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 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,
170 gf_sub_nr(b, d->y, d->x); /* 3+e */
172 gf_add_nr(b, d->x, d->y); /* 2+e */
173 gf_mul(d->y, e->a, b);
174 gf_mul(d->x, e->c, d->t);
175 gf_add_nr(c, a, d->y); /* 2+e */
176 gf_sub_nr(b, d->y, a); /* 3+e */
177 gf_add_nr(d->y, d->z, d->x); /* 2+e */
178 gf_sub_nr(a, d->z, d->x); /* 3+e */
179 gf_mul(d->z, a, d->y);
180 gf_mul(d->x, d->y, b);
186 static void add_pniels_to_pt(curve448_point_t p, const pniels_t pn,
191 gf_mul(L0, p->z, pn->z);
193 add_niels_to_pt(p, pn->n, before_double);
196 static void sub_pniels_from_pt(curve448_point_t p, const pniels_t pn,
201 gf_mul(L0, p->z, pn->z);
203 sub_niels_from_pt(p, pn->n, before_double);
206 c448_bool_t curve448_point_eq(const curve448_point_t p,
207 const curve448_point_t q)
211 /* equality mod 2-torsion compares x/y */
213 gf_mul(a, p->y, q->x);
214 gf_mul(b, q->y, p->x);
217 return mask_to_bool(succ);
220 c448_bool_t curve448_point_valid(const curve448_point_t p)
225 gf_mul(a, p->x, p->y);
226 gf_mul(b, p->z, p->t);
232 gf_mulw(c, b, TWISTED_D);
236 out &= ~gf_eq(p->z, ZERO);
237 return mask_to_bool(out);
240 static ossl_inline void constant_time_lookup_niels(niels_s * RESTRICT ni,
241 const niels_t * table,
244 constant_time_lookup(ni, table, sizeof(niels_s), nelts, idx);
247 void curve448_precomputed_scalarmul(curve448_point_t out,
248 const curve448_precomputed_s * table,
249 const curve448_scalar_t scalar)
253 const unsigned int n = COMBS_N, t = COMBS_T, s = COMBS_S;
256 curve448_scalar_t scalar1x;
257 curve448_scalar_add(scalar1x, scalar, precomputed_scalarmul_adjustment);
258 curve448_scalar_halve(scalar1x, scalar1x);
260 for (i = s - 1; 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 + s * (k + j * t);
270 if (bit < C448_SCALAR_BITS) {
272 (scalar1x->limb[bit / WBITS] >> (bit % WBITS) & 1) << k;
276 invert = (tab >> (t - 1)) - 1;
278 tab &= (1 << (t - 1)) - 1;
280 constant_time_lookup_niels(ni, &table->table[j << (t - 1)],
283 cond_neg_niels(ni, invert);
284 if ((i != (int)s - 1) || j) {
285 add_niels_to_pt(out, ni, j == n - 1 && i);
287 niels_to_pt(out, ni);
292 OPENSSL_cleanse(ni, sizeof(ni));
293 OPENSSL_cleanse(scalar1x, sizeof(scalar1x));
296 void curve448_point_mul_by_ratio_and_encode_like_eddsa(
297 uint8_t enc[EDDSA_448_PUBLIC_BYTES],
298 const curve448_point_t p)
301 /* The point is now on the twisted curve. Move it to untwisted. */
304 curve448_point_copy(q, p);
307 /* 4-isogeny: 2xy/(y^+x^2), (y^2-x^2)/(2z^2-y^2+x^2) */
313 gf_add(z, q->y, q->x);
323 OPENSSL_cleanse(u, sizeof(u));
332 enc[EDDSA_448_PRIVATE_BYTES - 1] = 0;
333 gf_serialize(enc, x, 1);
334 enc[EDDSA_448_PRIVATE_BYTES - 1] |= 0x80 & gf_lobit(t);
336 OPENSSL_cleanse(x, sizeof(x));
337 OPENSSL_cleanse(y, sizeof(y));
338 OPENSSL_cleanse(z, sizeof(z));
339 OPENSSL_cleanse(t, sizeof(t));
340 curve448_point_destroy(q);
