@noble/curves
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Audited & minimal JS implementation of elliptic curve cryptography
423 lines • 19.8 kB
JavaScript
/**
* 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';
import { concatBytes, randomBytes, utf8ToBytes, wrapConstructor } from '@noble/hashes/utils';
import { pippenger } from './abstract/curve.js';
import { twistedEdwards } from './abstract/edwards.js';
import { createHasher, expand_message_xof, } from './abstract/hash-to-curve.js';
import { Field, isNegativeLE, mod, pow2 } from './abstract/modular.js';
import { montgomery } from './abstract/montgomery.js';
import { bytesToHex, bytesToNumberLE, ensureBytes, equalBytes, numberToBytesLE, } from './abstract/utils.js';
const shake256_114 = wrapConstructor(() => shake256.create({ dkLen: 114 }));
const shake256_64 = wrapConstructor(() => shake256.create({ dkLen: 64 }));
const ed448P = BigInt('726838724295606890549323807888004534353641360687318060281490199180612328166730772686396383698676545930088884461843637361053498018365439');
// 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) {
const P = ed448P;
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) {
// Section 5: Likewise, for X448, set the two least significant bits of the first byte to 0, and the most
// significant bit of the last byte to 1.
bytes[0] &= 252; // 0b11111100
// and the most significant bit of the last byte to 1.
bytes[55] |= 128; // 0b10000000
// NOTE: is 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, v) {
const P = ed448P;
// 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 };
}
const Fp = Field(ed448P, 456, true);
const ED448_DEF = {
// Param: a
a: BigInt(1),
// -39081. Negative number is P - number
d: BigInt('726838724295606890549323807888004534353641360687318060281490199180612328166730772686396383698676545930088884461843637361053498018326358'),
// Finite field 𝔽p over which we'll do calculations; 2n**448n - 2n**224n - 1n
Fp,
// Subgroup order: how many points curve has;
// 2n**446n - 13818066809895115352007386748515426880336692474882178609894547503885n
n: BigInt('181709681073901722637330951972001133588410340171829515070372549795146003961539585716195755291692375963310293709091662304773755859649779'),
// RFC 7748 has 56-byte keys, RFC 8032 has 57-byte keys
nBitLength: 456,
// Cofactor
h: BigInt(4),
// Base point (x, y) aka generator point
Gx: BigInt('224580040295924300187604334099896036246789641632564134246125461686950415467406032909029192869357953282578032075146446173674602635247710'),
Gy: BigInt('298819210078481492676017930443930673437544040154080242095928241372331506189835876003536878655418784733982303233503462500531545062832660'),
// SHAKE256(dom4(phflag,context)||x, 114)
hash: shake256_114,
randomBytes,
adjustScalarBytes,
// dom4
domain: (data, ctx, phflag) => {
if (ctx.length > 255)
throw new Error('context must be smaller than 255, got: ' + ctx.length);
return concatBytes(utf8ToBytes('SigEd448'), new Uint8Array([phflag ? 1 : 0, ctx.length]), ctx, data);
},
uvRatio,
};
/**
* ed448 EdDSA curve and methods.
* @example
* import { ed448 } from '@noble/curves/ed448';
* const priv = ed448.utils.randomPrivateKey();
* const pub = ed448.getPublicKey(priv);
* const msg = new TextEncoder().encode('whatsup');
* const sig = ed448.sign(msg, priv);
* ed448.verify(sig, msg, pub);
*/
export const ed448 = /* @__PURE__ */ twistedEdwards(ED448_DEF);
// NOTE: there is no ed448ctx, since ed448 supports ctx by default
export const ed448ph = /* @__PURE__ */ twistedEdwards({
...ED448_DEF,
prehash: shake256_64,
});
/**
* ECDH using curve448 aka x448.
