/* * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved. * Copyright 2015-2016 Cryptography Research, Inc. * * Licensed under the OpenSSL license (the "License"). You may not use * this file except in compliance with the License. You can obtain a copy * in the file LICENSE in the source distribution or at * https://www.openssl.org/source/license.html * * Originally written by Mike Hamburg */ #include #include "word.h" #include "field.h" #include "point_448.h" #include "ed448.h" #include "curve448_lcl.h" #define COFACTOR 4 #define C448_WNAF_FIXED_TABLE_BITS 5 #define C448_WNAF_VAR_TABLE_BITS 3 #define EDWARDS_D (-39081) static const curve448_scalar_t precomputed_scalarmul_adjustment = { { { SC_LIMB(0xc873d6d54a7bb0cfULL), SC_LIMB(0xe933d8d723a70aadULL), SC_LIMB(0xbb124b65129c96fdULL), SC_LIMB(0x00000008335dc163ULL) } } }; #define TWISTED_D (EDWARDS_D - 1) #define WBITS C448_WORD_BITS /* NB this may be different from ARCH_WORD_BITS */ /* Inverse. */ static void gf_invert(gf y, const gf x, int assert_nonzero) { mask_t ret; gf t1, t2; gf_sqr(t1, x); /* o^2 */ ret = gf_isr(t2, t1); /* +-1/sqrt(o^2) = +-1/o */ (void)ret; if (assert_nonzero) assert(ret); gf_sqr(t1, t2); gf_mul(t2, t1, x); /* not direct to y in case of alias. */ gf_copy(y, t2); } /** identity = (0,1) */ const curve448_point_t curve448_point_identity = { {{{{0}}}, {{{1}}}, {{{1}}}, {{{0}}}} }; static void point_double_internal(curve448_point_t p, const curve448_point_t q, int before_double) { gf a, b, c, d; gf_sqr(c, q->x); gf_sqr(a, q->y); gf_add_nr(d, c, a); /* 2+e */ gf_add_nr(p->t, q->y, q->x); /* 2+e */ gf_sqr(b, p->t); gf_subx_nr(b, b, d, 3); /* 4+e */ gf_sub_nr(p->t, a, c); /* 3+e */ gf_sqr(p->x, q->z); gf_add_nr(p->z, p->x, p->x); /* 2+e */ gf_subx_nr(a, p->z, p->t, 4); /* 6+e */ if (GF_HEADROOM == 5) gf_weak_reduce(a); /* or 1+e */ gf_mul(p->x, a, b); gf_mul(p->z, p->t, a); gf_mul(p->y, p->t, d); if (!before_double) gf_mul(p->t, b, d); } void curve448_point_double(curve448_point_t p, const curve448_point_t q) { point_double_internal(p, q, 0); } /* Operations on [p]niels */ static ossl_inline void cond_neg_niels(niels_t n, mask_t neg) { gf_cond_swap(n->a, n->b, neg); gf_cond_neg(n->c, neg); } static void pt_to_pniels(pniels_t b, const curve448_point_t a) { gf_sub(b->n->a, a->y, a->x); gf_add(b->n->b, a->x, a->y); gf_mulw(b->n->c, a->t, 2 * TWISTED_D); gf_add(b->z, a->z, a->z); } static void pniels_to_pt(curve448_point_t e, const pniels_t d) { gf eu; gf_add(eu, d->n->b, d->n->a); gf_sub(e->y, d->n->b, d->n->a); gf_mul(e->t, e->y, eu); gf_mul(e->x, d->z, e->y); gf_mul(e->y, d->z, eu); gf_sqr(e->z, d->z); } static void niels_to_pt(curve448_point_t e, const niels_t n) { gf_add(e->y, n->b, n->a); gf_sub(e->x, n->b, n->a); gf_mul(e->t, e->y, e->x); gf_copy(e->z, ONE); } static