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 #define C448_WNAF_FIXED_TABLE_BITS 5
23 #define C448_WNAF_VAR_TABLE_BITS 3
25 #define EDWARDS_D (-39081)
27 static const curve448_scalar_t precomputed_scalarmul_adjustment = {
30 SC_LIMB(0xc873d6d54a7bb0cf), SC_LIMB(0xe933d8d723a70aad),
31 SC_LIMB(0xbb124b65129c96fd), SC_LIMB(0x00000008335dc163)
36 #define TWISTED_D (EDWARDS_D - 1)
38 #define WBITS C448_WORD_BITS /* NB this may be different from ARCH_WORD_BITS */
41 static void gf_invert(gf y, const gf x, int assert_nonzero)
46 gf_sqr(t1, x); /* o^2 */
47 ret = gf_isr(t2, t1); /* +-1/sqrt(o^2) = +-1/o */
52 gf_mul(t2, t1, x); /* not direct to y in case of alias. */
56 /** identity = (0,1) */
57 const curve448_point_t curve448_point_identity =
58 { {{{{0}}}, {{{1}}}, {{{1}}}, {{{0}}}} };
60 static void point_double_internal(curve448_point_t p, const curve448_point_t q,
67 gf_add_nr(d, c, a); /* 2+e */
68 gf_add_nr(p->t, q->y, q->x); /* 2+e */
70 gf_subx_nr(b, b, d, 3); /* 4+e */
71 gf_sub_nr(p->t, a, c); /* 3+e */
73 gf_add_nr(p->z, p->x, p->x); /* 2+e */
74 gf_subx_nr(a, p->z, p->t, 4); /* 6+e */
76 gf_weak_reduce(a); /* or 1+e */
78 gf_mul(p->z, p->t, a);
79 gf_mul(p->y, p->t, d);
84 void curve448_point_double(curve448_point_t p, const curve448_point_t q)
86 point_double_internal(p, q, 0);
89 /* Operations on [p]niels */
90 static ossl_inline void cond_neg_niels(niels_t n, mask_t neg)
92 gf_cond_swap(n->a, n->b, neg);
93 gf_cond_neg(n->c, neg);
96 static void pt_to_pniels(pniels_t b, const curve448_point_t a)
98 gf_sub(b->n->a, a->y, a->x);
99 gf_add(b->n->b, a->x, a->y);
100 gf_mulw(b->n->c, a->t, 2 * TWISTED_D);
101 gf_add(b->z, a->z, a->z);
104 static void pniels_to_pt(curve448_point_t e, const pniels_t d)
108 gf_add(eu, d->n->b, d->n->a);
109 gf_sub(e->y, d->n->b, d->n->a);
110 gf_mul(e->t, e->y, eu);
111 gf_mul(e->x, d->z, e->y);
112 gf_mul(e->y, d->z, eu);
116 static void niels_to_pt(curve448_point_t e, const niels_t n)
118 gf_add(e->y, n->b, n->a);
119 gf_sub(e->x, n->b, n->a);
120 gf_mul(e->t, e->y, e->x);
124 static void add_niels_to_pt(curve448_point_t d, const niels_t e,
129 gf_sub_nr(b, d->y, d->x); /* 3+e */
131 gf_add_nr(b, d->x, d->y); /* 2+e */
132 gf_mul(d->y, e->b, b);
133 gf_mul(d->x, e->c, d->t);
134 gf_add_nr(c, a, d->y); /* 2+e */
135 gf_sub_nr(b, d->y, a); /* 3+e */
136 gf_sub_nr(d->y, d->z, d->x); /* 3+e */
137 gf_add_nr(a, d->x, d->z); /* 2+e */
138 gf_mul(d->z, a, d->y);
139 gf_mul(d->x, d->y, b);
145 static void sub_niels_from_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->a, 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_add_nr(d->y, d->z, d->x); /* 2+e */
158 gf_sub_nr(a, d->z, d->x); /* 3+e */