*/
export const x448 = /* @__PURE__ */ (() => montgomery({
a: BigInt(156326),
// RFC 7748 has 56-byte keys, RFC 8032 has 57-byte keys
montgomeryBits: 448,
nByteLength: 56,
P: ed448P,
Gu: BigInt(5),
powPminus2: (x) => {
const P = ed448P;
const Pminus3div4 = ed448_pow_Pminus3div4(x);
const Pminus3 = pow2(Pminus3div4, BigInt(2), P);
return mod(Pminus3 * x, P); // Pminus3 * x = Pminus2
},
adjustScalarBytes,
randomBytes,
}))();
/**
* Converts edwards448 public key to x448 public key. Uses formula:
* * `(u, v) = ((y-1)/(y+1), sqrt(156324)*u/x)`
* * `(x, y) = (sqrt(156324)*u/v, (1+u)/(1-u))`
* @example
* const aPub = ed448.getPublicKey(utils.randomPrivateKey());
* x448.getSharedSecret(edwardsToMontgomery(aPub), edwardsToMontgomery(someonesPub))
*/
export function edwardsToMontgomeryPub(edwardsPub) {
const { y } = ed448.ExtendedPoint.fromHex(edwardsPub);
const _1n = BigInt(1);
return Fp.toBytes(Fp.create((y - _1n) * Fp.inv(y + _1n)));
}
export const edwardsToMontgomery = edwardsToMontgomeryPub; // deprecated
// TODO: add edwardsToMontgomeryPriv, similar to ed25519 version
// Hash To Curve Elligator2 Map
const ELL2_C1 = (Fp.ORDER - BigInt(3)) / BigInt(4); // 1. c1 = (q - 3) / 4 # Integer arithmetic
const ELL2_J = BigInt(156326);
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)
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) {
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 = Fp.invertBatch([xEd, yEd]); // batch division
return { x: Fp.mul(xEn, inv[0]), y: Fp.mul(yEn, inv[1]) }; // 38. return (xEn, xEd, yEn, yEd)
}
const htf = /* @__PURE__ */ (() => createHasher(ed448.ExtendedPoint, (scalars) => 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,
}))();
export const hashToCurve = /* @__PURE__ */ (() => htf.hashToCurve)();
export const encodeToCurve = /* @__PURE__ */ (() => htf.encodeToCurve)();
function assertDcfPoint(other) {
if (!(other instanceof DcfPoint))
throw new Error('DecafPoint expected');
}
// 1-d
const ONE_MINUS_D = BigInt('39082');
// 1-2d
const ONE_MINUS_TWO_D = BigInt('78163');
// √(-d)
const SQRT_MINUS_D = BigInt('98944233647732219769177004876929019128417576295529901074099889598043702116001257856802131563896515373927712232092845883226922417596214');
// 1 / √(-d)
const INVSQRT_MINUS_D = BigInt('315019913931389607337177038330951043522456072897266928557328499619017160722351061360252776265186336876723201881398623946864393857820716');
// Calculates 1/√(number)
const invertSqrt = (number) => uvRatio(_1n, number);
const MAX_448B = BigInt('0xffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffff');
const bytes448ToNumberLE = (bytes) => ed448.CURVE.Fp.create(bytesToNumberLE(bytes) & MAX_448B);
// Computes Elligator map for Decaf
// https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448-07#name-element-derivation-2
function calcElligatorDecafMap(r0) {
const { d } = ed448.CURVE;
const P = ed448.CURVE.Fp.ORDER;
const mod = ed448.CURVE.Fp.create;
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.ExtendedPoint(mod(W0 * W3), mod(W2 * W1), mod(W1 * W3), mod(W0 * W2));
}
/**
* Each ed448/ExtendedPoint 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 ExtendedPoint,
* but it should work in its own namespace: do not combine those two.
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448
*/
class DcfPoint {
// Private property to discourage combining ExtendedPoint + DecafPoint
// Always use Decaf encoding/decoding instead.
constructor(ep) {
this.ep = ep;
}
static fromAffine(ap) {
return new DcfPoint(ed448.ExtendedPoint.fromAffine(ap));
}
/**
* Takes uniform output of 112-byte hash function like shake256 and converts it to `DecafPoint`.
* The hash-to-group operation applies Elligator twice and adds the results.