343 c448_error_t curve448_point_decode_like_eddsa_and_mul_by_ratio(
345 const uint8_t enc[EDDSA_448_PUBLIC_BYTES])
347 uint8_t enc2[EDDSA_448_PUBLIC_BYTES];
351 memcpy(enc2, enc, sizeof(enc2));
353 low = ~word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1] & 0x80);
354 enc2[EDDSA_448_PRIVATE_BYTES - 1] &= ~0x80;
356 succ = gf_deserialize(p->y, enc2, 1, 0);
358 succ &= word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1]);
362 gf_sub(p->z, ONE, p->x); /* num = 1-y^2 */
363 gf_mulw(p->t, p->x, EDWARDS_D); /* dy^2 */
364 gf_sub(p->t, ONE, p->t); /* denom = 1-dy^2 or 1-d + dy^2 */
366 gf_mul(p->x, p->z, p->t);
367 succ &= gf_isr(p->t, p->x); /* 1/sqrt(num * denom) */
369 gf_mul(p->x, p->t, p->z); /* sqrt(num / denom) */
370 gf_cond_neg(p->x, gf_lobit(p->x) ^ low);
374 /* 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)
481 curve448_point_copy(q, p);
482 gf_invert(q->t, q->x, 0); /* 1/x */
483 gf_mul(q->z, q->t, q->y); /* y/x */
484 gf_sqr(q->y, q->z); /* (y/x)^2 */
485 gf_serialize(out, q->y, 1);
486 curve448_point_destroy(q);
489 void x448_derive_public_key(uint8_t out[X_PUBLIC_BYTES],
490 const uint8_t scalar[X_PRIVATE_BYTES])
492 /* Scalar conditioning */
493 uint8_t scalar2[X_PRIVATE_BYTES];
494 curve448_scalar_t the_scalar;
498 memcpy(scalar2, scalar, sizeof(scalar2));
499 scalar2[0] &= -(uint8_t)COFACTOR;
501 scalar2[X_PRIVATE_BYTES - 1] &= ~((0u - 1u) << ((X_PRIVATE_BITS + 7) % 8));
502 scalar2[X_PRIVATE_BYTES - 1] |= 1 << ((X_PRIVATE_BITS + 7) % 8);
504 curve448_scalar_decode_long(the_scalar, scalar2, sizeof(scalar2));
506 /* Compensate for the encoding ratio */
507 for (i = 1; i < X448_ENCODE_RATIO; i <<= 1) {
508 curve448_scalar_halve(the_scalar, the_scalar);
510 curve448_precomputed_scalarmul(p, curve448_precomputed_base, the_scalar);
511 curve448_point_mul_by_ratio_and_encode_like_x448(out, p);
512 curve448_point_destroy(p);
515 /* Control for variable-time scalar multiply algorithms. */
516 struct smvt_control {
520 #if defined(__GNUC__) || defined(__clang__)
521 # define NUMTRAILINGZEROS __builtin_ctz
523 # define NUMTRAILINGZEROS numtrailingzeros
524 static uint32_t numtrailingzeros(uint32_t i)
559 static int recode_wnaf(struct smvt_control *control,
560 /* [nbits/(table_bits + 1) + 3] */
561 const curve448_scalar_t scalar,
562 unsigned int table_bits)
564 unsigned int table_size = C448_SCALAR_BITS / (table_bits + 1) + 3;
565 int position = table_size - 1; /* at the end */
566 uint64_t current = scalar->limb[0] & 0xFFFF;
567 uint32_t mask = (1 << (table_bits + 1)) - 1;
569 const unsigned int B_OVER_16 = sizeof(scalar->limb[0]) / 2;
572 /* place the end marker */
573 control[position].power = -1;
574 control[position].addend = 0;
578 * PERF: Could negate scalar if it's large. But then would need more cases
579 * in the actual code that uses it, all for an expected reduction of like
580 * 1/5 op. Probably not worth it.
