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/**
* Edwards448 (also called 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 { concatBytes, hexToBytes, createHasher as wrapConstructor } from '@noble/hashes/utils.js';
import { eddsa, edwards, PrimeEdwardsPoint, } from "./abstract/edwards.js";
import { createFROST } from "./abstract/frost.js";
import { _DST_scalar, createHasher, expand_message_xof, } from "./abstract/hash-to-curve.js";
import { Field, FpInvertBatch, isNegativeLE, mod, pow2 } from "./abstract/modular.js";
import { montgomery } from "./abstract/montgomery.js";
import { createOPRF } from "./abstract/oprf.js";
import { abytes, asciiToBytes, bytesToNumberLE, equalBytes, } from "./utils.js";
// edwards448 curve
// a = 1n
// d = Fp.neg(39081n)
// Finite field 2n**448n - 2n**224n - 1n
// Subgroup order
// 2n**446n - 13818066809895115352007386748515426880336692474882178609894547503885n
const ed448_CURVE_p = /* @__PURE__ */ BigInt('0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffffff');
const ed448_CURVE = /* @__PURE__ */ (() => ({
p: ed448_CURVE_p,
n: BigInt('0x3fffffffffffffffffffffffffffffffffffffffffffffffffffffff7cca23e9c44edb49aed63690216cc2728dc58f552378c292ab5844f3'),
h: BigInt(4),
a: BigInt(1),
d: BigInt('0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffffffffffffffffffffffffffffffffffffffffffffffff6756'),
Gx: BigInt('0x4f1970c66bed0ded221d15a622bf36da9e146570470f1767ea6de324a3d3a46412ae1af72ab66511433b80e18b00938e2626a82bc70cc05e'),
Gy: BigInt('0x693f46716eb6bc248876203756c9c7624bea73736ca3984087789c1e05a0c2d73ad3ff1ce67c39c4fdbd132c4ed7c8ad9808795bf230fa14'),
}))();
// This is not RFC 8032 edwards448 / Goldilocks (`ed448` below, d = -39081).
// It is NIST SP 800-186 §3.2.3.3 E448, the Curve448-isomorphic Edwards model
// also described in draft-ietf-lwig-curve-representations-23 Appendix M, with
// d = 39082/39081 and Gy = 3/2.
// RFC 7748's literal Edwards point / birational map are wrong here: the literal
// point is the wrong-sign (Gx, -Gy) order-2*n variant. Keep the corrected
// prime-order (Gx, Gy) base so Point.BASE stays a subgroup generator, which is
// what noble's generic Edwards API expects.
const E448_CURVE = /* @__PURE__ */ (() => 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 = /* @__PURE__ */ BigInt(1), _2n = /* @__PURE__ */ BigInt(2), _3n = /* @__PURE__ */ BigInt(3), _4n = /* @__PURE__ */ BigInt(4), _11n = /* @__PURE__ */ BigInt(11);
// prettier-ignore
const _22n = /* @__PURE__ */ BigInt(22), _44n = /* @__PURE__ */ BigInt(44), _88n = /* @__PURE__ */ BigInt(88), _223n = /* @__PURE__ */ 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) {
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;
}
// Mutates and returns the provided buffer in place. The final `bytes[56] = 0`
// write is the Ed448 path; for 56-byte X448 inputs it is an out-of-bounds no-op.
function adjustScalarBytes(bytes) {
// 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 Ed448 decode helper for RFC 8032 §5.2.3 steps 2-3. Unlike
// `SQRT_RATIO_M1`, the returned `value` only has the documented meaning when
// `isValid` is true.
function uvRatio(u, v) {
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
// RFC 8032 encodes Ed448 field/scalar elements in 57 bytes even though field
// values fit in 448 bits and scalars in 446 bits. Noble models that with a
// 456-bit storage width so the final-octet x-sign bit (bit 455) still fits in
// the shared little-endian container.
const Fp = /* @__PURE__ */ (() => Field(ed448_CURVE_p, { BITS: 456, isLE: true }))();
// Same 57-byte container shape as `Fp`; canonical scalar encodings still have
// the top ten bits clear per RFC 8032.
const Fn = /* @__PURE__ */ (() => Field(ed448_CURVE.n, { BITS: 456, isLE: true }))();
// Generic 56-byte field shape used by decaf448 and raw X448 u-coordinates.