* **Note:** this is one-way map, there is no conversion from point to hash.
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448-07#name-element-derivation-2
* @param hex 112-byte output of a hash function
*/
static hashToCurve(hex) {
hex = ensureBytes('decafHash', hex, 112);
const r1 = bytes448ToNumberLE(hex.slice(0, 56));
const R1 = calcElligatorDecafMap(r1);
const r2 = bytes448ToNumberLE(hex.slice(56, 112));
const R2 = calcElligatorDecafMap(r2);
return new DcfPoint(R1.add(R2));
}
/**
* Converts decaf-encoded string to decaf point.
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448-07#name-decode-2
* @param hex Decaf-encoded 56 bytes. Not every 56-byte string is valid decaf encoding
*/
static fromHex(hex) {
hex = ensureBytes('decafHex', hex, 56);
const { d } = ed448.CURVE;
const P = ed448.CURVE.Fp.ORDER;
const mod = ed448.CURVE.Fp.create;
const emsg = 'DecafPoint.fromHex: the hex is not valid encoding of DecafPoint';
const s = bytes448ToNumberLE(hex);
// 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(numberToBytesLE(s, 56), hex) || isNegativeLE(s, P))
throw new Error(emsg);
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(emsg);
return new DcfPoint(new ed448.ExtendedPoint(x, y, _1n, t));
}
static msm(points, scalars) {
const Fn = Field(ed448.CURVE.n, ed448.CURVE.nBitLength);
return pippenger(DcfPoint, Fn, points, scalars);
}
/**
* Encodes decaf point to Uint8Array.
* https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448-07#name-encode-2
*/
toRawBytes() {
let { ex: x, ey: _y, ez: z, et: t } = this.ep;
const P = ed448.CURVE.Fp.ORDER;
const mod = ed448.CURVE.Fp.create;
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 numberToBytesLE(s, 56);
}
toHex() {
return bytesToHex(this.toRawBytes());
}
toString() {
return this.toHex();
}
// Compare one point to another.
// https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-ristretto255-decaf448-07#name-equals-2
equals(other) {
assertDcfPoint(other);
const { ex: X1, ey: Y1 } = this.ep;
const { ex: X2, ey: Y2 } = other.ep;
const mod = ed448.CURVE.Fp.create;
// (x1 * y2 == y1 * x2)
return mod(X1 * Y2) === mod(Y1 * X2);
}
add(other) {
assertDcfPoint(other);
return new DcfPoint(this.ep.add(other.ep));
}
subtract(other) {
assertDcfPoint(other);
return new DcfPoint(this.ep.subtract(other.ep));
}
multiply(scalar) {
return new DcfPoint(this.ep.multiply(scalar));
}
multiplyUnsafe(scalar) {
return new DcfPoint(this.ep.multiplyUnsafe(scalar));
}
double() {
return new DcfPoint(this.ep.double());
}
negate() {
return new DcfPoint(this.ep.negate());
}
}
export const DecafPoint = /* @__PURE__ */ (() => {
// decaf448 base point is ed448 base x 2
// https://github.com/dalek-cryptography/curve25519-dalek/blob/59837c6ecff02b77b9d5ff84dbc239d0cf33ef90/vendor/ristretto.sage#L699
if (!DcfPoint.BASE)
DcfPoint.BASE = new DcfPoint(ed448.ExtendedPoint.BASE).multiply(_2n);
if (!DcfPoint.ZERO)
DcfPoint.ZERO = new DcfPoint(ed448.ExtendedPoint.ZERO);
return DcfPoint;
})();
// Hashing to decaf448. https://www.rfc-editor.org/rfc/rfc9380#appendix-C
export const hashToDecaf448 = (msg, options) => {
const d = options.DST;
const DST = typeof d === 'string' ? utf8ToBytes(d) : d;
const uniform_bytes = expand_message_xof(msg, DST, 112, 224, shake256);
const P = DcfPoint.hashToCurve(uniform_bytes);
return P;
};
export const hash_to_decaf448 = hashToDecaf448; // legacy
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