void add_niels_to_pt(curve448_point_t d, const niels_t e, int before_double) { gf a, b, c; gf_sub_nr(b, d->y, d->x); /* 3+e */ gf_mul(a, e->a, b); gf_add_nr(b, d->x, d->y); /* 2+e */ gf_mul(d->y, e->b, b); gf_mul(d->x, e->c, d->t); gf_add_nr(c, a, d->y); /* 2+e */ gf_sub_nr(b, d->y, a); /* 3+e */ gf_sub_nr(d->y, d->z, d->x); /* 3+e */ gf_add_nr(a, d->x, d->z); /* 2+e */ gf_mul(d->z, a, d->y); gf_mul(d->x, d->y, b); gf_mul(d->y, a, c); if (!before_double) gf_mul(d->t, b, c); } static void sub_niels_from_pt(curve448_point_t d, const niels_t e, int before_double) { gf a, b, c; gf_sub_nr(b, d->y, d->x); /* 3+e */ gf_mul(a, e->b, b); gf_add_nr(b, d->x, d->y); /* 2+e */ gf_mul(d->y, e->a, b); gf_mul(d->x, e->c, d->t); gf_add_nr(c, a, d->y); /* 2+e */ gf_sub_nr(b, d->y, a); /* 3+e */ gf_add_nr(d->y, d->z, d->x); /* 2+e */ gf_sub_nr(a, d->z, d->x); /* 3+e */ gf_mul(d->z, a, d->y); gf_mul(d->x, d->y, b); gf_mul(d->y, a, c); if (!before_double) gf_mul(d->t, b, c); } static void add_pniels_to_pt(curve448_point_t p, const pniels_t pn, int before_double) { gf L0; gf_mul(L0, p->z, pn->z); gf_copy(p->z, L0); add_niels_to_pt(p, pn->n, before_double); } static void sub_pniels_from_pt(curve448_point_t p, const pniels_t pn, int before_double) { gf L0; gf_mul(L0, p->z, pn->z); gf_copy(p->z, L0); sub_niels_from_pt(p, pn->n, before_double); } c448_bool_t curve448_point_eq(const curve448_point_t p, const curve448_point_t q) { mask_t succ; gf a, b; /* equality mod 2-torsion compares x/y */ gf_mul(a, p->y, q->x); gf_mul(b, q->y, p->x); succ = gf_eq(a, b); return mask_to_bool(succ); } c448_bool_t curve448_point_valid(const curve448_point_t p) { mask_t out; gf a, b, c; gf_mul(a, p->x, p->y); gf_mul(b, p->z, p->t); out = gf_eq(a, b); gf_sqr(a, p->x); gf_sqr(b, p->y); gf_sub(a, b, a); gf_sqr(b, p->t); gf_mulw(c, b, TWISTED_D); gf_sqr(b, p->z); gf_add(b, b, c); out &= gf_eq(a, b); out &= ~gf_eq(p->z, ZERO); return mask_to_bool(out); } static ossl_inline void constant_time_lookup_niels(niels_s * RESTRICT ni, const niels_t * table, int nelts, int idx) { constant_time_lookup(ni, table, sizeof(niels_s), nelts, idx); } void curve448_precomputed_scalarmul(curve448_point_t out, const curve448_precomputed_s * table, const curve448_scalar_t scalar) { unsigned int i, j, k; const unsigned int n = COMBS_N, t = COMBS_T, s = COMBS_S; niels_t ni; curve448_scalar_t scalar1x; curve448_scalar_add(scalar1x, scalar, precomputed_scalarmul_adjustment); curve448_scalar_halve(scalar1x, scalar1x); for (i = s; i > 0; i--) { if (i != s) point_double_internal(out, out, 0); for (j = 0; j < n; j++) { int tab = 0; mask_t invert; for (k = 0; k < t; k++) { unsigned int bit = (i - 1) + s * (k + j * t); if (bit < C448_SCALAR_BITS) tab |= (scalar1x->limb[bit / WBITS] >> (bit % WBITS) & 1) << k; } invert = (tab >> (t - 1)) - 1; tab ^= invert; tab &= (1 << (t - 1)) - 1; constant_time_lookup_niels(ni, &table->table[j << (t - 1)], 1 << (t - 1), tab); cond_neg_niels(ni, invert); if ((i != s) || j != 0) add_niels_to_pt(out, ni, j == n - 1 && i != 1); else niels_to_pt(out, ni); } } OPENSSL_cleanse(ni, sizeof(ni)); OPENSSL_cleanse(scalar1x, sizeof(scalar1x)); } void curve448_point_mul_by_ratio_and_encode_like_eddsa( uint8_t enc[EDDSA_448_PUBLIC_BYTES], const curve448_point_t p) { gf x, y, z, t; curve448_point_t q; /* The point is now on the twisted curve. Move it to untwisted. */ curve448_point_copy(q, p); { /* 4-isogeny: 2xy/(y^+x^2), (y^2-x^2)/(2z^2-y^2+x^2) */ gf u; gf_sqr(x, q->x); gf_sqr(t, q->y); gf_add(u, x, t); gf_add(z, q->y, q->x); gf_sqr(y, z); gf_sub(y, y, u); gf_sub(z, t, x); gf_sqr(x, q->z); gf_add(t, x, x); gf_sub(t, t, z); gf_mul(x, t, y); gf_mul(y, z, u); gf_mul(z, u, t); OPENSSL_cleanse(u, sizeof(u)); } /* Affinize */ gf_invert(z, z, 1); gf_mul(t, x, z); gf_mul(x, y, z); /* Encode */ enc[EDDSA_448_PRIVATE_BYTES - 1] = 0; gf_serialize(enc, x, 1); enc[EDDSA_448_PRIVATE_BYTES - 1] |= 0x80 & gf_lobit(t); OPENSSL_cleanse(x, sizeof(x)); OPENSSL_cleanse(y, sizeof(y)); OPENSSL_cleanse(z, sizeof(z)); OPENSSL_cleanse(t, sizeof(t)); curve448_point_destroy(q); } c448_error_t curve448_point_decode_like_eddsa_and_mul_by_ratio( curve448_point_t p, const uint8_t enc[EDDSA_448_PUBLIC_BYTES]) { uint8_t enc2[EDDSA_448_PUBLIC_BYTES]; mask_t low; mask_t succ; memcpy(enc2, enc, sizeof(enc2)); low = ~word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1] & 0x80); enc2[EDDSA_448_PRIVATE_BYTES - 1] &= ~0x80; succ = gf_deserialize(p->y, enc2, 1, 0); succ &= word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1]); gf_sqr(p->x, p->y); gf_sub(p->z, ONE, p->x); /* num = 1-y^2 */ gf_mulw(p->t, p->x, EDWARDS_D); /* dy^2 */ gf_sub(p->t, ONE, p->t); /* denom = 1-dy^2 or 1-d + dy^2 */ gf_mul(p->x, p->z, p->t); succ &= gf_isr(p->t, p->x); /* 1/sqrt(num * denom) */ gf_mul(p->x, p->t, p->z); /* sqrt(num / denom) */ gf_cond_neg(p->x, gf_lobit(p->x) ^ low); gf_copy(p->z, ONE); { gf a, b, c, d; /* 4-isogeny 2xy/(y^2-ax^2), (y^2+ax^2)/(2-y^2-ax^2) */ gf_sqr(c, p->x); gf_sqr(a, p->y); gf_add(d, c, a); gf_add(p->t, p->y, p->x); gf_sqr(b, p->t); gf_sub(b, b, d); gf_sub(p->t, a, c); gf_sqr(p->x, p->z); gf_add(p->z, p->x, p->x); gf_sub(a, p->z, d); gf_mul(p->x, a, b); gf_mul(p->z, p->t, a); gf_mul(p->y, p->t, d); gf_mul(p->t, b, d); OPENSSL_cleanse(a, sizeof(a)); OPENSSL_cleanse(b, sizeof(b)); OPENSSL_cleanse(c, sizeof(c)); OPENSSL_cleanse(d, sizeof(d)); } OPENSSL_cleanse(enc2, sizeof(enc2)); assert(curve448_point_valid(p) || ~succ); return c448_succeed_if(mask_to_bool(succ)); } c448_error_t x448_int(uint8_t out[X_PUBLIC_BYTES], const