2 * Copyright 2017 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 DECAF_WINDOW_BITS 5
27 #define DECAF_WNAF_FIXED_TABLE_BITS 5
28 #define DECAF_WNAF_VAR_TABLE_BITS 3
30 static const int EDWARDS_D = -39081;
31 static const curve448_scalar_t precomputed_scalarmul_adjustment = { {{
43 const uint8_t decaf_x448_base_point[DECAF_X448_PUBLIC_BYTES] = { 0x05 };
45 #define TWISTED_D ((EDWARDS_D)-1)
47 #define EFF_D (-(TWISTED_D))
50 /* End of template stuff */
52 #define WBITS DECAF_WORD_BITS /* NB this may be different from ARCH_WORD_BITS */
54 /* Projective Niels coordinates */
57 } niels_s, niels_t[1];
61 } VECTOR_ALIGNED pniels_s, pniels_t[1];
63 /* Precomputed base */
64 struct curve448_precomputed_s {
65 niels_t table[COMBS_N << (COMBS_T - 1)];
68 extern const gf curve448_precomputed_base_as_fe[];
69 const curve448_precomputed_s *curve448_precomputed_base =
70 (const curve448_precomputed_s *)&curve448_precomputed_base_as_fe;
73 static void gf_invert(gf y, const gf x, int assert_nonzero)
78 gf_sqr(t1, x); /* o^2 */
79 ret = gf_isr(t2, t1); /* +-1/sqrt(o^2) = +-1/o */
84 gf_mul(t2, t1, x); /* not direct to y in case of alias. */
88 /** identity = (0,1) */
89 const curve448_point_t curve448_point_identity =
90 { {{{{0}}}, {{{1}}}, {{{1}}}, {{{0}}}} };
93 point_double_internal(curve448_point_t p,
94 const curve448_point_t q, int before_double)
99 gf_add_nr(d, c, a); /* 2+e */
100 gf_add_nr(p->t, q->y, q->x); /* 2+e */
102 gf_subx_nr(b, b, d, 3); /* 4+e */
103 gf_sub_nr(p->t, a, c); /* 3+e */
105 gf_add_nr(p->z, p->x, p->x); /* 2+e */
106 gf_subx_nr(a, p->z, p->t, 4); /* 6+e */
107 if (GF_HEADROOM == 5)
108 gf_weak_reduce(a); /* or 1+e */
110 gf_mul(p->z, p->t, a);
111 gf_mul(p->y, p->t, d);
116 void curve448_point_double(curve448_point_t p, const curve448_point_t q)
118 point_double_internal(p, q, 0);
121 /* Operations on [p]niels */
122 static ossl_inline void cond_neg_niels(niels_t n, mask_t neg)
124 gf_cond_swap(n->a, n->b, neg);
125 gf_cond_neg(n->c, neg);
128 static void pt_to_pniels(pniels_t b, const curve448_point_t a)
130 gf_sub(b->n->a, a->y, a->x);
131 gf_add(b->n->b, a->x, a->y);
132 gf_mulw(b->n->c, a->t, 2 * TWISTED_D);
133 gf_add(b->z, a->z, a->z);
136 static void pniels_to_pt(curve448_point_t e, const pniels_t d)
139 gf_add(eu, d->n->b, d->n->a);
140 gf_sub(e->y, d->n->b, d->n->a);
141 gf_mul(e->t, e->y, eu);
142 gf_mul(e->x, d->z, e->y);
143 gf_mul(e->y, d->z, eu);
147 static void niels_to_pt(curve448_point_t e, const niels_t n)
149 gf_add(e->y, n->b, n->a);
150 gf_sub(e->x, n->b, n->a);
151 gf_mul(e->t, e->y, e->x);
156 add_niels_to_pt(curve448_point_t d, const niels_t e, int before_double)
159 gf_sub_nr(b, d->y, d->x); /* 3+e */
161 gf_add_nr(b, d->x, d->y); /* 2+e */
162 gf_mul(d->y, e->b, b);
163 gf_mul(d->x, e->c, d->t);
164 gf_add_nr(c, a, d->y); /* 2+e */
165 gf_sub_nr(b, d->y, a); /* 3+e */
166 gf_sub_nr(d->y, d->z, d->x); /* 3+e */
167 gf_add_nr(a, d->x, d->z); /* 2+e */
