2 * Copyright 2017-2018 The OpenSSL Project Authors. All Rights Reserved.
3 * Copyright 2015-2016 Cryptography Research, Inc.
5 * Licensed under the OpenSSL license (the "License"). You may not use
6 * this file except in compliance with the License. You can obtain a copy
7 * in the file LICENSE in the source distribution or at
8 * https://www.openssl.org/source/license.html
10 * Originally written by Mike Hamburg
12 #include <openssl/crypto.h>
16 #include "point_448.h"
18 #include "curve448_lcl.h"
22 /* Comb config: number of combs, n, t, s. */
26 #define C448_WNAF_FIXED_TABLE_BITS 5
27 #define C448_WNAF_VAR_TABLE_BITS 3
29 static const int EDWARDS_D = -39081;
30 static const curve448_scalar_t precomputed_scalarmul_adjustment = {
33 SC_LIMB(0xc873d6d54a7bb0cf), SC_LIMB(0xe933d8d723a70aad),
34 SC_LIMB(0xbb124b65129c96fd), SC_LIMB(0x00000008335dc163)
39 #define TWISTED_D ((EDWARDS_D)-1)
41 #define WBITS C448_WORD_BITS /* NB this may be different from ARCH_WORD_BITS */
43 /* Projective Niels coordinates */
46 } niels_s, niels_t[1];
50 } VECTOR_ALIGNED pniels_t[1];
52 /* Precomputed base */
53 struct curve448_precomputed_s {
54 niels_t table[COMBS_N << (COMBS_T - 1)];
57 extern const gf curve448_precomputed_base_as_fe[];
58 const curve448_precomputed_s *curve448_precomputed_base =
59 (const curve448_precomputed_s *)&curve448_precomputed_base_as_fe;
62 static void gf_invert(gf y, const gf x, int assert_nonzero)
67 gf_sqr(t1, x); /* o^2 */
68 ret = gf_isr(t2, t1); /* +-1/sqrt(o^2) = +-1/o */
73 gf_mul(t2, t1, x); /* not direct to y in case of alias. */
77 /** identity = (0,1) */
78 const curve448_point_t curve448_point_identity =
79 { {{{{0}}}, {{{1}}}, {{{1}}}, {{{0}}}} };
81 static void point_double_internal(curve448_point_t p, const curve448_point_t q,
88 gf_add_nr(d, c, a); /* 2+e */
89 gf_add_nr(p->t, q->y, q->x); /* 2+e */
91 gf_subx_nr(b, b, d, 3); /* 4+e */
92 gf_sub_nr(p->t, a, c); /* 3+e */
94 gf_add_nr(p->z, p->x, p->x); /* 2+e */
95 gf_subx_nr(a, p->z, p->t, 4); /* 6+e */
97 gf_weak_reduce(a); /* or 1+e */
99 gf_mul(p->z, p->t, a);
100 gf_mul(p->y, p->t, d);
105 void curve448_point_double(curve448_point_t p, const curve448_point_t q)
107 point_double_internal(p, q, 0);
110 /* Operations on [p]niels */
111 static ossl_inline void cond_neg_niels(niels_t n, mask_t neg)
113 gf_cond_swap(n->a, n->b, neg);
114 gf_cond_neg(n->c, neg);
117 static void pt_to_pniels(pniels_t b, const curve448_point_t a)
119 gf_sub(b->n->a, a->y, a->x);
120 gf_add(b->n->b, a->x, a->y);
121 gf_mulw(b->n->c, a->t, 2 * TWISTED_D);
122 gf_add(b->z, a->z, a->z);
125 static void pniels_to_pt(curve448_point_t e, const pniels_t d)
129 gf_add(eu, d->n->b, d->n->a);
130 gf_sub(e->y, d->n->b, d->n->a);
131 gf_mul(e->t, e->y, eu);
132 gf_mul(e->x, d->z, e->y);
133 gf_mul(e->y, d->z, eu);
137 static void niels_to_pt(curve448_point_t e, const niels_t n)
139 gf_add(e->y, n->b, n->a);