159 gf_mul(d->z, a, d->y);
160 gf_mul(d->x, d->y, b);
166 static void add_pniels_to_pt(curve448_point_t p, const pniels_t pn,
171 gf_mul(L0, p->z, pn->z);
173 add_niels_to_pt(p, pn->n, before_double);
176 static void sub_pniels_from_pt(curve448_point_t p, const pniels_t pn,
181 gf_mul(L0, p->z, pn->z);
183 sub_niels_from_pt(p, pn->n, before_double);
186 c448_bool_t curve448_point_eq(const curve448_point_t p,
187 const curve448_point_t q)
192 /* equality mod 2-torsion compares x/y */
193 gf_mul(a, p->y, q->x);
194 gf_mul(b, q->y, p->x);
197 return mask_to_bool(succ);
200 c448_bool_t curve448_point_valid(const curve448_point_t p)
205 gf_mul(a, p->x, p->y);
206 gf_mul(b, p->z, p->t);
212 gf_mulw(c, b, TWISTED_D);
216 out &= ~gf_eq(p->z, ZERO);
217 return mask_to_bool(out);
220 static ossl_inline void constant_time_lookup_niels(niels_s * RESTRICT ni,
221 const niels_t * table,
224 constant_time_lookup(ni, table, sizeof(niels_s), nelts, idx);
227 void curve448_precomputed_scalarmul(curve448_point_t out,
228 const curve448_precomputed_s * table,
229 const curve448_scalar_t scalar)
231 unsigned int i, j, k;
232 const unsigned int n = COMBS_N, t = COMBS_T, s = COMBS_S;
234 curve448_scalar_t scalar1x;
236 curve448_scalar_add(scalar1x, scalar, precomputed_scalarmul_adjustment);
237 curve448_scalar_halve(scalar1x, scalar1x);
239 for (i = s; i > 0; i--) {
241 point_double_internal(out, out, 0);
243 for (j = 0; j < n; j++) {
247 for (k = 0; k < t; k++) {
248 unsigned int bit = (i - 1) + s * (k + j * t);
250 if (bit < C448_SCALAR_BITS)
252 (scalar1x->limb[bit / WBITS] >> (bit % WBITS) & 1) << k;
255 invert = (tab >> (t - 1)) - 1;
257 tab &= (1 << (t - 1)) - 1;
259 constant_time_lookup_niels(ni, &table->table[j << (t - 1)],
262 cond_neg_niels(ni, invert);
263 if ((i != s) || j != 0)
264 add_niels_to_pt(out, ni, j == n - 1 && i != 1);
266 niels_to_pt(out, ni);
270 OPENSSL_cleanse(ni, sizeof(ni));
271 OPENSSL_cleanse(scalar1x, sizeof(scalar1x));
274 void curve448_point_mul_by_ratio_and_encode_like_eddsa(
275 uint8_t enc[EDDSA_448_PUBLIC_BYTES],
276 const curve448_point_t p)
281 /* The point is now on the twisted curve. Move it to untwisted. */
282 curve448_point_copy(q, p);
285 /* 4-isogeny: 2xy/(y^+x^2), (y^2-x^2)/(2z^2-y^2+x^2) */
291 gf_add(z, q->y, q->x);
301 OPENSSL_cleanse(u, sizeof(u));
310 enc[EDDSA_448_PRIVATE_BYTES - 1] = 0;
311 gf_serialize(enc, x, 1);
312 enc[EDDSA_448_PRIVATE_BYTES - 1] |= 0x80 & gf_lobit(t);
314 OPENSSL_cleanse(x, sizeof(x));
315 OPENSSL_cleanse(y, sizeof(y));
316 OPENSSL_cleanse(z, sizeof(z));
317 OPENSSL_cleanse(t, sizeof(t));
318 curve448_point_destroy(q);