uint8_t base[X_PUBLIC_BYTES], const uint8_t scalar[X_PRIVATE_BYTES]) { gf x1, x2, z2, x3, z3, t1, t2; int t; mask_t swap = 0; mask_t nz; (void)gf_deserialize(x1, base, 1, 0); gf_copy(x2, ONE); gf_copy(z2, ZERO); gf_copy(x3, x1); gf_copy(z3, ONE); for (t = X_PRIVATE_BITS - 1; t >= 0; t--) { uint8_t sb = scalar[t / 8]; mask_t k_t; /* Scalar conditioning */ if (t / 8 == 0) sb &= -(uint8_t)COFACTOR; else if (t == X_PRIVATE_BITS - 1) sb = -1; k_t = (sb >> (t % 8)) & 1; k_t = 0 - k_t; /* set to all 0s or all 1s */ swap ^= k_t; gf_cond_swap(x2, x3, swap); gf_cond_swap(z2, z3, swap); swap = k_t; /* * The "_nr" below skips coefficient reduction. In the following * comments, "2+e" is saying that the coefficients are at most 2+epsilon * times the reduction limit. */ gf_add_nr(t1, x2, z2); /* A = x2 + z2 */ /* 2+e */ gf_sub_nr(t2, x2, z2); /* B = x2 - z2 */ /* 3+e */ gf_sub_nr(z2, x3, z3); /* D = x3 - z3 */ /* 3+e */ gf_mul(x2, t1, z2); /* DA */ gf_add_nr(z2, z3, x3); /* C = x3 + z3 */ /* 2+e */ gf_mul(x3, t2, z2); /* CB */ gf_sub_nr(z3, x2, x3); /* DA-CB */ /* 3+e */ gf_sqr(z2, z3); /* (DA-CB)^2 */ gf_mul(z3, x1, z2); /* z3 = x1(DA-CB)^2 */ gf_add_nr(z2, x2, x3); /* (DA+CB) */ /* 2+e */ gf_sqr(x3, z2); /* x3 = (DA+CB)^2 */ gf_sqr(z2, t1); /* AA = A^2 */ gf_sqr(t1, t2); /* BB = B^2 */ gf_mul(x2, z2, t1); /* x2 = AA*BB */ gf_sub_nr(t2, z2, t1); /* E = AA-BB */ /* 3+e */ gf_mulw(t1, t2, -EDWARDS_D); /* E*-d = a24*E */ gf_add_nr(t1, t1, z2); /* AA + a24*E */ /* 2+e */ gf_mul(z2, t2, t1); /* z2 = E(AA+a24*E) */ } /* Finish */ gf_cond_swap(x2, x3, swap); gf_cond_swap(z2, z3, swap); gf_invert(z2, z2, 0); gf_mul(x1, x2, z2); gf_serialize(out, x1, 1); nz = ~gf_eq(x1, ZERO); OPENSSL_cleanse(x1, sizeof(x1)); OPENSSL_cleanse(x2, sizeof(x2)); OPENSSL_cleanse(z2, sizeof(z2)); OPENSSL_cleanse(x3, sizeof(x3)); OPENSSL_cleanse(z3, sizeof(z3)); OPENSSL_cleanse(t1, sizeof(t1)); OPENSSL_cleanse(t2, sizeof(t2)); return c448_succeed_if(mask_to_bool(nz)); } void curve448_point_mul_by_ratio_and_encode_like_x448(uint8_t out[X_PUBLIC_BYTES], const curve448_point_t p) { curve448_point_t q; curve448_point_copy(q, p); gf_invert(q->t, q->x, 0); /* 1/x */ gf_mul(q->z, q->t, q->y); /* y/x */ gf_sqr(q->y, q->z); /* (y/x)^2 */ gf_serialize(out, q->y, 1); curve448_point_destroy(q); } void x448_derive_public_key(uint8_t out[X_PUBLIC_BYTES], const uint8_t scalar[X_PRIVATE_BYTES]) { /* Scalar conditioning */ uint8_t scalar2[X_PRIVATE_BYTES]; curve448_scalar_t the_scalar; curve448_point_t p; unsigned int i; memcpy(scalar2, scalar, sizeof(scalar2)); scalar2[0] &= -(uint8_t)COFACTOR; scalar2[X_PRIVATE_BYTES - 1] &= ~((0u - 1u) << ((X_PRIVATE_BITS + 7) % 8)); scalar2[X_PRIVATE_BYTES - 1] |= 1 << ((X_PRIVATE_BITS + 7) % 8); curve448_scalar_decode_long(the_scalar, scalar2, sizeof(scalar2)); /* Compensate