168 gf_mul(d->z, a, d->y);
169 gf_mul(d->x, d->y, b);
176 sub_niels_from_pt(curve448_point_t d, const niels_t e, int before_double)
179 gf_sub_nr(b, d->y, d->x); /* 3+e */
181 gf_add_nr(b, d->x, d->y); /* 2+e */
182 gf_mul(d->y, e->a, b);
183 gf_mul(d->x, e->c, d->t);
184 gf_add_nr(c, a, d->y); /* 2+e */
185 gf_sub_nr(b, d->y, a); /* 3+e */
186 gf_add_nr(d->y, d->z, d->x); /* 2+e */
187 gf_sub_nr(a, d->z, d->x); /* 3+e */
188 gf_mul(d->z, a, d->y);
189 gf_mul(d->x, d->y, b);
196 add_pniels_to_pt(curve448_point_t p, const pniels_t pn, int before_double)
199 gf_mul(L0, p->z, pn->z);
201 add_niels_to_pt(p, pn->n, before_double);
205 sub_pniels_from_pt(curve448_point_t p, const pniels_t pn, int before_double)
208 gf_mul(L0, p->z, pn->z);
210 sub_niels_from_pt(p, pn->n, before_double);
213 decaf_bool_t curve448_point_eq(const curve448_point_t p,
214 const curve448_point_t q)
218 /* equality mod 2-torsion compares x/y */
220 gf_mul(a, p->y, q->x);
221 gf_mul(b, q->y, p->x);
224 return mask_to_bool(succ);
227 decaf_bool_t curve448_point_valid(const curve448_point_t p)
232 gf_mul(a, p->x, p->y);
233 gf_mul(b, p->z, p->t);
239 gf_mulw(c, b, TWISTED_D);
243 out &= ~gf_eq(p->z, ZERO);
244 return mask_to_bool(out);
247 static ossl_inline void
248 constant_time_lookup_niels(niels_s * __restrict__ ni,
249 const niels_t * table, int nelts, int idx)
251 constant_time_lookup(ni, table, sizeof(niels_s), nelts, idx);
254 void curve448_precomputed_scalarmul(curve448_point_t out,
255 const curve448_precomputed_s * table,
256 const curve448_scalar_t scalar)
260 const unsigned int n = COMBS_N, t = COMBS_T, s = COMBS_S;
263 curve448_scalar_t scalar1x;
264 curve448_scalar_add(scalar1x, scalar, precomputed_scalarmul_adjustment);
265 curve448_scalar_halve(scalar1x, scalar1x);
267 for (i = s - 1; i >= 0; i--) {
269 point_double_internal(out, out, 0);
271 for (j = 0; j < n; j++) {
275 for (k = 0; k < t; k++) {
276 unsigned int bit = i + s * (k + j * t);
277 if (bit < DECAF_448_SCALAR_BITS) {
279 (scalar1x->limb[bit / WBITS] >> (bit % WBITS) & 1) << k;
283 invert = (tab >> (t - 1)) - 1;
285 tab &= (1 << (t - 1)) - 1;
287 constant_time_lookup_niels(ni, &table->table[j << (t - 1)],
290 cond_neg_niels(ni, invert);
291 if ((i != (int)s - 1) || j) {
292 add_niels_to_pt(out, ni, j == n - 1 && i);
294 niels_to_pt(out, ni);
299 OPENSSL_cleanse(ni, sizeof(ni));
300 OPENSSL_cleanse(scalar1x, sizeof(scalar1x));
303 void curve448_point_mul_by_ratio_and_encode_like_eddsa(uint8_t
305 [DECAF_EDDSA_448_PUBLIC_BYTES],
306 const curve448_point_t p)
309 /* The point is now on the twisted curve. Move it to untwisted. */
312 curve448_point_copy(q, p);
315 /* 4-isogeny: 2xy/(y^+x^2), (y^2-x^2)/(2z^2-y^2+x^2) */
320 gf_add(z, q->y, q->x);
330 OPENSSL_cleanse(u, sizeof(u));
339 enc[DECAF_EDDSA_448_PRIVATE_BYTES - 1] = 0;
340 gf_serialize(enc, x, 1);
341 enc[DECAF_EDDSA_448_PRIVATE_BYTES - 1] |= 0x80 & gf_lobit(t);