321 c448_error_t curve448_point_decode_like_eddsa_and_mul_by_ratio(
323 const uint8_t enc[EDDSA_448_PUBLIC_BYTES])
325 uint8_t enc2[EDDSA_448_PUBLIC_BYTES];
329 memcpy(enc2, enc, sizeof(enc2));
331 low = ~word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1] & 0x80);
332 enc2[EDDSA_448_PRIVATE_BYTES - 1] &= ~0x80;
334 succ = gf_deserialize(p->y, enc2, 1, 0);
335 succ &= word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1]);
338 gf_sub(p->z, ONE, p->x); /* num = 1-y^2 */
339 gf_mulw(p->t, p->x, EDWARDS_D); /* dy^2 */
340 gf_sub(p->t, ONE, p->t); /* denom = 1-dy^2 or 1-d + dy^2 */
342 gf_mul(p->x, p->z, p->t);
343 succ &= gf_isr(p->t, p->x); /* 1/sqrt(num * denom) */
345 gf_mul(p->x, p->t, p->z); /* sqrt(num / denom) */
346 gf_cond_neg(p->x, gf_lobit(p->x) ^ low);
352 /* 4-isogeny 2xy/(y^2-ax^2), (y^2+ax^2)/(2-y^2-ax^2) */
356 gf_add(p->t, p->y, p->x);
361 gf_add(p->z, p->x, p->x);
364 gf_mul(p->z, p->t, a);
365 gf_mul(p->y, p->t, d);
367 OPENSSL_cleanse(a, sizeof(a));
368 OPENSSL_cleanse(b, sizeof(b));
369 OPENSSL_cleanse(c, sizeof(c));
370 OPENSSL_cleanse(d, sizeof(d));
373 OPENSSL_cleanse(enc2, sizeof(enc2));
374 assert(curve448_point_valid(p) || ~succ);
376 return c448_succeed_if(mask_to_bool(succ));
379 c448_error_t x448_int(uint8_t out[X_PUBLIC_BYTES],
380 const uint8_t base[X_PUBLIC_BYTES],
381 const uint8_t scalar[X_PRIVATE_BYTES])
383 gf x1, x2, z2, x3, z3, t1, t2;
388 (void)gf_deserialize(x1, base, 1, 0);
394 for (t = X_PRIVATE_BITS - 1; t >= 0; t--) {
395 uint8_t sb = scalar[t / 8];
398 /* Scalar conditioning */
400 sb &= -(uint8_t)COFACTOR;
401 else if (t == X_PRIVATE_BITS - 1)
404 k_t = (sb >> (t % 8)) & 1;
405 k_t = 0 - k_t; /* set to all 0s or all 1s */
408 gf_cond_swap(x2, x3, swap);
409 gf_cond_swap(z2, z3, swap);
413 * The "_nr" below skips coefficient reduction. In the following
414 * comments, "2+e" is saying that the coefficients are at most 2+epsilon
415 * times the reduction limit.
417 gf_add_nr(t1, x2, z2); /* A = x2 + z2 */ /* 2+e */
418 gf_sub_nr(t2, x2, z2); /* B = x2 - z2 */ /* 3+e */
419 gf_sub_nr(z2, x3, z3); /* D = x3 - z3 */ /* 3+e */
420 gf_mul(x2, t1, z2); /* DA */
421 gf_add_nr(z2, z3, x3); /* C = x3 + z3 */ /* 2+e */
422 gf_mul(x3, t2, z2); /* CB */
423 gf_sub_nr(z3, x2, x3); /* DA-CB */ /* 3+e */
424 gf_sqr(z2, z3); /* (DA-CB)^2 */
425 gf_mul(z3, x1, z2); /* z3 = x1(DA-CB)^2 */
426 gf_add_nr(z2, x2, x3); /* (DA+CB) */ /* 2+e */
427 gf_sqr(x3, z2); /* x3 = (DA+CB)^2 */
429 gf_sqr(z2, t1); /* AA = A^2 */
430 gf_sqr(t1, t2); /* BB = B^2 */
431 gf_mul(x2, z2, t1); /* x2 = AA*BB */
432 gf_sub_nr(t2, z2, t1); /* E = AA-BB */ /* 3+e */
434 gf_mulw(t1, t2, -EDWARDS_D); /* E*-d = a24*E */
435 gf_add_nr(t1, t1, z2); /* AA + a24*E */ /* 2+e */