140 gf_sub(e->x, n->b, n->a);
141 gf_mul(e->t, e->y, e->x);
145 static void add_niels_to_pt(curve448_point_t d, const niels_t e,
150 gf_sub_nr(b, d->y, d->x); /* 3+e */
152 gf_add_nr(b, d->x, d->y); /* 2+e */
153 gf_mul(d->y, e->b, b);
154 gf_mul(d->x, e->c, d->t);
155 gf_add_nr(c, a, d->y); /* 2+e */
156 gf_sub_nr(b, d->y, a); /* 3+e */
157 gf_sub_nr(d->y, d->z, d->x); /* 3+e */
158 gf_add_nr(a, d->x, d->z); /* 2+e */
159 gf_mul(d->z, a, d->y);
160 gf_mul(d->x, d->y, b);
166 static void sub_niels_from_pt(curve448_point_t d, const niels_t e,
171 gf_sub_nr(b, d->y, d->x); /* 3+e */
173 gf_add_nr(b, d->x, d->y); /* 2+e */
174 gf_mul(d->y, e->a, b);
175 gf_mul(d->x, e->c, d->t);
176 gf_add_nr(c, a, d->y); /* 2+e */
177 gf_sub_nr(b, d->y, a); /* 3+e */
178 gf_add_nr(d->y, d->z, d->x); /* 2+e */
179 gf_sub_nr(a, d->z, d->x); /* 3+e */
180 gf_mul(d->z, a, d->y);
181 gf_mul(d->x, d->y, b);
187 static void add_pniels_to_pt(curve448_point_t p, const pniels_t pn,
192 gf_mul(L0, p->z, pn->z);
194 add_niels_to_pt(p, pn->n, before_double);
197 static void sub_pniels_from_pt(curve448_point_t p, const pniels_t pn,
202 gf_mul(L0, p->z, pn->z);
204 sub_niels_from_pt(p, pn->n, before_double);
207 c448_bool_t curve448_point_eq(const curve448_point_t p,
208 const curve448_point_t q)
213 /* equality mod 2-torsion compares x/y */
214 gf_mul(a, p->y, q->x);
215 gf_mul(b, q->y, p->x);
218 return mask_to_bool(succ);
221 c448_bool_t curve448_point_valid(const curve448_point_t p)
226 gf_mul(a, p->x, p->y);
227 gf_mul(b, p->z, p->t);
233 gf_mulw(c, b, TWISTED_D);
237 out &= ~gf_eq(p->z, ZERO);
238 return mask_to_bool(out);
241 static ossl_inline void constant_time_lookup_niels(niels_s * RESTRICT ni,
242 const niels_t * table,
245 constant_time_lookup(ni, table, sizeof(niels_s), nelts, idx);
248 void curve448_precomputed_scalarmul(curve448_point_t out,
249 const curve448_precomputed_s * table,
250 const curve448_scalar_t scalar)
252 unsigned int i, j, k;
253 const unsigned int n = COMBS_N, t = COMBS_T, s = COMBS_S;
255 curve448_scalar_t scalar1x;
257 curve448_scalar_add(scalar1x, scalar, precomputed_scalarmul_adjustment);
258 curve448_scalar_halve(scalar1x, scalar1x);
260 for (i = s; i > 0; i--) {
262 point_double_internal(out, out, 0);
264 for (j = 0; j < n; j++) {
268 for (k = 0; k < t; k++) {
269 unsigned int bit = (i - 1) + s * (k + j * t);
271 if (bit < C448_SCALAR_BITS) {
273 (scalar1x->limb[bit / WBITS] >> (bit % WBITS) & 1) << k;
277 invert = (tab >> (t - 1)) - 1;
279 tab &= (1 << (t - 1)) - 1;
281 constant_time_lookup_niels(ni, &table->table[j << (t - 1)],
284 cond_neg_niels(ni, invert);
285 if ((i != s) || j != 0) {
286 add_niels_to_pt(out, ni, j == n - 1 && i != 1);
288 niels_to_pt(out, ni);
293 OPENSSL_cleanse(ni, sizeof(ni));
294 OPENSSL_cleanse(scalar1x, sizeof(scalar1x));
297 void curve448_point_mul_by_ratio_and_encode_like_eddsa(