// Plain `Field` decoding stays canonical here, so callers that want RFC 7748's
// modulo-p acceptance must reduce externally.
const Fp448 = /* @__PURE__ */ (() => Field(ed448_CURVE_p, { BITS: 448, isLE: true }))();
// Strict 56-byte scalar parser matching RFC 9496's recommended canonical form.
const Fn448 = /* @__PURE__ */ (() => Field(ed448_CURVE.n, { BITS: 448, isLE: true }))();
// SHAKE256(dom4(phflag,context)||x, 114)
// RFC 8032 `dom4` prefix. Empty contexts are valid; the accepted length range
// is 0..255 octets inclusive.
function dom4(data, ctx, phflag) {
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_Point = /* @__PURE__ */ edwards(ed448_CURVE, { Fp, Fn, uvRatio });
// Shared internal factory for both `ed448` and `ed448ph`; callers are only
// expected to override narrow family options such as prehashing.
function ed4(opts) {
return eddsa(ed448_Point, shake256_114, Object.assign({ adjustScalarBytes, domain: dom4 }, opts));
}
/**
* ed448 EdDSA curve and methods.
* @example
* Generate one Ed448 keypair, sign a message, and verify it.
*
* ```js
* import { ed448 } from '@noble/curves/ed448.js';
* const { secretKey, publicKey } = ed448.keygen();
* // const publicKey = ed448.getPublicKey(secretKey);
* const msg = new TextEncoder().encode('hello noble');
* const sig = ed448.sign(msg, secretKey);
* const isValid = ed448.verify(sig, msg, publicKey);
* ```
*/
export const ed448 = /* @__PURE__ */ ed4({});
// There is no ed448ctx, since ed448 supports ctx by default
/**
* Prehashed version of ed448. See {@link ed448}
* @example
* Use the prehashed Ed448 variant for one message.
*
* ```ts
* const { secretKey, publicKey } = ed448ph.keygen();
* const msg = new TextEncoder().encode('hello noble');
* const sig = ed448ph.sign(msg, secretKey);
* const isValid = ed448ph.verify(sig, msg, publicKey);
* ```
*/
export const ed448ph = /* @__PURE__ */ ed4({ prehash: shake256_64 });
/**
* E448 here is NIST SP 800-186 §3.2.3.3 E448, the Edwards representation of
* Curve448, not RFC 8032 edwards448 / Goldilocks.
* Goldilocks is the separate 4-isogenous curve exposed as `ed448`.
* We keep the corrected prime-order base here; RFC 7748's literal Edwards
* point / map are wrong for this curve model, and the literal point is the
* wrong-sign order-2*n variant.
* @param X - Projective X coordinate.
* @param Y - Projective Y coordinate.
* @param Z - Projective Z coordinate.
* @param T - Projective T coordinate.
* @example
* Multiply the E448 base point.
*
* ```ts
* const point = E448.BASE.multiply(2n);
* ```
*/
export const E448 = /* @__PURE__ */ edwards(E448_CURVE);
/**
* ECDH using curve448 aka x448.
* The wrapper aborts on all-zero shared secrets by default, and seeded
* `keygen(seed)` reuses the provided 56-byte seed buffer instead of copying it.
*
* @example
* Derive one shared secret between two X448 peers.
*
* ```js
* import { x448 } from '@noble/curves/ed448.js';
* const alice = x448.keygen();
* const bob = x448.keygen();
* const shared = x448.getSharedSecret(alice.secretKey, bob.publicKey);
* ```
*/
export const x448 = /* @__PURE__ */ (() => {
const P = ed448_CURVE_p;
return montgomery({
P,
type: 'x448',
powPminus2: (x) => {
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
// 1. c1 = (q - 3) / 4 # Integer arithmetic
const ELL2_C1 = /* @__PURE__ */ (() => (ed448_CURVE_p - BigInt(3)) / BigInt(4))();
const ELL2_J = /* @__PURE__ */ BigInt(156326);
// Returns RFC 9380 Appendix G.2.3 rational Montgomery numerators/denominators
// `{ xn, xd, yn, yd }`, not an affine point.
function map_to_curve_elligator2_curve448(u) {
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)
// 17. x2n = -tv1 * x1n # x2 = x2n / xd = -1 * u^2 * x1n / xd
let x2n = Fp.mul(x1n, Fp.neg(tv1));
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)
}
// Returns affine `{ x, y }` after inverting the Appendix G.2.4 denominators.
function map_to_curve_elligator2_edwards448(u) {
// 1. (xn, xd, yn, yd) = map_to_curve_elligator2_curve448(u)
let { 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.