343 OPENSSL_cleanse(x, sizeof(x));
344 OPENSSL_cleanse(y, sizeof(y));
345 OPENSSL_cleanse(z, sizeof(z));
346 OPENSSL_cleanse(t, sizeof(t));
347 curve448_point_destroy(q);
350 decaf_error_t curve448_point_decode_like_eddsa_and_mul_by_ratio(curve448_point_t
354 [DECAF_EDDSA_448_PUBLIC_BYTES]
357 uint8_t enc2[DECAF_EDDSA_448_PUBLIC_BYTES];
361 memcpy(enc2, enc, sizeof(enc2));
363 low = ~word_is_zero(enc2[DECAF_EDDSA_448_PRIVATE_BYTES - 1] & 0x80);
364 enc2[DECAF_EDDSA_448_PRIVATE_BYTES - 1] &= ~0x80;
366 succ = gf_deserialize(p->y, enc2, 1, 0);
368 succ &= word_is_zero(enc2[DECAF_EDDSA_448_PRIVATE_BYTES - 1]);
372 gf_sub(p->z, ONE, p->x); /* num = 1-y^2 */
373 gf_mulw(p->t, p->x, EDWARDS_D); /* dy^2 */
374 gf_sub(p->t, ONE, p->t); /* denom = 1-dy^2 or 1-d + dy^2 */
376 gf_mul(p->x, p->z, p->t);
377 succ &= gf_isr(p->t, p->x); /* 1/sqrt(num * denom) */
379 gf_mul(p->x, p->t, p->z); /* sqrt(num / denom) */
380 gf_cond_neg(p->x, gf_lobit(p->x) ^ low);
384 /* 4-isogeny 2xy/(y^2-ax^2), (y^2+ax^2)/(2-y^2-ax^2) */
389 gf_add(p->t, p->y, p->x);
394 gf_add(p->z, p->x, p->x);
397 gf_mul(p->z, p->t, a);
398 gf_mul(p->y, p->t, d);
400 OPENSSL_cleanse(a, sizeof(a));
401 OPENSSL_cleanse(b, sizeof(b));
402 OPENSSL_cleanse(c, sizeof(c));
403 OPENSSL_cleanse(d, sizeof(d));
406 OPENSSL_cleanse(enc2, sizeof(enc2));
407 assert(curve448_point_valid(p) || ~succ);
409 return decaf_succeed_if(mask_to_bool(succ));
412 decaf_error_t decaf_x448(uint8_t out[X_PUBLIC_BYTES],
413 const uint8_t base[X_PUBLIC_BYTES],
414 const uint8_t scalar[X_PRIVATE_BYTES]
417 gf x1, x2, z2, x3, z3, t1, t2;
422 ignore_result(gf_deserialize(x1, base, 1, 0));
428 for (t = X_PRIVATE_BITS - 1; t >= 0; t--) {
429 uint8_t sb = scalar[t / 8];
432 /* Scalar conditioning */
434 sb &= -(uint8_t)COFACTOR;
435 else if (t == X_PRIVATE_BITS - 1)
438 k_t = (sb >> (t % 8)) & 1;
439 k_t = -k_t; /* set to all 0s or all 1s */
442 gf_cond_swap(x2, x3, swap);
443 gf_cond_swap(z2, z3, swap);
446 gf_add_nr(t1, x2, z2); /* A = x2 + z2 *//* 2+e */
447 gf_sub_nr(t2, x2, z2); /* B = x2 - z2 *//* 3+e */
448 gf_sub_nr(z2, x3, z3); /* D = x3 - z3 *//* 3+e */
449 gf_mul(x2, t1, z2); /* DA */
450 gf_add_nr(z2, z3, x3); /* C = x3 + z3 *//* 2+e */
451 gf_mul(x3, t2, z2); /* CB */
452 gf_sub_nr(z3, x2, x3); /* DA-CB *//* 3+e */
453 gf_sqr(z2, z3); /* (DA-CB)^2 */
454 gf_mul(z3, x1, z2); /* z3 = x1(DA-CB)^2 */
455 gf_add_nr(z2, x2, x3); /* (DA+CB) *//* 2+e */
456 gf_sqr(x3, z2); /* x3 = (DA+CB)^2 */
458 gf_sqr(z2, t1); /* AA = A^2 */
459 gf_sqr(t1, t2); /* BB = B^2 */
460 gf_mul(x2, z2, t1); /* x2 = AA*BB */
461 gf_sub_nr(t2, z2, t1); /* E = AA-BB *//* 3+e */
463 gf_mulw(t1, t2, -EDWARDS_D); /* E*-d = a24*E */
464 gf_add_nr(t1, t1, z2); /* AA + a24*E *//* 2+e */
465 gf_mul(z2, t2, t1); /* z2 = E(AA+a24*E) */
469 gf_cond_swap(x2, x3, swap);
470 gf_cond_swap(z2, z3, swap);
471 gf_invert(z2, z2, 0);
473 gf_serialize(out, x1, 1);
474 nz = ~gf_eq(x1, ZERO);