298 uint8_t enc[EDDSA_448_PUBLIC_BYTES],
299 const curve448_point_t p)
304 /* The point is now on the twisted curve. Move it to untwisted. */
305 curve448_point_copy(q, p);
308 /* 4-isogeny: 2xy/(y^+x^2), (y^2-x^2)/(2z^2-y^2+x^2) */
314 gf_add(z, q->y, q->x);
324 OPENSSL_cleanse(u, sizeof(u));
333 enc[EDDSA_448_PRIVATE_BYTES - 1] = 0;
334 gf_serialize(enc, x, 1);
335 enc[EDDSA_448_PRIVATE_BYTES - 1] |= 0x80 & gf_lobit(t);
337 OPENSSL_cleanse(x, sizeof(x));
338 OPENSSL_cleanse(y, sizeof(y));
339 OPENSSL_cleanse(z, sizeof(z));
340 OPENSSL_cleanse(t, sizeof(t));
341 curve448_point_destroy(q);
344 c448_error_t curve448_point_decode_like_eddsa_and_mul_by_ratio(
346 const uint8_t enc[EDDSA_448_PUBLIC_BYTES])
348 uint8_t enc2[EDDSA_448_PUBLIC_BYTES];
352 memcpy(enc2, enc, sizeof(enc2));
354 low = ~word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1] & 0x80);
355 enc2[EDDSA_448_PRIVATE_BYTES - 1] &= ~0x80;
357 succ = gf_deserialize(p->y, enc2, 1, 0);
358 succ &= word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1]);
361 gf_sub(p->z, ONE, p->x); /* num = 1-y^2 */
362 gf_mulw(p->t, p->x, EDWARDS_D); /* dy^2 */
363 gf_sub(p->t, ONE, p->t); /* denom = 1-dy^2 or 1-d + dy^2 */
365 gf_mul(p->x, p->z, p->t);
366 succ &= gf_isr(p->t, p->x); /* 1/sqrt(num * denom) */
368 gf_mul(p->x, p->t, p->z); /* sqrt(num / denom) */
369 gf_cond_neg(p->x, gf_lobit(p->x) ^ low);
375 /* 4-isogeny 2xy/(y^2-ax^2), (y^2+ax^2)/(2-y^2-ax^2) */
379 gf_add(p->t, p->y, p->x);
384 gf_add(p->z, p->x, p->x);
387 gf_mul(p->z, p->t, a);
388 gf_mul(p->y, p->t, d);
390 OPENSSL_cleanse(a, sizeof(a));
391 OPENSSL_cleanse(b, sizeof(b));
392 OPENSSL_cleanse(c, sizeof(c));
393 OPENSSL_cleanse(d, sizeof(d));
396 OPENSSL_cleanse(enc2, sizeof(enc2));
397 assert(curve448_point_valid(p) || ~succ);
399 return c448_succeed_if(mask_to_bool(succ));
402 c448_error_t x448_int(uint8_t out[X_PUBLIC_BYTES],
403 const uint8_t base[X_PUBLIC_BYTES],
404 const uint8_t scalar[X_PRIVATE_BYTES])
406 gf x1, x2, z2, x3, z3, t1, t2;
411 ignore_result(gf_deserialize(x1, base, 1, 0));
417 for (t = X_PRIVATE_BITS - 1; t >= 0; t--) {
418 uint8_t sb = scalar[t / 8];
421 /* Scalar conditioning */
423 sb &= -(uint8_t)COFACTOR;
424 else if (t == X_PRIVATE_BITS - 1)
427 k_t = (sb >> (t % 8)) & 1;
428 k_t = 0 - k_t; /* set to all 0s or all 1s */
431 gf_cond_swap(x2, x3, swap);
432 gf_cond_swap(z2, z3, swap);
435 gf_add_nr(t1, x2, z2); /* A = x2 + z2 *//* 2+e */
436 gf_sub_nr(t2, x2, z2); /* B = x2 - z2 *//* 3+e */
437 gf_sub_nr(z2, x3, z3); /* D = x3 - z3 *//* 3+e */
438 gf_mul(x2, t1, z2); /* DA */
439 gf_add_nr(z2, z3, x3); /* C = x3 + z3 *//* 2+e */
440 gf_mul(x3, t2, z2); /* CB */
441 gf_sub_nr(z3, x2, x3); /* DA-CB *//* 3+e */
442 gf_sqr(z2, z3); /* (DA-CB)^2 */
443 gf_mul(z3, x1, z2); /* z3 = x1(DA-CB)^2 */
444 gf_add_nr(z2, x2, x3); /* (DA+CB) *//* 2+e */
445 gf_sqr(x3, z2); /* x3 = (DA+CB)^2 */
447 gf_sqr(z2, t1); /* AA = A^2 */