for the encoding ratio */ for (i = 1; i < X448_ENCODE_RATIO; i <<= 1) curve448_scalar_halve(the_scalar, the_scalar); curve448_precomputed_scalarmul(p, curve448_precomputed_base, the_scalar); curve448_point_mul_by_ratio_and_encode_like_x448(out, p); curve448_point_destroy(p); } /* Control for variable-time scalar multiply algorithms. */ struct smvt_control { int power, addend; }; #if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ > 3)) # define NUMTRAILINGZEROS __builtin_ctz #else # define NUMTRAILINGZEROS numtrailingzeros static uint32_t numtrailingzeros(uint32_t i) { uint32_t tmp; uint32_t num = 31; if (i == 0) return 32; tmp = i << 16; if (tmp != 0) { i = tmp; num -= 16; } tmp = i << 8; if (tmp != 0) { i = tmp; num -= 8; } tmp = i << 4; if (tmp != 0) { i = tmp; num -= 4; } tmp = i << 2; if (tmp != 0) { i = tmp; num -= 2; } tmp = i << 1; if (tmp != 0) num--; return num; } #endif static int recode_wnaf(struct smvt_control *control, /* [nbits/(table_bits + 1) + 3] */ const curve448_scalar_t scalar, unsigned int table_bits) { unsigned int table_size = C448_SCALAR_BITS / (table_bits + 1) + 3; int position = table_size - 1; /* at the end */ uint64_t current = scalar->limb[0] & 0xFFFF; uint32_t mask = (1 << (table_bits + 1)) - 1; unsigned int w; const unsigned int B_OVER_16 = sizeof(scalar->limb[0]) / 2; unsigned int n, i; /* place the end marker */ control[position].power = -1; control[position].addend = 0; position--; /* * PERF: Could negate scalar if it's large. But then would need more cases * in the actual code that uses it, all for an expected reduction of like * 1/5 op. Probably not worth it. */ for (w = 1; w < (C448_SCALAR_BITS - 1) / 16 + 3; w++) { if (w < (C448_SCALAR_BITS - 1) / 16 + 1) { /* Refill the 16 high bits of current */ current += (uint32_t)((scalar->limb[w / B_OVER_16] >> (16 * (w % B_OVER_16))) << 16); } while (current & 0xFFFF) { uint32_t pos = NUMTRAILINGZEROS((uint32_t)current); uint32_t odd = (uint32_t)current >> pos; int32_t delta = odd & mask; assert(position >= 0); if (odd & (1 << (table_bits + 1))) delta -= (1 << (table_bits + 1)); current -= delta * (1 << pos); control[position].power = pos + 16 * (w - 1); control[position].addend = delta; position--; } current >>= 16; } assert(current == 0); position++; n = table_size - position; for (i = 0; i < n; i++) control[i] = control[i + position]; return n - 1; } static void prepare_wnaf_table(pniels_t * output, const curve448_point_t working, unsigned int tbits) { curve448_point_t tmp; int i; pniels_t twop; pt_to_pniels(output[0], working); if (tbits == 0) return; curve448_point_double(tmp, working); pt_to_pniels(twop, tmp); add_pniels_to_pt(tmp, output[0], 0); pt_to_pniels(output[1], tmp); for (i = 2; i < 1 << tbits; i++) { add_pniels_to_pt(tmp, twop, 0); pt_to_pniels(output[i], tmp); } curve448_point_destroy(tmp); OPENSSL_cleanse(twop, sizeof(twop)); } void curve448_base_double_scalarmul_non_secret(curve448_point_t combo, const curve448_scalar_t scalar1, const curve448_point_t base2, const curve448_scalar_t scalar2) { const int table_bits_var = C448_WNAF_VAR_TABLE_BITS; const int table_bits_pre = C448_WNAF_FIXED_TABLE_BITS; struct smvt_control control_var[C448_SCALAR_BITS / (C448_WNAF_VAR_TABLE_BITS + 1) + 3]; struct smvt_control control_pre[C448_SCALAR_BITS / (C448_WNAF_FIXED_TABLE_BITS + 1) + 3]; int ncb_pre = recode_wnaf(control_pre, scalar1, table_bits_pre); int ncb_var = recode_wnaf(control_var, scalar2, table_bits_var); pniels_t precmp_var[1 << C448_WNAF_VAR_TABLE_BITS]; int contp = 0, contv = 0, i; prepare_wnaf_table(precmp_var, base2, table_bits_var); i = control_var[0].power; if (i < 0) { curve448_point_copy(combo, curve448_point_identity); return; } if (i > control_pre[0].power) { pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]); contv++; } else if (i == control_pre[0].power && i >= 0) { pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]); add_niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1], i); contv++; contp++; } else { i = control_pre[0].power; niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1]); contp++; } for (i--; i >= 0; i--) { int cv = (i == control_var[contv].power); int cp = (i == control_pre[contp].power); point_double_internal(combo, combo, i && !(cv || cp)); if (cv) { assert(control_var[contv].addend); if (control_var[contv].addend > 0) add_pniels_to_pt(combo, precmp_var[control_var[contv].addend >> 1], i && !cp); else sub_pniels_from_pt(combo, precmp_var[(-control_var[contv].addend) >> 1], i && !cp); contv++; } if (cp) { assert(control_pre[contp].addend); if (control_pre[contp].addend > 0) add_niels_to_pt(combo, curve448_wnaf_base[control_pre[contp].addend >> 1], i); else sub_niels_from_pt(combo, curve448_wnaf_base[(-control_pre [contp].addend) >> 1], i); contp++; } } /* This function is non-secret, but whatever this is cheap. */ OPENSSL_cleanse(control_var, sizeof(control_var)); OPENSSL_cleanse(control_pre, sizeof(control_pre)); OPENSSL_cleanse(precmp_var, sizeof(precmp_var)); assert(contv == ncb_var); (void)ncb_var; assert(contp == ncb_pre); (void)ncb_pre; } void curve448_point_destroy(curve448_point_t point) { OPENSSL_cleanse(point, sizeof(curve448_point_t)); } int X448(uint8_t out_shared_key[56], const uint8_t private_key[56], const uint8_t peer_public_value[56]) { return x448_int(out_shared_key, peer_public_value, private_key) == C448_SUCCESS; } void X448_public_from_private(uint8_t out_public_value[56], const uint8_t private_key[56]) { x448_derive_public_key(out_public_value, private_key); }