476 OPENSSL_cleanse(x1, sizeof(x1));
477 OPENSSL_cleanse(x2, sizeof(x2));
478 OPENSSL_cleanse(z2, sizeof(z2));
479 OPENSSL_cleanse(x3, sizeof(x3));
480 OPENSSL_cleanse(z3, sizeof(z3));
481 OPENSSL_cleanse(t1, sizeof(t1));
482 OPENSSL_cleanse(t2, sizeof(t2));
484 return decaf_succeed_if(mask_to_bool(nz));
487 /* Thanks Johan Pascal */
488 void decaf_ed448_convert_public_key_to_x448(uint8_t x[DECAF_X448_PUBLIC_BYTES],
490 ed[DECAF_EDDSA_448_PUBLIC_BYTES]
494 const uint8_t mask = (uint8_t)(0xFE << (7));
495 ignore_result(gf_deserialize(y, ed, 1, mask));
500 /* u = y^2 * (1-dy^2) / (1-y^2) */
501 gf_sqr(n, y); /* y^2 */
502 gf_sub(d, ONE, n); /* 1-y^2 */
503 gf_invert(d, d, 0); /* 1/(1-y^2) */
504 gf_mul(y, n, d); /* y^2 / (1-y^2) */
505 gf_mulw(d, n, EDWARDS_D); /* dy^2 */
506 gf_sub(d, ONE, d); /* 1-dy^2 */
507 gf_mul(n, y, d); /* y^2 * (1-dy^2) / (1-y^2) */
508 gf_serialize(x, n, 1);
510 OPENSSL_cleanse(y, sizeof(y));
511 OPENSSL_cleanse(n, sizeof(n));
512 OPENSSL_cleanse(d, sizeof(d));
516 void curve448_point_mul_by_ratio_and_encode_like_x448(uint8_t
518 const curve448_point_t p)
521 curve448_point_copy(q, p);
522 gf_invert(q->t, q->x, 0); /* 1/x */
523 gf_mul(q->z, q->t, q->y); /* y/x */
524 gf_sqr(q->y, q->z); /* (y/x)^2 */
525 gf_serialize(out, q->y, 1);
526 curve448_point_destroy(q);
529 void decaf_x448_derive_public_key(uint8_t out[X_PUBLIC_BYTES],
530 const uint8_t scalar[X_PRIVATE_BYTES]
533 /* Scalar conditioning */
534 uint8_t scalar2[X_PRIVATE_BYTES];
535 curve448_scalar_t the_scalar;
539 memcpy(scalar2, scalar, sizeof(scalar2));
540 scalar2[0] &= -(uint8_t)COFACTOR;
542 scalar2[X_PRIVATE_BYTES - 1] &= ~(-1u << ((X_PRIVATE_BITS + 7) % 8));
543 scalar2[X_PRIVATE_BYTES - 1] |= 1 << ((X_PRIVATE_BITS + 7) % 8);
545 curve448_scalar_decode_long(the_scalar, scalar2, sizeof(scalar2));
547 /* Compensate for the encoding ratio */
548 for (i = 1; i < DECAF_X448_ENCODE_RATIO; i <<= 1) {
549 curve448_scalar_halve(the_scalar, the_scalar);
551 curve448_precomputed_scalarmul(p, curve448_precomputed_base, the_scalar);
552 curve448_point_mul_by_ratio_and_encode_like_x448(out, p);
553 curve448_point_destroy(p);
558 * Control for variable-time scalar multiply algorithms.
560 struct smvt_control {
564 static int recode_wnaf(struct smvt_control *control, /* [nbits/(table_bits+1) +
566 const curve448_scalar_t scalar, unsigned int table_bits)
568 unsigned int table_size = DECAF_448_SCALAR_BITS / (table_bits + 1) + 3;
569 int position = table_size - 1; /* at the end */
570 uint64_t current = scalar->limb[0] & 0xFFFF;
571 uint32_t mask = (1 << (table_bits + 1)) - 1;
573 const unsigned int B_OVER_16 = sizeof(scalar->limb[0]) / 2;
576 /* place the end marker */
577 control[position].power = -1;
578 control[position].addend = 0;
582 * PERF: Could negate scalar if it's large. But then would need more cases
583 * in the actual code that uses it, all for an expected reduction of like
584 * 1/5 op. Probably not worth it.