448 gf_sqr(t1, t2); /* BB = B^2 */
449 gf_mul(x2, z2, t1); /* x2 = AA*BB */
450 gf_sub_nr(t2, z2, t1); /* E = AA-BB *//* 3+e */
452 gf_mulw(t1, t2, -EDWARDS_D); /* E*-d = a24*E */
453 gf_add_nr(t1, t1, z2); /* AA + a24*E *//* 2+e */
454 gf_mul(z2, t2, t1); /* z2 = E(AA+a24*E) */
458 gf_cond_swap(x2, x3, swap);
459 gf_cond_swap(z2, z3, swap);
460 gf_invert(z2, z2, 0);
462 gf_serialize(out, x1, 1);
463 nz = ~gf_eq(x1, ZERO);
465 OPENSSL_cleanse(x1, sizeof(x1));
466 OPENSSL_cleanse(x2, sizeof(x2));
467 OPENSSL_cleanse(z2, sizeof(z2));
468 OPENSSL_cleanse(x3, sizeof(x3));
469 OPENSSL_cleanse(z3, sizeof(z3));
470 OPENSSL_cleanse(t1, sizeof(t1));
471 OPENSSL_cleanse(t2, sizeof(t2));
473 return c448_succeed_if(mask_to_bool(nz));
476 void curve448_point_mul_by_ratio_and_encode_like_x448(uint8_t
478 const curve448_point_t p)
482 curve448_point_copy(q, p);
483 gf_invert(q->t, q->x, 0); /* 1/x */
484 gf_mul(q->z, q->t, q->y); /* y/x */
485 gf_sqr(q->y, q->z); /* (y/x)^2 */
486 gf_serialize(out, q->y, 1);
487 curve448_point_destroy(q);
490 void x448_derive_public_key(uint8_t out[X_PUBLIC_BYTES],
491 const uint8_t scalar[X_PRIVATE_BYTES])
493 /* Scalar conditioning */
494 uint8_t scalar2[X_PRIVATE_BYTES];
495 curve448_scalar_t the_scalar;
499 memcpy(scalar2, scalar, sizeof(scalar2));
500 scalar2[0] &= -(uint8_t)COFACTOR;
502 scalar2[X_PRIVATE_BYTES - 1] &= ~((0u - 1u) << ((X_PRIVATE_BITS + 7) % 8));
503 scalar2[X_PRIVATE_BYTES - 1] |= 1 << ((X_PRIVATE_BITS + 7) % 8);
505 curve448_scalar_decode_long(the_scalar, scalar2, sizeof(scalar2));
507 /* Compensate for the encoding ratio */
508 for (i = 1; i < X448_ENCODE_RATIO; i <<= 1) {
509 curve448_scalar_halve(the_scalar, the_scalar);
511 curve448_precomputed_scalarmul(p, curve448_precomputed_base, the_scalar);
512 curve448_point_mul_by_ratio_and_encode_like_x448(out, p);
513 curve448_point_destroy(p);
516 /* Control for variable-time scalar multiply algorithms. */
517 struct smvt_control {
521 #if defined(__GNUC__) || defined(__clang__)
522 # define NUMTRAILINGZEROS __builtin_ctz
524 # define NUMTRAILINGZEROS numtrailingzeros
525 static uint32_t numtrailingzeros(uint32_t i)
561 static int recode_wnaf(struct smvt_control *control,
562 /* [nbits/(table_bits + 1) + 3] */
563 const curve448_scalar_t scalar,
564 unsigned int table_bits)
566 unsigned int table_size = C448_SCALAR_BITS / (table_bits + 1) + 3;
567 int position = table_size - 1; /* at the end */
568 uint64_t current = scalar->limb[0] & 0xFFFF;
569 uint32_t mask = (1 << (table_bits + 1)) - 1;
571 const unsigned int B_OVER_16 = sizeof(scalar->limb[0]) / 2;
574 /* place the end marker */
575 control[position].power = -1;
576 control[position].addend = 0;
580 * PERF: Could negate scalar if it's large. But then would need more cases
581 * in the actual code that uses it, all for an expected reduction of like
582 * 1/5 op. Probably not worth it.
