From 26cbc55f783b5ab6541f24c8f69f595595125aca Mon Sep 17 00:00:00 2001 From: Matthias Schiffer Date: Fri, 18 Mar 2016 23:37:56 +0100 Subject: Switch internal point representation to the Ed25519 curve The Ed25519 curve allows slightly more efficient addition. --- src/ec25519.c | 193 ++++++++++++++++++++++++++++++++-------------------------- 1 file changed, 108 insertions(+), 85 deletions(-) diff --git a/src/ec25519.c b/src/ec25519.c index 82ff709..ad43f30 100644 --- a/src/ec25519.c +++ b/src/ec25519.c @@ -41,7 +41,7 @@ * which is the curve used by the Ed25519 algorithm. The functions for this curve * have the suffix \em ed25519. * - * Internally, libuecc always uses the former representation for its \em work structure. + * Internally, libuecc always uses the latter representation for its \em work structure. * * The curves are equivalent to the Montgomery Curve used in D. J. Bernstein's * Curve25519 Diffie-Hellman algorithm. @@ -62,58 +62,73 @@ const ecc_25519_work_t ecc_25519_work_identity = {{0}, {1}, {1}, {0}}; const ecc_25519_work_t ecc_25519_work_base_legacy = { - {0xd4, 0x6b, 0xfe, 0x7f, 0x39, 0xfa, 0x8c, 0x22, - 0xe1, 0x96, 0x23, 0xeb, 0x26, 0xb7, 0x8e, 0x6a, - 0x34, 0x74, 0x8b, 0x66, 0xd6, 0xa3, 0x26, 0xdd, - 0x19, 0x5e, 0x9f, 0x21, 0x50, 0x43, 0x7c, 0x54}, + {0x1a, 0xd5, 0x25, 0x8f, 0x60, 0x2d, 0x56, 0xc9, + 0xb2, 0xa7, 0x25, 0x95, 0x60, 0xc7, 0x2c, 0x69, + 0x5c, 0xdc, 0xd6, 0xfd, 0x31, 0xe2, 0xa4, 0xc0, + 0xfe, 0x53, 0x6e, 0xcd, 0xd3, 0x36, 0x69, 0x21}, {0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}, {1}, - {0x47, 0x56, 0x98, 0x99, 0xc7, 0x61, 0x0a, 0x82, - 0x1a, 0xdf, 0x82, 0x22, 0x1f, 0x2c, 0x72, 0x88, - 0xc3, 0x29, 0x09, 0x52, 0x78, 0xe9, 0x1e, 0xe4, - 0x47, 0x4b, 0x4c, 0x81, 0xa6, 0x02, 0xfd, 0x29} + {0xa3, 0xdd, 0xb7, 0xa5, 0xb3, 0x8a, 0xde, 0x6d, + 0xf5, 0x52, 0x51, 0x77, 0x80, 0x9f, 0xf0, 0x20, + 0x7d, 0xe3, 0xab, 0x64, 0x8e, 0x4e, 0xea, 0x66, + 0x65, 0x76, 0x8b, 0xd7, 0x0f, 0x5f, 0x87, 0x67}, }; const ecc_25519_work_t ecc_25519_work_default_base = { - {0xd4, 0x6b, 0xfe, 0x7f, 0x39, 0xfa, 0x8c, 0x22, - 0xe1, 0x96, 0x23, 0xeb, 0x26, 0xb7, 0x8e, 0x6a, - 0x34, 0x74, 0x8b, 0x66, 0xd6, 0xa3, 0x26, 0xdd, - 0x19, 0x5e, 0x9f, 0x21, 0x50, 0x43, 0x7c, 0x54}, + {0x1a, 0xd5, 0x25, 0x8f, 0x60, 0x2d, 0x56, 0xc9, + 0xb2, 0xa7, 0x25, 0x95, 0x60, 0xc7, 0x2c, 0x69, + 0x5c, 0xdc, 0xd6, 0xfd, 0x31, 0xe2, 0xa4, 0xc0, + 0xfe, 0x53, 0x6e, 0xcd, 0xd3, 0x36, 0x69, 0x21}, {0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}, {1}, - {0x47, 0x56, 0x98, 0x99, 0xc7, 0x61, 0x0a, 0x82, - 0x1a, 0xdf, 0x82, 0x22, 0x1f, 0x2c, 0x72, 0x88, - 0xc3, 0x29, 0x09, 0x52, 0x78, 0xe9, 0x1e, 0xe4, - 0x47, 0x4b, 0x4c, 0x81, 0xa6, 0x02, 0xfd, 0x29} + {0xa3, 0xdd, 0xb7, 0xa5, 0xb3, 0x8a, 0xde, 0x6d, + 0xf5, 0x52, 0x51, 0x77, 0x80, 0x9f, 0xf0, 0x20, + 0x7d, 0xe3, 0xab, 0x64, 0x8e, 0x4e, 0xea, 0x66, + 0x65, 0x76, 0x8b, 0xd7, 0x0f, 0x5f, 0x87, 0x67}, }; const ecc_25519_work_t ecc_25519_work_base_ed25519 = { - {0xd4, 0x6b, 0xfe, 0x7f, 0x39, 0xfa, 0x8c, 0x22, - 0xe1, 0x96, 0x23, 0xeb, 0x26, 0xb7, 0x8e, 0x6a, - 0x34, 0x74, 0x8b, 0x66, 0xd6, 