* Public `mapToCurve()` consumes one field element bigint for `m = 1`, and RFC
* Appendix J vectors use the special `QUUX-V01-*` test DST overrides rather
* than the default suite IDs below.
* @example
* Hash one message onto the ed448 curve.
*
* ```ts
* const point = ed448_hasher.hashToCurve(new TextEncoder().encode('hello noble'));
* ```
*/
export const ed448_hasher = /* @__PURE__ */ (() => createHasher(ed448_Point, (scalars) => map_to_curve_elligator2_edwards448(scalars[0]), {
DST: 'edwards448_XOF:SHAKE256_ELL2_RO_',
encodeDST: 'edwards448_XOF:SHAKE256_ELL2_NU_',
p: ed448_CURVE_p,
m: 1,
k: 224,
expand: 'xof',
hash: shake256,
}))();
/**
* FROST threshold signatures over ed448. RFC 9591.
* @example
* Create one trusted-dealer package for 2-of-3 ed448 signing.
*
* ```ts
* const alice = ed448_FROST.Identifier.derive('alice@example.com');
* const bob = ed448_FROST.Identifier.derive('bob@example.com');
* const carol = ed448_FROST.Identifier.derive('carol@example.com');
* const deal = ed448_FROST.trustedDealer({ min: 2, max: 3 }, [alice, bob, carol]);
* ```
*/
export const ed448_FROST = /* @__PURE__ */ (() => createFROST({
name: 'FROST-ED448-SHAKE256-v1',
Point: ed448_Point,
validatePoint: (p) => {
p.assertValidity();
if (!p.isTorsionFree())
throw new Error('bad point: not torsion-free');
},
// Group: edwards448 [RFC8032], where Ne = 57 and Ns = 57.
// Fn is 57 bytes, Fp is 57 bytes too
Fn,
hash: shake256_114,
H2: 'SigEd448\0\0',
}))();
// 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');
// RFC 9496 `SQRT_RATIO_M1` must return `CT_ABS(s)`, i.e. the nonnegative root.
// Keep this Decaf-local: RFC 9496 decode/encode/map formulas depend on that
// canonical representative, while ordinary Ed448 decoding still uses `uvRatio()`
// plus the public sign bit from RFC 8032.
const sqrtRatioM1 = (u, v) => {
const P = ed448_CURVE_p;
const { isValid, value } = uvRatio(u, v);
return { isValid, value: isNegativeLE(value, P) ? Fp448.create(-value) : value };
};
const invertSqrt = (number) => sqrtRatioM1(_1n, number);
/**
* Elligator map for hash-to-curve of decaf448.
* Primary formula source is RFC 9496 §5.3.4. Step 1 intentionally reduces the
* input modulo `p`, and the return value is the internal Edwards
* representation, not a public decaf encoding.
*/
function calcElligatorDecafMap(r0) {
const { d, p: P } = ed448_CURVE;
const mod = (n) => Fp448.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 } = sqrtRatioM1(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));
}
// Keep the Decaf448 base representative literal here: deriving it with
// `new _DecafPoint(ed448_Point.BASE).multiplyUnsafe(2)` forces eager WNAF precomputes and
// adds about 100ms to `ed448.js` import time.
const DECAF_BASE_X = /* @__PURE__ */ BigInt('0xaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa955555555555555555555555555555555555555555555555555555555');
const DECAF_BASE_Y = /* @__PURE__ */ BigInt('0xae05e9634ad7048db359d6205086c2b0036ed7a035884dd7b7e36d728ad8c4b80d6565833a2a3098bbbcb2bed1cda06bdaeafbcdea9386ed');
const DECAF_BASE_T = /* @__PURE__ */ BigInt('0x696d84643374bace9d70983a12aa9d461da74d2d5c35e8d97ba72c3aba4450a5d29274229bd22c1d5e3a6474ee4ffb0e7a9e200a28eee402');
/**
* 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 {
// The following gymnastics is done because typescript strips comments otherwise
// prettier-ignore
static BASE =
/* @__PURE__ */ (() => new _DecafPoint(new ed448_Point(DECAF_BASE_X, DECAF_BASE_Y, _1n, DECAF_BASE_T)))();
// prettier-ignore
static ZERO =
/* @__PURE__ */ (() => new _DecafPoint(ed448_Point.ZERO))();
// prettier-ignore
static Fp =
/* @__PURE__ */ (() => Fp448)();
// prettier-ignore
static Fn =
/* @__PURE__ */ (() => Fn448)();
constructor(ep) {
super(ep);
}
/**
* Create one Decaf448 point from affine Edwards coordinates.