587 for (w = 1; w < (DECAF_448_SCALAR_BITS - 1) / 16 + 3; w++) {
588 if (w < (DECAF_448_SCALAR_BITS - 1) / 16 + 1) {
589 /* Refill the 16 high bits of current */
591 (uint32_t)((scalar->limb[w / B_OVER_16] >> (16 *
597 while (current & 0xFFFF) {
598 uint32_t pos = __builtin_ctz((uint32_t)current), odd =
599 (uint32_t)current >> pos;
600 int32_t delta = odd & mask;
602 assert(position >= 0);
603 if (odd & 1 << (table_bits + 1))
604 delta -= (1 << (table_bits + 1));
605 current -= delta << pos;
606 control[position].power = pos + 16 * (w - 1);
607 control[position].addend = delta;
612 assert(current == 0);
615 n = table_size - position;
616 for (i = 0; i < n; i++) {
617 control[i] = control[i + position];
623 prepare_wnaf_table(pniels_t * output,
624 const curve448_point_t working, unsigned int tbits)
626 curve448_point_t tmp;
630 pt_to_pniels(output[0], working);
635 curve448_point_double(tmp, working);
636 pt_to_pniels(twop, tmp);
638 add_pniels_to_pt(tmp, output[0], 0);
639 pt_to_pniels(output[1], tmp);
641 for (i = 2; i < 1 << tbits; i++) {
642 add_pniels_to_pt(tmp, twop, 0);
643 pt_to_pniels(output[i], tmp);
646 curve448_point_destroy(tmp);
647 OPENSSL_cleanse(twop, sizeof(twop));
650 extern const gf curve448_precomputed_wnaf_as_fe[];
651 static const niels_t *curve448_wnaf_base =
652 (const niels_t *)curve448_precomputed_wnaf_as_fe;
654 void curve448_base_double_scalarmul_non_secret(curve448_point_t combo,
655 const curve448_scalar_t scalar1,
656 const curve448_point_t base2,
657 const curve448_scalar_t scalar2)
659 const int table_bits_var = DECAF_WNAF_VAR_TABLE_BITS,
660 table_bits_pre = DECAF_WNAF_FIXED_TABLE_BITS;
661 struct smvt_control control_var[DECAF_448_SCALAR_BITS /
662 (DECAF_WNAF_VAR_TABLE_BITS + 1) + 3];
663 struct smvt_control control_pre[DECAF_448_SCALAR_BITS /
664 (DECAF_WNAF_FIXED_TABLE_BITS + 1) + 3];
665 int ncb_pre = recode_wnaf(control_pre, scalar1, table_bits_pre);
666 int ncb_var = recode_wnaf(control_var, scalar2, table_bits_var);
667 pniels_t precmp_var[1 << DECAF_WNAF_VAR_TABLE_BITS];
668 int contp = 0, contv = 0, i;
670 prepare_wnaf_table(precmp_var, base2, table_bits_var);
671 i = control_var[0].power;
674 curve448_point_copy(combo, curve448_point_identity);
676 } else if (i > control_pre[0].power) {
677 pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
679 } else if (i == control_pre[0].power && i >= 0) {
680 pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
681 add_niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1],
686 i = control_pre[0].power;
687 niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1]);
691 for (i--; i >= 0; i--) {
692 int cv = (i == control_var[contv].power), cp =
693 (i == control_pre[contp].power);
694 point_double_internal(combo, combo, i && !(cv || cp));
697 assert(control_var[contv].addend);
699 if (control_var[contv].addend > 0) {
700 add_pniels_to_pt(combo,
701 precmp_var[control_var[contv].addend >> 1], i
704 sub_pniels_from_pt(combo,
705 precmp_var[(-control_var[contv].addend) >>
712 assert(control_pre[contp].addend);
714 if (control_pre[contp].addend > 0) {
715 add_niels_to_pt(combo,
716 curve448_wnaf_base[control_pre[contp].addend >>
719 sub_niels_from_pt(combo,
720 curve448_wnaf_base[(-control_pre
721 [contp].addend) >> 1], i);
727 /* This function is non-secret, but whatever this is cheap. */
728 OPENSSL_cleanse(control_var, sizeof(control_var));
729 OPENSSL_cleanse(control_pre, sizeof(control_pre));
730 OPENSSL_cleanse(precmp_var, sizeof(precmp_var));
732 assert(contv == ncb_var);
734 assert(contp == ncb_pre);
738 void curve448_point_destroy(curve448_point_t point)
740 OPENSSL_cleanse(point, sizeof(curve448_point_t));
743 int X448(uint8_t out_shared_key[56], const uint8_t private_key[56],
744 const uint8_t peer_public_value[56])
746 return decaf_x448(out_shared_key, peer_public_value, private_key)
750 void X448_public_from_private(uint8_t out_public_value[56],
751 const uint8_t private_key[56])
753 decaf_x448_derive_public_key(out_public_value, private_key);