
const DST = options?.DST || 'decaf448_XOF:SHAKE256_D448MAP_RO_';
return decaf448_map(expand_message_xof(msg, DST, 112, 224, shake256));
},
// Warning: has big modulo bias of 2^-64.
// RFC is invalid. RFC says "use 64-byte xof", while for 2^-112 bias
// it must use 84-byte xof (56+56/2), not 64.
hashToScalar(msg: Uint8Array, options: htfBasicOpts = { DST: _DST_scalar }) {
// Can't use `Fn448.fromBytes()`. 64-byte input => 56-byte field element
const xof = expand_message_xof(msg, options.DST, 64, 256, shake256);
return Fn448.create(bytesToNumberLE(xof));
},
};
// export const decaf448_oprf: OPRF = createORPF({
// name: 'decaf448-SHAKE256',
// Point: DecafPoint,
// hash: (msg: Uint8Array) => shake256(msg, { dkLen: 64 }),
// hashToGroup: decaf448_hasher.hashToCurve,
// hashToScalar: decaf448_hasher.hashToScalar,
// });
/**
* Weird / bogus points, useful for debugging.
* Unlike ed25519, there is no ed448 generator point which can produce full T subgroup.
* Instead, there is a Klein four-group, which spans over 2 independent 2-torsion points:
* (0, 1), (0, -1), (-1, 0), (1, 0).
*/
export const ED448_TORSION_SUBGROUP: string[] = [
'010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000',
'fefffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffff00',
'000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000',
'000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000080',
];
type DcfHasher = (msg: Uint8Array, options: htfBasicOpts) => _DecafPoint;
/** @deprecated use `decaf448.Point` */
export const DecafPoint: typeof _DecafPoint = _DecafPoint;
/** @deprecated use `import { ed448_hasher } from '@noble/curves/ed448.js';` */
export const hashToCurve: H2CMethod<bigint> = /* @__PURE__ */ (() => ed448_hasher.hashToCurve)();
/** @deprecated use `import { ed448_hasher } from '@noble/curves/ed448.js';` */
export const encodeToCurve: H2CMethod<bigint> = /* @__PURE__ */ (() =>
ed448_hasher.encodeToCurve)();
/** @deprecated use `import { decaf448_hasher } from '@noble/curves/ed448.js';` */
export const hashToDecaf448: DcfHasher = /* @__PURE__ */ (() =>
decaf448_hasher.hashToCurve as DcfHasher)();
/** @deprecated use `import { decaf448_hasher } from '@noble/curves/ed448.js';` */
export const hash_to_decaf448: DcfHasher = /* @__PURE__ */ (() =>
decaf448_hasher.hashToCurve as DcfHasher)();
/** @deprecated use `ed448.utils.toMontgomery` */
export function edwardsToMontgomeryPub(edwardsPub: string | Uint8Array): Uint8Array {
return ed448.utils.toMontgomery(ensureBytes('pub', edwardsPub));
}
/** @deprecated use `ed448.utils.toMontgomery` */
export const edwardsToMontgomery: typeof edwardsToMontgomeryPub = edwardsToMontgomeryPub;
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