void ge_montx_to_p2(ge_p2* p, const fe u, const unsigned char ed_sign_bit) { fe x, y, A, v, v2, iv, nx; fe_frombytes(A, A_bytes); /* given u, recover edwards y */ /* given u, recover v */ /* given u and v, recover edwards x */ fe_montx_to_edy(y, u); /* y = (u - 1) / (u + 1) */ fe_mont_rhs(v2, u); /* v^2 = u(u^2 + Au + 1) */ fe_sqrt(v, v2); /* v = sqrt(v^2) */ fe_mul(x, u, A); /* x = u * sqrt(-(A+2)) */ fe_invert(iv, v); /* 1/v */ fe_mul(x, x, iv); /* x = (u/v) * sqrt(-(A+2)) */ fe_neg(nx, x); /* negate x to match sign bit */ fe_cmov(x, nx, fe_isnegative(x) ^ ed_sign_bit); fe_copy(p->X, x); fe_copy(p->Y, y); fe_1(p->Z); /* POSTCONDITION: check that p->X and p->Y satisfy the Ed curve equation */ /* -x^2 + y^2 = 1 + dx^2y^2 */ #ifndef NDEBUG { fe one, d, x2, y2, x2y2, dx2y2; unsigned char dbytes[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 }; fe_frombytes(d, dbytes); fe_1(one); fe_sq(x2, p->X); /* x^2 */ fe_sq(y2, p->Y); /* y^2 */ fe_mul(dx2y2, x2, y2); /* x^2y^2 */ fe_mul(dx2y2, dx2y2, d); /* dx^2y^2 */ fe_add(dx2y2, dx2y2, one); /* dx^2y^2 + 1 */ fe_neg(x2y2, x2); /* -x^2 */ fe_add(x2y2, x2y2, y2); /* -x^2 + y^2 */ assert(fe_isequal(x2y2, dx2y2)); } #endif }
void fe_mont_rhs(fe v2, fe u) { fe A, one; fe u2, Au, inner; fe_1(one); fe_0(A); A[0] = 486662; /* A = 486662 */ fe_sq(u2, u); /* u^2 */ fe_mul(Au, A, u); /* Au */ fe_add(inner, u2, Au); /* u^2 + Au */ fe_add(inner, inner, one); /* u^2 + Au + 1 */ fe_mul(v2, u, inner); /* u(u^2 + Au + 1) */ }
void fe_sqrt(fe out, const fe a) { fe exp, b, b2, bi, i; #ifndef NDEBUG fe legendre, zero, one; #endif fe_frombytes(i, i_bytes); fe_pow22523(exp, a); /* b = a^(q-5)/8 */ /* PRECONDITION: legendre symbol == 1 (square) or 0 (a == zero) */ #ifndef NDEBUG fe_sq(legendre, exp); /* in^((q-5)/4) */ fe_sq(legendre, legendre); /* in^((q-5)/2) */ fe_mul(legendre, legendre, a); /* in^((q-3)/2) */ fe_mul(legendre, legendre, a); /* in^((q-1)/2) */ fe_0(zero); fe_1(one); assert(fe_isequal(legendre, zero) || fe_isequal(legendre, one)); #endif fe_mul(b, a, exp); /* b = a * a^(q-5)/8 */ fe_sq(b2, b); /* b^2 = a * a^(q-1)/4 */ /* note b^4 == a^2, so b^2 == a or -a * if b^2 != a, multiply it by sqrt(-1) */ fe_mul(bi, b, i); fe_cmov(b, bi, 1 ^ fe_isequal(b2, a)); fe_copy(out, b); /* PRECONDITION: out^2 == a */ #ifndef NDEBUG fe_sq(b2, out); assert(fe_isequal(a, b2)); #endif }
static int crypto_scalarmult_curve25519_ref10(unsigned char *q, const unsigned char *n, const unsigned char *p) { unsigned char e[32]; unsigned int i; fe x1; fe x2; fe z2; fe x3; fe z3; fe tmp0; fe tmp1; int pos; unsigned int swap; unsigned int b; for (i = 0;i < 32;++i) e[i] = n[i]; e[0] &= 248; e[31] &= 127; e[31] |= 64; fe_frombytes(x1,p); fe_1(x2); fe_0(z2); fe_copy(x3,x1); fe_1(z3); swap = 0; for (pos = 254;pos >= 0;--pos) { b = e[pos / 8] >> (pos & 7); b &= 1; swap ^= b; fe_cswap(x2,x3,swap); fe_cswap(z2,z3,swap); swap = b; fe_sub(tmp0,x3,z3); fe_sub(tmp1,x2,z2); fe_add(x2,x2,z2); fe_add(z2,x3,z3); fe_mul(z3,tmp0,x2); fe_mul(z2,z2,tmp1); fe_sq(tmp0,tmp1); fe_sq(tmp1,x2); fe_add(x3,z3,z2); fe_sub(z2,z3,z2); fe_mul(x2,tmp1,tmp0); fe_sub(tmp1,tmp1,tmp0); fe_sq(z2,z2); fe_mul121666(z3,tmp1); fe_sq(x3,x3); fe_add(tmp0,tmp0,z3); fe_mul(z3,x1,z2); fe_mul(z2,tmp1,tmp0); } fe_cswap(x2,x3,swap); fe_cswap(z2,z3,swap); fe_invert(z2,z2); fe_mul(x2,x2,z2); fe_tobytes(q,x2); return 0; }
void ed25519_key_exchange(unsigned char *shared_secret, const unsigned char *public_key, const unsigned char *private_key) { unsigned char e[32]; unsigned int i; fe x1; fe x2; fe z2; fe x3; fe z3; fe tmp0; fe tmp1; int pos; unsigned int swap; unsigned int b; /* copy the private key and make sure it's valid */ for (i = 0; i < 32; ++i) { e[i] = private_key[i]; } e[0] &= 248; e[31] &= 63; e[31] |= 64; /* unpack the public key and convert edwards to montgomery */ /* due to CodesInChaos: montgomeryX = (edwardsY + 1)*inverse(1 - edwardsY) mod p */ fe_frombytes(x1, public_key); fe_1(tmp1); fe_add(tmp0, x1, tmp1); fe_sub(tmp1, tmp1, x1); fe_invert(tmp1, tmp1); fe_mul(x1, tmp0, tmp1); fe_1(x2); fe_0(z2); fe_copy(x3, x1); fe_1(z3); swap = 0; for (pos = 254; pos >= 0; --pos) { b = e[pos / 8] >> (pos & 7); b &= 1; swap ^= b; fe_cswap(x2, x3, swap); fe_cswap(z2, z3, swap); swap = b; /* from montgomery.h */ fe_sub(tmp0, x3, z3); fe_sub(tmp1, x2, z2); fe_add(x2, x2, z2); fe_add(z2, x3, z3); fe_mul(z3, tmp0, x2); fe_mul(z2, z2, tmp1); fe_sq(tmp0, tmp1); fe_sq(tmp1, x2); fe_add(x3, z3, z2); fe_sub(z2, z3, z2); fe_mul(x2, tmp1, tmp0); fe_sub(tmp1, tmp1, tmp0); fe_sq(z2, z2); fe_mul121666(z3, tmp1); fe_sq(x3, x3); fe_add(tmp0, tmp0, z3); fe_mul(z3, x1, z2); fe_mul(z2, tmp1, tmp0); } fe_cswap(x2, x3, swap); fe_cswap(z2, z3, swap); fe_invert(z2, z2); fe_mul(x2, x2, z2); fe_tobytes(shared_secret, x2); }