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hash-to-curve: elligator in 25519, 448. Stark: adjust type

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This commit is contained in:
Paul Miller 2022-12-31 06:51:12 +00:00
parent 12da04a2bb
commit 910c508da9
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GPG Key ID: 697079DA6878B89B
3 changed files with 168 additions and 11 deletions
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85
src/ed25519.ts
View File

@ -3,7 +3,7 @@ import { sha512 } from '@noble/hashes/sha512';
import { concatBytes, randomBytes, utf8ToBytes } from '@noble/hashes/utils';
import { twistedEdwards, ExtendedPointType } from './abstract/edwards.js';
import { montgomery } from './abstract/montgomery.js';
import { mod, pow2, isNegativeLE, Fp as Field } from './abstract/modular.js';
import { mod, pow2, isNegativeLE, Fp as Field, FpSqrtEven } from './abstract/modular.js';
import {
ensureBytes,
equalBytes,
@ -91,7 +91,80 @@ export const ED25519_TORSION_SUBGROUP = [
'c7176a703d4dd84fba3c0b760d10670f2a2053fa2c39ccc64ec7fd7792ac03fa',
];
const Fp = Field(ED25519_P);
const Fp = Field(ED25519_P, undefined, true);
// Hash To Curve Elligator2 Map (NOTE: different from ristretto255 elligator)
// NOTE: very important part is usage of FpSqrtEven for ELL2_C1_EDWARDS, since
// SageMath returns different root first and everything falls apart
const ELL2_C1 = (Fp.ORDER + BigInt(3)) / BigInt(8); // 1. c1 = (q + 3) / 8 # Integer arithmetic
const ELL2_C2 = Fp.pow(_2n, ELL2_C1); // 2. c2 = 2^c1
const ELL2_C3 = Fp.sqrt(Fp.negate(Fp.ONE)); // 3. c3 = sqrt(-1)
const ELL2_C4 = (Fp.ORDER - BigInt(5)) / BigInt(8); // 4. c4 = (q - 5) / 8 # Integer arithmetic
const ELL2_J = BigInt(486662);
// prettier-ignore
function map_to_curve_elligator2_curve25519(u: bigint) {
let tv1 = Fp.square(u); // 1. tv1 = u^2
tv1 = Fp.mul(tv1, _2n); // 2. tv1 = 2 * tv1
let xd = Fp.add(tv1, Fp.ONE); // 3. xd = tv1 + 1 # Nonzero: -1 is square (mod p), tv1 is not
let x1n = Fp.negate(ELL2_J); // 4. x1n = -J # x1 = x1n / xd = -J / (1 + 2 * u^2)
let tv2 = Fp.square(xd); // 5. tv2 = xd^2
let gxd = Fp.mul(tv2, xd); // 6. gxd = tv2 * xd # gxd = xd^3
let gx1 = Fp.mul(tv1, ELL2_J); // 7. gx1 = J * tv1 # x1n + J * xd
gx1 = Fp.mul(gx1, x1n); // 8. gx1 = gx1 * x1n # x1n^2 + J * x1n * xd
gx1 = Fp.add(gx1, tv2); // 9. gx1 = gx1 + tv2 # x1n^2 + J * x1n * xd + xd^2
gx1 = Fp.mul(gx1, x1n); // 10. gx1 = gx1 * x1n # x1n^3 + J * x1n^2 * xd + x1n * xd^2
let tv3 = Fp.square(gxd); // 11. tv3 = gxd^2
tv2 = Fp.square(tv3); // 12. tv2 = tv3^2 # gxd^4
tv3 = Fp.mul(tv3, gxd); // 13. tv3 = tv3 * gxd # gxd^3
tv3 = Fp.mul(tv3, gx1); // 14. tv3 = tv3 * gx1 # gx1 * gxd^3
tv2 = Fp.mul(tv2, tv3); // 15. tv2 = tv2 * tv3 # gx1 * gxd^7
let y11 = Fp.pow(tv2, ELL2_C4); // 16. y11 = tv2^c4 # (gx1 * gxd^7)^((p - 5) / 8)
y11 = Fp.mul(y11, tv3); // 17. y11 = y11 * tv3 # gx1*gxd^3*(gx1*gxd^7)^((p-5)/8)
let y12 = Fp.mul(y11, ELL2_C3); // 18. y12 = y11 * c3