585 for (w = 1; w < (C448_SCALAR_BITS - 1) / 16 + 3; w++) {
586 if (w < (C448_SCALAR_BITS - 1) / 16 + 1) {
587 /* Refill the 16 high bits of current */
588 current += (uint32_t)((scalar->limb[w / B_OVER_16]
589 >> (16 * (w % B_OVER_16))) << 16);
592 while (current & 0xFFFF) {
593 uint32_t pos = NUMTRAILINGZEROS((uint32_t)current);
594 uint32_t odd = (uint32_t)current >> pos;
595 int32_t delta = odd & mask;
597 assert(position >= 0);
598 if (odd & (1 << (table_bits + 1)))
599 delta -= (1 << (table_bits + 1));
600 current -= delta << pos;
601 control[position].power = pos + 16 * (w - 1);
602 control[position].addend = delta;
607 assert(current == 0);
610 n = table_size - position;
611 for (i = 0; i < n; i++)
612 control[i] = control[i + position];
617 static void prepare_wnaf_table(pniels_t * output,
618 const curve448_point_t working,
621 curve448_point_t tmp;
625 pt_to_pniels(output[0], working);
630 curve448_point_double(tmp, working);
631 pt_to_pniels(twop, tmp);
633 add_pniels_to_pt(tmp, output[0], 0);
634 pt_to_pniels(output[1], tmp);
636 for (i = 2; i < 1 << tbits; i++) {
637 add_pniels_to_pt(tmp, twop, 0);
638 pt_to_pniels(output[i], tmp);
641 curve448_point_destroy(tmp);
642 OPENSSL_cleanse(twop, sizeof(twop));
645 extern const gf curve448_precomputed_wnaf_as_fe[];
646 static const niels_t *curve448_wnaf_base =
647 (const niels_t *)curve448_precomputed_wnaf_as_fe;
649 void curve448_base_double_scalarmul_non_secret(curve448_point_t combo,
650 const curve448_scalar_t scalar1,
651 const curve448_point_t base2,
652 const curve448_scalar_t scalar2)
654 const int table_bits_var = C448_WNAF_VAR_TABLE_BITS;
655 const int table_bits_pre = C448_WNAF_FIXED_TABLE_BITS;
656 struct smvt_control control_var[C448_SCALAR_BITS /
657 (C448_WNAF_VAR_TABLE_BITS + 1) + 3];
658 struct smvt_control control_pre[C448_SCALAR_BITS /
659 (C448_WNAF_FIXED_TABLE_BITS + 1) + 3];
660 int ncb_pre = recode_wnaf(control_pre, scalar1, table_bits_pre);
661 int ncb_var = recode_wnaf(control_var, scalar2, table_bits_var);
662 pniels_t precmp_var[1 << C448_WNAF_VAR_TABLE_BITS];
663 int contp = 0, contv = 0, i;
665 prepare_wnaf_table(precmp_var, base2, table_bits_var);
666 i = control_var[0].power;
669 curve448_point_copy(combo, curve448_point_identity);
671 } else if (i > control_pre[0].power) {
672 pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
674 } else if (i == control_pre[0].power && i >= 0) {
675 pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
676 add_niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1],
681 i = control_pre[0].power;
682 niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1]);
686 for (i--; i >= 0; i--) {
687 int cv = (i == control_var[contv].power);
688 int cp = (i == control_pre[contp].power);
690 point_double_internal(combo, combo, i && !(cv || cp));
693 assert(control_var[contv].addend);
695 if (control_var[contv].addend > 0)
696 add_pniels_to_pt(combo,
697 precmp_var[control_var[contv].addend >> 1],
700 sub_pniels_from_pt(combo,
701 precmp_var[(-control_var[contv].addend)
707 assert(control_pre[contp].addend);
709 if (control_pre[contp].addend > 0)
710 add_niels_to_pt(combo,
711 curve448_wnaf_base[control_pre[contp].addend
714 sub_niels_from_pt(combo,
715 curve448_wnaf_base[(-control_pre
716 [contp].addend) >> 1], i);
721 /* This function is non-secret, but whatever this is cheap. */
722 OPENSSL_cleanse(control_var, sizeof(control_var));
723 OPENSSL_cleanse(control_pre, sizeof(control_pre));
724 OPENSSL_cleanse(precmp_var, sizeof(precmp_var));
726 assert(contv == ncb_var);
728 assert(contp == ncb_pre);
732 void curve448_point_destroy(curve448_point_t point)
734 OPENSSL_cleanse(point, sizeof(curve448_point_t));
737 int X448(uint8_t out_shared_key[56], const uint8_t private_key[56],
738 const uint8_t peer_public_value[56])
740 return x448_int(out_shared_key, peer_public_value, private_key)
744 void X448_public_from_private(uint8_t out_public_value[56],
745 const uint8_t private_key[56])
747 x448_derive_public_key(out_public_value, private_key);