436 gf_mul(z2, t2, t1); /* z2 = E(AA+a24*E) */
440 gf_cond_swap(x2, x3, swap);
441 gf_cond_swap(z2, z3, swap);
442 gf_invert(z2, z2, 0);
444 gf_serialize(out, x1, 1);
445 nz = ~gf_eq(x1, ZERO);
447 OPENSSL_cleanse(x1, sizeof(x1));
448 OPENSSL_cleanse(x2, sizeof(x2));
449 OPENSSL_cleanse(z2, sizeof(z2));
450 OPENSSL_cleanse(x3, sizeof(x3));
451 OPENSSL_cleanse(z3, sizeof(z3));
452 OPENSSL_cleanse(t1, sizeof(t1));
453 OPENSSL_cleanse(t2, sizeof(t2));
455 return c448_succeed_if(mask_to_bool(nz));
458 void curve448_point_mul_by_ratio_and_encode_like_x448(uint8_t
460 const curve448_point_t p)
464 curve448_point_copy(q, p);
465 gf_invert(q->t, q->x, 0); /* 1/x */
466 gf_mul(q->z, q->t, q->y); /* y/x */
467 gf_sqr(q->y, q->z); /* (y/x)^2 */
468 gf_serialize(out, q->y, 1);
469 curve448_point_destroy(q);
472 void x448_derive_public_key(uint8_t out[X_PUBLIC_BYTES],
473 const uint8_t scalar[X_PRIVATE_BYTES])
475 /* Scalar conditioning */
476 uint8_t scalar2[X_PRIVATE_BYTES];
477 curve448_scalar_t the_scalar;
481 memcpy(scalar2, scalar, sizeof(scalar2));
482 scalar2[0] &= -(uint8_t)COFACTOR;
484 scalar2[X_PRIVATE_BYTES - 1] &= ~((0u - 1u) << ((X_PRIVATE_BITS + 7) % 8));
485 scalar2[X_PRIVATE_BYTES - 1] |= 1 << ((X_PRIVATE_BITS + 7) % 8);
487 curve448_scalar_decode_long(the_scalar, scalar2, sizeof(scalar2));
489 /* Compensate for the encoding ratio */
490 for (i = 1; i < X448_ENCODE_RATIO; i <<= 1)
491 curve448_scalar_halve(the_scalar, the_scalar);
493 curve448_precomputed_scalarmul(p, curve448_precomputed_base, the_scalar);
494 curve448_point_mul_by_ratio_and_encode_like_x448(out, p);
495 curve448_point_destroy(p);
498 /* Control for variable-time scalar multiply algorithms. */
499 struct smvt_control {
503 #if defined(__GNUC__) || defined(__clang__)
504 # define NUMTRAILINGZEROS __builtin_ctz
506 # define NUMTRAILINGZEROS numtrailingzeros
507 static uint32_t numtrailingzeros(uint32_t i)
543 static int recode_wnaf(struct smvt_control *control,
544 /* [nbits/(table_bits + 1) + 3] */
545 const curve448_scalar_t scalar,
546 unsigned int table_bits)
548 unsigned int table_size = C448_SCALAR_BITS / (table_bits + 1) + 3;
549 int position = table_size - 1; /* at the end */
550 uint64_t current = scalar->limb[0] & 0xFFFF;
551 uint32_t mask = (1 << (table_bits + 1)) - 1;
553 const unsigned int B_OVER_16 = sizeof(scalar->limb[0]) / 2;
556 /* place the end marker */
557 control[position].power = -1;
558 control[position].addend = 0;
562 * PERF: Could negate scalar if it's large. But then would need more cases
563 * in the actual code that uses it, all for an expected reduction of like
564 * 1/5 op. Probably not worth it.