tv2 = Fp.square(y11); // 19. tv2 = y11^2
tv2 = Fp.mul(tv2, gxd); // 20. tv2 = tv2 * gxd
let e1 = Fp.equals(tv2, gx1); // 21. e1 = tv2 == gx1
let y1 = Fp.cmov(y12, y11, e1); // 22. y1 = CMOV(y12, y11, e1) # If g(x1) is square, this is its sqrt
let x2n = Fp.mul(x1n, tv1); // 23. x2n = x1n * tv1 # x2 = x2n / xd = 2 * u^2 * x1n / xd
let y21 = Fp.mul(y11, u); // 24. y21 = y11 * u
y21 = Fp.mul(y21, ELL2_C2); // 25. y21 = y21 * c2
let y22 = Fp.mul(y21, ELL2_C3); // 26. y22 = y21 * c3
let gx2 = Fp.mul(gx1, tv1); // 27. gx2 = gx1 * tv1 # g(x2) = gx2 / gxd = 2 * u^2 * g(x1)
tv2 = Fp.square(y21); // 28. tv2 = y21^2
tv2 = Fp.mul(tv2, gxd); // 29. tv2 = tv2 * gxd
let e2 = Fp.equals(tv2, gx2); // 30. e2 = tv2 == gx2
let y2 = Fp.cmov(y22, y21, e2); // 31. y2 = CMOV(y22, y21, e2) # If g(x2) is square, this is its sqrt
tv2 = Fp.square(y1); // 32. tv2 = y1^2
tv2 = Fp.mul(tv2, gxd); // 33. tv2 = tv2 * gxd
let e3 = Fp.equals(tv2, gx1); // 34. e3 = tv2 == gx1
let xn = Fp.cmov(x2n, x1n, e3); // 35. xn = CMOV(x2n, x1n, e3) # If e3, x = x1, else x = x2
let y = Fp.cmov(y2, y1, e3); // 36. y = CMOV(y2, y1, e3) # If e3, y = y1, else y = y2
let e4 = Fp.isOdd(y); // 37. e4 = sgn0(y) == 1 # Fix sign of y
y = Fp.cmov(y, Fp.negate(y), e3 !== e4); // 38. y = CMOV(y, -y, e3 XOR e4)
return { xMn: xn, xMd: xd, yMn: y, yMd: 1n }; // 39. return (xn, xd, y, 1)
}
const ELL2_C1_EDWARDS = FpSqrtEven(Fp, Fp.negate(BigInt(486664))); // sgn0(c1) MUST equal 0
function map_to_curve_elligator2_edwards25519(u: bigint) {
const { xMn, xMd, yMn, yMd } = map_to_curve_elligator2_curve25519(u); // 1. (xMn, xMd, yMn, yMd) = map_to_curve_elligator2_curve25519(u)
let xn = Fp.mul(xMn, yMd); // 2. xn = xMn * yMd
xn = Fp.mul(xn, ELL2_C1_EDWARDS); // 3. xn = xn * c1
let xd = Fp.mul(xMd, yMn); // 4. xd = xMd * yMn # xn / xd = c1 * xM / yM
let yn = Fp.sub(xMn, xMd); // 5. yn = xMn - xMd
let yd = Fp.add(xMn, xMd); // 6. yd = xMn + xMd # (n / d - 1) / (n / d + 1) = (n - d) / (n + d)
let tv1 = Fp.mul(xd, yd); // 7. tv1 = xd * yd
let e = Fp.equals(tv1, Fp.ZERO); // 8. e = tv1 == 0
xn = Fp.cmov(xn, Fp.ZERO, e); // 9. xn = CMOV(xn, 0, e)
xd = Fp.cmov(xd, Fp.ONE, e); // 10. xd = CMOV(xd, 1, e)
yn = Fp.cmov(yn, Fp.ONE, e); // 11. yn = CMOV(yn, 1, e)
yd = Fp.cmov(yd, Fp.ONE, e); // 12. yd = CMOV(yd, 1, e)
const inv = Fp.invertBatch([xd, yd]); // batch division
return { x: Fp.mul(xn, inv[0]), y: Fp.mul(yn, inv[1]) }; // 13. return (xn, xd, yn, yd)
}
const ED25519_DEF = {
// Param: a
@ -121,14 +194,10 @@ const ED25519_DEF = {
p: Fp.ORDER,
m: 1,
k: 128,
expand: true,
expand: 'xmd',
hash: sha512,
},
mapToCurve: (scalars: bigint[]): { x: bigint; y: bigint } => {
throw new Error('Not supported yet');
// const { x, y } = calcElligatorRistrettoMap(scalars[0]).toAffine();
// return { x, y };
},
mapToCurve: (scalars: bigint[]) => map_to_curve_elligator2_edwards25519(scalars[0]),
} as const;
export const ed25519 = twistedEdwards(ED25519_DEF);