583 for (w = 1; w < (C448_SCALAR_BITS - 1) / 16 + 3; w++) {
584 if (w < (C448_SCALAR_BITS - 1) / 16 + 1) {
585 /* Refill the 16 high bits of current */
586 current += (uint32_t)((scalar->limb[w / B_OVER_16]
587 >> (16 * (w % B_OVER_16))) << 16);
590 while (current & 0xFFFF) {
591 uint32_t pos = NUMTRAILINGZEROS((uint32_t)current);
592 uint32_t odd = (uint32_t)current >> pos;
593 int32_t delta = odd & mask;
595 assert(position >= 0);
596 if (odd & 1 << (table_bits + 1))
597 delta -= (1 << (table_bits + 1));
598 current -= delta << pos;
599 control[position].power = pos + 16 * (w - 1);
600 control[position].addend = delta;
605 assert(current == 0);
608 n = table_size - position;
609 for (i = 0; i < n; i++) {
610 control[i] = control[i + position];
615 static void prepare_wnaf_table(pniels_t * output,
616 const curve448_point_t working,
619 curve448_point_t tmp;
623 pt_to_pniels(output[0], working);
628 curve448_point_double(tmp, working);
629 pt_to_pniels(twop, tmp);
631 add_pniels_to_pt(tmp, output[0], 0);
632 pt_to_pniels(output[1], tmp);
634 for (i = 2; i < 1 << tbits; i++) {
635 add_pniels_to_pt(tmp, twop, 0);
636 pt_to_pniels(output[i], tmp);
639 curve448_point_destroy(tmp);
640 OPENSSL_cleanse(twop, sizeof(twop));
643 extern const gf curve448_precomputed_wnaf_as_fe[];
644 static const niels_t *curve448_wnaf_base =
645 (const niels_t *)curve448_precomputed_wnaf_as_fe;
647 void curve448_base_double_scalarmul_non_secret(curve448_point_t combo,
648 const curve448_scalar_t scalar1,
649 const curve448_point_t base2,
650 const curve448_scalar_t scalar2)
652 const int table_bits_var = C448_WNAF_VAR_TABLE_BITS,
653 table_bits_pre = C448_WNAF_FIXED_TABLE_BITS;
654 struct smvt_control control_var[C448_SCALAR_BITS /
655 (C448_WNAF_VAR_TABLE_BITS + 1) + 3];
656 struct smvt_control control_pre[C448_SCALAR_BITS /
657 (C448_WNAF_FIXED_TABLE_BITS + 1) + 3];
658 int ncb_pre = recode_wnaf(control_pre, scalar1, table_bits_pre);
659 int ncb_var = recode_wnaf(control_var, scalar2, table_bits_var);
660 pniels_t precmp_var[1 << C448_WNAF_VAR_TABLE_BITS];
661 int contp = 0, contv = 0, i;
663 prepare_wnaf_table(precmp_var, base2, table_bits_var);
664 i = control_var[0].power;
667 curve448_point_copy(combo, curve448_point_identity);
669 } else if (i > control_pre[0].power) {
670 pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
672 } else if (i == control_pre[0].power && i >= 0) {
673 pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
674 add_niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1],
679 i = control_pre[0].power;
680 niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1]);
684 for (i--; i >= 0; i--) {
685 int cv = (i == control_var[contv].power), cp =
686 (i == control_pre[contp].power);
687 point_double_internal(combo, combo, i && !(cv || cp));
690 assert(control_var[contv].addend);
692 if (control_var[contv].addend > 0) {
693 add_pniels_to_pt(combo,
694 precmp_var[control_var[contv].addend >> 1],
697 sub_pniels_from_pt(combo,
698 precmp_var[(-control_var[contv].addend)
705 assert(control_pre[contp].addend);
707 if (control_pre[contp].addend > 0) {
708 add_niels_to_pt(combo,
709 curve448_wnaf_base[control_pre[contp].addend
712 sub_niels_from_pt(combo,
713 curve448_wnaf_base[(-control_pre
714 [contp].addend) >> 1], i);
720 /* This function is non-secret, but whatever this is cheap. */
721 OPENSSL_cleanse(control_var, sizeof(control_var));
722 OPENSSL_cleanse(control_pre, sizeof(control_pre));
723 OPENSSL_cleanse(precmp_var, sizeof(precmp_var));
725 assert(contv == ncb_var);
727 assert(contp == ncb_pre);
731 void curve448_point_destroy(curve448_point_t point)
733 OPENSSL_cleanse(point, sizeof(curve448_point_t));
736 int X448(uint8_t out_shared_key[56], const uint8_t private_key[56],
737 const uint8_t peer_public_value[56])
739 return x448_int(out_shared_key, peer_public_value, private_key)
743 void X448_public_from_private(uint8_t out_public_value[56],
744 const uint8_t private_key[56])
746 x448_derive_public_key(out_public_value, private_key);