* This wraps the internal Edwards representative directly and is not a
* canonical decaf448 decoding path.
* Use `toBytes()` / `fromBytes()` if canonical decaf448 bytes matter.
*/
static fromAffine(ap) {
return new _DecafPoint(ed448_Point.fromAffine(ap));
}
assertSame(other) {
if (!(other instanceof _DecafPoint))
throw new Error('DecafPoint expected');
}
init(ep) {
return new _DecafPoint(ep);
}
static fromBytes(bytes) {
abytes(bytes, 56);
const { d, p: P } = ed448_CURVE;
const mod = (n) => 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) {
return _DecafPoint.fromBytes(hexToBytes(hex));
}
/**
* Encodes decaf point to Uint8Array.
* Described in [RFC9496](https://www.rfc-editor.org/rfc/rfc9496#name-encode-2).
*/
toBytes() {
const { X, Z, T } = this.ep;
const P = ed448_CURVE.p;
const mod = (n) => Fp448.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) {
this.assertSame(other);
const { X: X1, Y: Y1 } = this.ep;
const { X: X2, Y: Y2 } = other.ep;
// (x1 * y2 == y1 * x2)
return Fp448.create(X1 * Y2) === Fp448.create(Y1 * X2);
}
is0() {
return this.equals(_DecafPoint.ZERO);
}
}
Object.freeze(_DecafPoint.BASE);
Object.freeze(_DecafPoint.ZERO);
Object.freeze(_DecafPoint.prototype);
Object.freeze(_DecafPoint);
/** Prime-order Decaf448 group bundle. */
export const decaf448 = /* @__PURE__ */ Object.freeze({ Point: _DecafPoint });
/**
* Hashing to decaf448 points / field. RFC 9380 methods.
* `hashToCurve()` is RFC 9380 `hash_to_decaf448`, `deriveToCurve()` is RFC
* 9496 element derivation, and `hashToScalar()` is a library helper layered on
* top of RFC 9496 scalar reduction.
* @example
* Hash one message onto decaf448.
*
* ```ts
* const point = decaf448_hasher.hashToCurve(new TextEncoder().encode('hello noble'));
* ```
*/
export const decaf448_hasher = Object.freeze({
Point: _DecafPoint,
hashToCurve(msg, options) {
// Preserve explicit empty/invalid DST overrides so expand_message_xof() can reject them.
const DST = options?.DST === undefined ? 'decaf448_XOF:SHAKE256_D448MAP_RO_' : options.DST;
return decaf448_hasher.deriveToCurve(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, options = { 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));
},
/**
* HashToCurve-like construction based on RFC 9496 (Element Derivation).
* Converts 112 uniform random bytes into a curve point.
*
* WARNING: This represents an older hash-to-curve construction from before
* RFC 9380 was finalized.
* It was later reused as a component in the newer
* `hash_to_decaf448` function defined in RFC 9380.
*/
deriveToCurve(bytes) {
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));
},
});
/**
* decaf448 OPRF, defined in RFC 9497.
* @example
* Run one blind/evaluate/finalize OPRF round over decaf448.
*
* ```ts
* const input = new TextEncoder().encode('hello noble');
* const keys = decaf448_oprf.oprf.generateKeyPair();
* const blind = decaf448_oprf.oprf.blind(input);
* const evaluated = decaf448_oprf.oprf.blindEvaluate(keys.secretKey, blind.blinded);
* const output = decaf448_oprf.oprf.finalize(input, blind.blind, evaluated);
* ```
*/
export const decaf448_oprf = /* @__PURE__ */ (() => createOPRF({
name: 'decaf448-SHAKE256',
Point: _DecafPoint,
hash: (msg) => 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, the torsion subgroup here is cyclic of order 4, generated by
* `(1, 0)`, and the array below lists that subgroup set (Klein four-group).
* @example
* Decode one known torsion point for debugging.
*
* ```ts
* import { ED448_TORSION_SUBGROUP, ed448 } from '@noble/curves/ed448.js';
* const point = ed448.Point.fromHex(ED448_TORSION_SUBGROUP[1]);
* ```
*/
export const ED448_TORSION_SUBGROUP = /* @__PURE__ */ Object.freeze([
'010000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000',
'fefffffffffffffffffffffffffffffffffffffffffffffffffffffffeffffffffffffffffffffffffffffffffffffffffffffffffffffff00',
'000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000',
'000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000080',
]);
//# sourceMappingURL=ed448.js.map
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