90
src/ed448.ts
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@ -2,7 +2,7 @@
import { shake256 } from '@noble/hashes/sha3';
import { concatBytes, randomBytes, utf8ToBytes, wrapConstructor } from '@noble/hashes/utils';
import { twistedEdwards } from './abstract/edwards.js';
import { mod, pow2, Fp } from './abstract/modular.js';
import { mod, pow2, Fp as Field } from './abstract/modular.js';
import { montgomery } from './abstract/montgomery.js';
/**
@ -52,6 +52,83 @@ function adjustScalarBytes(bytes: Uint8Array): Uint8Array {
return bytes;
}
const Fp = Field(ed448P, 456, true);
// 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: bigint) {
let tv1 = Fp.square(u); // 1. tv1 = u^2
let e1 = Fp.equals(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.negate(ELL2_J); // 5. x1n = -J
let tv2 = Fp.square(xd); // 6. tv2 = xd^2
let gxd = Fp.mul(tv2, xd); // 7. gxd = tv2 * xd # gxd = xd^3
let gx1 = Fp.mul(tv1, Fp.negate(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.square(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.negate(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.square(y1); // 20. tv2 = y1^2
tv2 = Fp.mul(tv2, gxd); // 21. tv2 = tv2 * gxd
let e2 = Fp.equals(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.negate(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.square(xn); // 2. xn2 = xn^2
let xd2 = Fp.square(xd); // 3. xd2 = xd^2
let xd4 = Fp.square(xd2); // 4. xd4 = xd2^2
let yn2 = Fp.square(yn); // 5. yn2 = yn^2
let yd2 = Fp.square(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.equals(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 ED448_DEF = {
// Param: a
a: BigInt(1),
@ -60,7 +137,7 @@ const ED448_DEF = {
'726838724295606890549323807888004534353641360687318060281490199180612328166730772686396383698676545930088884461843637361053498018326358'
),
// Finite field 𝔽p over which we'll do calculations; 2n ** 448n - 2n ** 224n - 1n
Fp: Fp(ed448P, 456),
Fp,
// Subgroup order: how many points ed448 has; 2n**446n - 13818066809895115352007386748515426880336692474882178609894547503885n
n: BigInt(
'181709681073901722637330951972001133588410340171829515070372549795146003961539585716195755291692375963310293709091662304773755859649779'
@ -111,6 +188,15 @@ const ED448_DEF = {
// square root exists, and the decoding fails.
return { isValid: mod(x2 * v, P) === u, value: x };
},
htfDefaults: {
DST: 'edwards448_XOF:SHAKE256_ELL2_RO_',
p: Fp.ORDER,
m: 1,
k: 224,
expand: 'xof',
hash: shake256,
},
mapToCurve: (scalars: bigint[]) => map_to_curve_elligator2_edwards448(scalars[0]),
} as const;
export const ed448 = twistedEdwards(ED448_DEF);

4
src/stark.ts
View File

@ -97,7 +97,7 @@ function getSharedSecret0x(privKeyA: Hex, pubKeyB: Hex) {
return starkCurve.getSharedSecret(normalizePrivateKey(privKeyA), pubKeyB);
}
function sign0x(msgHash: Hex, privKey: Hex, opts: any) {
function sign0x(msgHash: Hex, privKey: Hex, opts?: any) {
if (typeof privKey === 'string') privKey = strip0x(privKey).padStart(64, '0');
return starkCurve.sign(ensureBytes0x(msgHash), normalizePrivateKey(privKey), opts);
}
@ -203,6 +203,8 @@ function pedersenPrecompute(p1: ProjectivePoint, p2: ProjectivePoint): Projectiv
out.push(p);
p = p.double();
}
// NOTE: we cannot use wNAF here, because last 4 bits will require full 248 bits multiplication
// We can add support for this to wNAF, but it will complicate wNAF.
p = p2;
for (let i = 0; i < 4; i++) {
out.push(p);