0xa3, 0x26, 0xdd, - 0x19, 0x5e, 0x9f, 0x21, 0x50, 0x43, 0x7c, 0x54}, + {0x1a, 0xd5, 0x25, 0x8f, 0x60, 0x2d, 0x56, 0xc9, + 0xb2, 0xa7, 0x25, 0x95, 0x60, 0xc7, 0x2c, 0x69, + 0x5c, 0xdc, 0xd6, 0xfd, 0x31, 0xe2, 0xa4, 0xc0, + 0xfe, 0x53, 0x6e, 0xcd, 0xd3, 0x36, 0x69, 0x21}, {0x58, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66, 0x66}, {1}, - {0x47, 0x56, 0x98, 0x99, 0xc7, 0x61, 0x0a, 0x82, - 0x1a, 0xdf, 0x82, 0x22, 0x1f, 0x2c, 0x72, 0x88, - 0xc3, 0x29, 0x09, 0x52, 0x78, 0xe9, 0x1e, 0xe4, - 0x47, 0x4b, 0x4c, 0x81, 0xa6, 0x02, 0xfd, 0x29} + {0xa3, 0xdd, 0xb7, 0xa5, 0xb3, 0x8a, 0xde, 0x6d, + 0xf5, 0x52, 0x51, 0x77, 0x80, 0x9f, 0xf0, 0x20, + 0x7d, 0xe3, 0xab, 0x64, 0x8e, 0x4e, 0xea, 0x66, + 0x65, 0x76, 0x8b, 0xd7, 0x0f, 0x5f, 0x87, 0x67}, }; static const uint32_t zero[32] = {0}; static const uint32_t one[32] = {1}; +static const uint32_t minus1[32] = { + 0xec, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, + 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, + 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, + 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x7f, +}; + +/** Ed25519 parameter -(121665/121666) */ +static const uint32_t d[32] = { + 0xa3, 0x78, 0x59, 0x13, 0xca, 0x4d, 0xeb, 0x75, + 0xab, 0xd8, 0x41, 0x41, 0x4d, 0x0a, 0x70, 0x00, + 0x98, 0xe8, 0x79, 0x77, 0x79, 0x40, 0xc7, 0x8c, + 0x73, 0xfe, 0x6f, 0x2b, 0xee, 0x6c, 0x03, 0x52, +}; + /** Factor to multiply the X coordinate with to convert from the legacy to the Ed25519 curve */ static const uint32_t legacy_to_ed25519[32] = { @@ -233,7 +248,7 @@ static void freeze(uint32_t a[32]) { * * The input must be \em squeezed. */ -static int parity(uint32_t a[32]) { +static int parity(const uint32_t a[32]) { uint32_t b[32]; add(b, a, minusp); @@ -395,13 +410,6 @@ static void select(uint32_t out[32], const uint32_t r[32], const uint32_t s[32], * If the given integer has no square root, 0 is returned, 1 otherwise. */ static int square_root(uint32_t out[32], const uint32_t z[32]) { - static const uint32_t minus1[32] = { - 0xec, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, - 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, - 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, - 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0xff, 0x7f - }; - static const uint32_t rho_s[32] = { 0xb0, 0xa0, 0x0e, 0x4a, 0x27, 0x1b, 0xee, 0xc4, 0x78, 0xe4, 0x2f, 0xad, 0x06, 0x18, 0x43, 0x2f, @@ -557,20 +565,22 @@ static void recip(uint32_t out[32], const uint32_t z[32]) { /** * Checks if the X and Y coordinates of a work structure represent a valid point of the curve * - * Also fills out the T coordinate. + * Also fills in the T coordinate. */ static int check_load_xy(ecc_25519_work_t *val) { - uint32_t X2[32], Y2[32], aX2[32], dX2[32], dX2Y2[32], aX2_Y2[32], _1_dX2Y2[32], r[32]; + uint32_t X2[32], Y2[32], dX2[32], dX2Y2[32], Y2_X2[32], Y2_X2_1[32], r[32]; /* Check validity */ square(X2, val->X); square(Y2, val->Y); - mult_int(aX2, UINT32_C(486664), X2); - mult_int(dX2, UINT32_C(486660), X2); + + mult(dX2, d, X2); mult(dX2Y2, dX2, Y2); - add(aX2_Y2, aX2, Y2); - add(_1_dX2Y2, one, dX2Y2); - sub(r, aX2_Y2, _1_dX2Y2); + + sub(Y2_X2, Y2, X2); + sub(Y2_X2_1, Y2_X2, one); + + sub(r, Y2_X2_1, dX2Y2); squeeze(r); if (!check_zero(r)) @@ -583,28 +593,28 @@ static int check_load_xy(ecc_25519_work_t *val) { int ecc_25519_load_xy_ed25519(ecc_25519_work_t *out, const ecc_int256_t *x, const ecc_int256_t *y) { int i; - uint32_t tmp[32]; for (i = 0; i < 32; i++) { - tmp[i] = x->p[i]; + out->X[i] = x->p[i]; out->Y[i] = y->p[i]; out->Z[i] = (i == 0); } - mult(out->X, tmp, ed25519_to_legacy); - return check_load_xy(out); } int ecc_25519_load_xy_legacy(ecc_25519_work_t *out, const ecc_int256_t *x, const ecc_int256_t *y) { int i; + uint32_t tmp[32]; for (i = 0; i < 32; i++) { - out->X[i] = x->p[i]; + tmp[i] = x->p[i]; out->Y[i] = y->p[i]; out->Z[i] = (i == 0); } + mult(out->X, tmp, legacy_to_ed25519); + return check_load_xy(out); } @@ -614,14 +624,13 @@ int ecc_25519_load_xy(ecc_25519_work_t *out, const ecc_int256_t *x, const ecc_in void ecc_25519_store_xy_ed25519(ecc_int256_t *x, ecc_int256_t *y, const ecc_25519_work_t *in) { - uint32_t X[32], tmp[32], Y[32], Z[32]; + uint32_t X[32], Y[32], Z[32]; int i; recip(Z, in->Z); if (x) { - mult(tmp, Z, in->X); - mult(X, tmp, legacy_to_ed25519); + mult(X, Z, in->X); freeze(X); for (i = 0; i < 32; i++) x->p[i] = X[i]; @@ -636,13 +645,14 @@ void ecc_25519_store_xy_ed25519(ecc_int256_t *x, ecc_int256_t *y, const ecc_2551 } void ecc_25519_store_xy_legacy(ecc_int256_t *x, ecc_int256_t *y, const ecc_25519_work_t *in) { - uint32_t X[32], Y[32], Z[32]; + uint32_t X[32], tmp[32], Y[32], Z[32]; int i; recip(Z, in->Z); if (x) { - mult(X, Z, in->X); + mult(tmp, Z, in->X); + mult(X, tmp, ed25519_to_legacy); freeze(X); for (i = 0; i < 32; i++) x->p[i] = X[i]; @@ -662,10 +672,9 @@ void ecc_25519_store_xy(ecc_int256_t *x, ecc_int256_t *y, const ecc_25519_work_t int ecc_25519_load_packed_ed25519(ecc_25519_work_t *out, const ecc_int256_t *in) { - static const uint32_t a[32] = {UINT32_C(486664)}; int i; - uint32_t Y2[32] /* Y^2 */, dY2[32] /* dY^2 */, _1_Y2[32] /* 1-Y^2 */, a_dY2[32] /* a-dY^2 */, _1_a_dY2[32] /* 1/(a-dY^2) */; - uint32_t X2[32] /* X^2 */, X[32], Xt[32], X_ed25519[32]; + uint32_t Y2[32] /* Y^2 */, dY2[32] /* dY^2 */, Y2_1[32] /* Y^2-1 */, dY2_1[32] /* dY^2+1 */, _1_dY2_1[32] /* 1/(dY^2+1) */; + uint32_t X2[32] /* X^2 */, X[32], Xt[32]; for (i = 0; i < 32; i++) { out->Y[i] = in->p[i]; @@ -675,21 +684,19 @@ int ecc_25519_load_packed_ed25519(ecc_25519_work_t *out, const ecc_int256_t *in) out->Y[31] &= 0x7f; square(Y2, out->Y); - mult_int(dY2, UINT32_C(486660), Y2); - sub(_1_Y2, one, Y2); - sub(a_dY2, a, dY2); - recip(_1_a_dY2, a_dY2); - mult(X2, _1_Y2, _1_a_dY2); + mult(dY2, d, Y2); + sub(Y2_1, Y2, one); + add(dY2_1, dY2, one); + recip(_1_dY2_1, dY2_1); + mult(X2, Y2_1, _1_dY2_1); if (!square_root(X, X2)) return 0; - mult(X_ed25519, X, legacy_to_ed25519); - /* No squeeze is necessary after subtractions from zero if the subtrahend is squeezed */ sub(Xt, zero, X); - select(out->X, X, Xt, (in->p[31] >> 7) ^ parity(X_ed25519)); + select(out->X, X, Xt, (in->p[31] >> 7) ^ parity(X)); mult(out->T, out->X, out->Y); @@ -699,16 +706,16 @@ int ecc_25519_load_packed_ed25519(ecc_25519_work_t *out, const ecc_int256_t *in) int ecc_25519_load_packed_legacy(ecc_25519_work_t *out, const ecc_int256_t *in) { int i; uint32_t X2[32] /* X^2 */, aX2[32] /* aX^2 */, dX2[32] /* dX^2 */, _1_aX2[32] /* 1-aX^2 */, _1_dX2[32] /* 1-aX^2 */; - uint32_t _1_1_dX2[32] /* 1/(1-aX^2) */, Y2[32] /* Y^2 */, Y[32], Yt[32]; + uint32_t _1_1_dX2[32] /* 1/(1-aX^2) */, Y2[32] /* Y^2 */, Y[32], Yt[32], X_legacy[32]; for (i = 0; i < 32; i++) { - out->X[i] = in->p[i]; + X_legacy[i] = in->p[i]; out->Z[i] = (i == 0); } - out->X[31] &= 0x7f; + X_legacy[31] &= 0x7f; - square(X2, out->X); + square(X2, X_legacy); mult_int(aX2, UINT32_C(486664), X2); mult_int(dX2, UINT32_C(486660), X2); sub(_1_aX2, one, aX2); @@ -724,6 +731,7 @@ int ecc_25519_load_packed_legacy(ecc_25519_work_t *out, const ecc_int256_t *in) select(out->Y, Y, Yt, (in->p[31] >> 7) ^ parity(Y)); + mult(out->X, X_legacy, legacy_to_ed25519); mult(out->T, out->X, out->Y); return 1; @@ -776,20 +784,26 @@ void ecc_25519_negate(ecc_25519_work_t *out, const ecc_25519_work_t *in) { } void ecc_25519_double(ecc_25519_work_t *out, const ecc_25519_work_t *in) { - uint32_t A[32], B[32], C[32], D[32], E[32], F[32], G[32], H[32], t0[32], t1[32], t2[32], t3[32]; + uint32_t A[32], B[32], C[32], D[32], E[32], F[32], G[32], H[32], t0[32], t1[32]; square(A, in->X); + square(B, in->Y); + square(t0, in->Z); mult_int(C, 2, t0); - mult_int(D, UINT32_C(486664), A); - add(t1, in->X, in->Y); - square(t2, t1); - sub(t3, t2, A); - sub(E, t3, B); + + sub(D, zero, A); + + add(t0, in->X, in->Y); + square(t1, t0); + sub(t0, t1, A); + sub(E, t0, B); + add(G, D, B); sub(F, G, C); sub(H, D, B); + mult(out->X, E, F); mult(out->Y, G, H); mult(out->T, E, H); @@ -797,22 +811,31 @@ void ecc_25519_double(ecc_25519_work_t *out, const ecc_25519_work_t *in) { } void ecc_25519_add(ecc_25519_work_t *out, const ecc_25519_work_t *in1, const ecc_25519_work_t *in2) { - uint32_t A[32], B[32], C[32], D[32], E[32], F[32], G[32], H[32], t0[32], t1[32], t2[32], t3[32], t4[32], t5[32]; + const uint32_t j = UINT32_C(60833); + const uint32_t k = UINT32_C(121665); + uint32_t A[32], B[32], C[32], D[32], E[32], F[32], G[32], H[32], t0[32], t1[32]; - mult(A, in1->X, in2->X); - mult(B, in1->Y, in2->Y); - mult_int(t0, UINT32_C(486660), in2->T); + sub(t0, in1->Y, in1->X); + mult_int(t1, j, t0); + sub(t0, in2->Y, in2->X); + mult(A, t0, t1); + + add(t0, in1->Y, in1->X); + mult_int(t1, j, t0); + add(t0, in2->Y, in2->X); + mult(B, t0, t1); + + mult_int(t0, k, in2->T); mult(C, in1->T, t0); - mult(D, in1->Z, in2->Z); - add(t1, in1->X, in1->Y); - add(t2, in2->X, in2->Y); - mult(t3, t1, t2); - sub(t4, t3, A); - sub(E, t4, B); - sub(F, D, C); - add(G, D, C); - mult_int(t5, UINT32_C(486664), A); - sub(H, B, t5); + + mult_int(t0, 2*j, in2->Z); + mult(D, in1->Z, t0); + + sub(E, B, A); + add(F, D, C); + sub(G, D, C); + add(H, B, A); + mult(out->X, E, F); mult(out->Y, G, H); mult(out->T, E, H); -- cgit v1.2.3