567 for (w = 1; w < (C448_SCALAR_BITS - 1) / 16 + 3; w++) {
568 if (w < (C448_SCALAR_BITS - 1) / 16 + 1) {
569 /* Refill the 16 high bits of current */
570 current += (uint32_t)((scalar->limb[w / B_OVER_16]
571 >> (16 * (w % B_OVER_16))) << 16);
574 while (current & 0xFFFF) {
575 uint32_t pos = NUMTRAILINGZEROS((uint32_t)current);
576 uint32_t odd = (uint32_t)current >> pos;
577 int32_t delta = odd & mask;
579 assert(position >= 0);
580 if (odd & (1 << (table_bits + 1)))
581 delta -= (1 << (table_bits + 1));
582 current -= delta << pos;
583 control[position].power = pos + 16 * (w - 1);
584 control[position].addend = delta;
589 assert(current == 0);
592 n = table_size - position;
593 for (i = 0; i < n; i++)
594 control[i] = control[i + position];
599 static void prepare_wnaf_table(pniels_t * output,
600 const curve448_point_t working,
603 curve448_point_t tmp;
607 pt_to_pniels(output[0], working);
612 curve448_point_double(tmp, working);
613 pt_to_pniels(twop, tmp);
615 add_pniels_to_pt(tmp, output[0], 0);
616 pt_to_pniels(output[1], tmp);
618 for (i = 2; i < 1 << tbits; i++) {
619 add_pniels_to_pt(tmp, twop, 0);
620 pt_to_pniels(output[i], tmp);
623 curve448_point_destroy(tmp);
624 OPENSSL_cleanse(twop, sizeof(twop));
627 void curve448_base_double_scalarmul_non_secret(curve448_point_t combo,
628 const curve448_scalar_t scalar1,
629 const curve448_point_t base2,
630 const curve448_scalar_t scalar2)
632 const int table_bits_var = C448_WNAF_VAR_TABLE_BITS;
633 const int table_bits_pre = C448_WNAF_FIXED_TABLE_BITS;
634 struct smvt_control control_var[C448_SCALAR_BITS /
635 (C448_WNAF_VAR_TABLE_BITS + 1) + 3];
636 struct smvt_control control_pre[C448_SCALAR_BITS /
637 (C448_WNAF_FIXED_TABLE_BITS + 1) + 3];
638 int ncb_pre = recode_wnaf(control_pre, scalar1, table_bits_pre);
639 int ncb_var = recode_wnaf(control_var, scalar2, table_bits_var);
640 pniels_t precmp_var[1 << C448_WNAF_VAR_TABLE_BITS];
641 int contp = 0, contv = 0, i;
643 prepare_wnaf_table(precmp_var, base2, table_bits_var);
644 i = control_var[0].power;
647 curve448_point_copy(combo, curve448_point_identity);
650 if (i > control_pre[0].power) {
651 pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
653 } else if (i == control_pre[0].power && i >= 0) {
654 pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
655 add_niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1],
660 i = control_pre[0].power;
661 niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1]);
665 for (i--; i >= 0; i--) {
666 int cv = (i == control_var[contv].power);
667 int cp = (i == control_pre[contp].power);
669 point_double_internal(combo, combo, i && !(cv || cp));
672 assert(control_var[contv].addend);
674 if (control_var[contv].addend > 0)
675 add_pniels_to_pt(combo,
676 precmp_var[control_var[contv].addend >> 1],
679 sub_pniels_from_pt(combo,
680 precmp_var[(-control_var[contv].addend)
686 assert(control_pre[contp].addend);
688 if (control_pre[contp].addend > 0)
689 add_niels_to_pt(combo,
690 curve448_wnaf_base[control_pre[contp].addend
693 sub_niels_from_pt(combo,
694 curve448_wnaf_base[(-control_pre
695 [contp].addend) >> 1], i);
700 /* This function is non-secret, but whatever this is cheap. */
701 OPENSSL_cleanse(control_var, sizeof(control_var));
702 OPENSSL_cleanse(control_pre, sizeof(control_pre));
703 OPENSSL_cleanse(precmp_var, sizeof(precmp_var));
705 assert(contv == ncb_var);
707 assert(contp == ncb_pre);
711 void curve448_point_destroy(curve448_point_t point)
713 OPENSSL_cleanse(point, sizeof(curve448_point_t));
716 int X448(uint8_t out_shared_key[56], const uint8_t private_key[56],
717 const uint8_t peer_public_value[56])
719 return x448_int(out_shared_key, peer_public_value, private_key)
723 void X448_public_from_private(uint8_t out_public_value[56],
724 const uint8_t private_key[56])
726 x448_derive_public_key(out_public_value, private_key);