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 static const int EDWARDS_D = -39081;
26 static const curve448_scalar_t precomputed_scalarmul_adjustment = {
29 SC_LIMB(0xc873d6d54a7bb0cf), SC_LIMB(0xe933d8d723a70aad),
30 SC_LIMB(0xbb124b65129c96fd), SC_LIMB(0x00000008335dc163)
35 #define TWISTED_D ((EDWARDS_D)-1)
37 #define WBITS C448_WORD_BITS /* NB this may be different from ARCH_WORD_BITS */
40 static void gf_invert(gf y, const gf x, int assert_nonzero)
45 gf_sqr(t1, x); /* o^2 */
46 ret = gf_isr(t2, t1); /* +-1/sqrt(o^2) = +-1/o */
51 gf_mul(t2, t1, x); /* not direct to y in case of alias. */
55 /** identity = (0,1) */
56 const curve448_point_t curve448_point_identity =
57 { {{{{0}}}, {{{1}}}, {{{1}}}, {{{0}}}} };
59 static void point_double_internal(curve448_point_t p, const curve448_point_t q,
66 gf_add_nr(d, c, a); /* 2+e */
67 gf_add_nr(p->t, q->y, q->x); /* 2+e */
69 gf_subx_nr(b, b, d, 3); /* 4+e */
70 gf_sub_nr(p->t, a, c); /* 3+e */
72 gf_add_nr(p->z, p->x, p->x); /* 2+e */
73 gf_subx_nr(a, p->z, p->t, 4); /* 6+e */
75 gf_weak_reduce(a); /* or 1+e */
77 gf_mul(p->z, p->t, a);
78 gf_mul(p->y, p->t, d);
83 void curve448_point_double(curve448_point_t p, const curve448_point_t q)
85 point_double_internal(p, q, 0);
88 /* Operations on [p]niels */
89 static ossl_inline void cond_neg_niels(niels_t n, mask_t neg)
91 gf_cond_swap(n->a, n->b, neg);
92 gf_cond_neg(n->c, neg);
95 static void pt_to_pniels(pniels_t b, const curve448_point_t a)
97 gf_sub(b->n->a, a->y, a->x);
98 gf_add(b->n->b, a->x, a->y);
99 gf_mulw(b->n->c, a->t, 2 * TWISTED_D);
100 gf_add(b->z, a->z, a->z);
103 static void pniels_to_pt(curve448_point_t e, const pniels_t d)
107 gf_add(eu, d->n->b, d->n->a);
108 gf_sub(e->y, d->n->b, d->n->a);
109 gf_mul(e->t, e->y, eu);
110 gf_mul(e->x, d->z, e->y);
111 gf_mul(e->y, d->z, eu);
115 static void niels_to_pt(curve448_point_t e, const niels_t n)
117 gf_add(e->y, n->b, n->a);
118 gf_sub(e->x, n->b, n->a);
119 gf_mul(e->t, e->y, e->x);
123 static void add_niels_to_pt(curve448_point_t d, const niels_t e,
128 gf_sub_nr(b, d->y, d->x); /* 3+e */
130 gf_add_nr(b, d->x, d->y); /* 2+e */
131 gf_mul(d->y, e->b, b);
132 gf_mul(d->x, e->c, d->t);
133 gf_add_nr(c, a, d->y); /* 2+e */
134 gf_sub_nr(b, d->y, a); /* 3+e */
135 gf_sub_nr(d->y, d->z, d->x); /* 3+e */
136 gf_add_nr(a, d->x, d->z); /* 2+e */
137 gf_mul(d->z, a, d->y);
138 gf_mul(d->x, d->y, b);
144 static void sub_niels_from_pt(curve448_point_t d, const niels_t e,
149 gf_sub_nr(b, d->y, d->x); /* 3+e */
151 gf_add_nr(b, d->x, d->y); /* 2+e */
152 gf_mul(d->y, e->a, b);
153 gf_mul(d->x, e->c, d->t);
154 gf_add_nr(c, a, d->y); /* 2+e */
155 gf_sub_nr(b, d->y, a); /* 3+e */
156 gf_add_nr(d->y, d->z, d->x); /* 2+e */
157 gf_sub_nr(a, d->z, d->x); /* 3+e */
158 gf_mul(d->z, a, d->y);
159 gf_mul(d->x, d->y, b);
165 static void add_pniels_to_pt(curve448_point_t p, const pniels_t pn,
170 gf_mul(L0, p->z, pn->z);
172 add_niels_to_pt(p, pn->n, before_double);
175 static void sub_pniels_from_pt(curve448_point_t p, const pniels_t pn,
180 gf_mul(L0, p->z, pn->z);
182 sub_niels_from_pt(p, pn->n, before_double);
185 c448_bool_t curve448_point_eq(const curve448_point_t p,
186 const curve448_point_t q)
191 /* equality mod 2-torsion compares x/y */
192 gf_mul(a, p->y, q->x);
193 gf_mul(b, q->y, p->x);
196 return mask_to_bool(succ);
199 c448_bool_t curve448_point_valid(const curve448_point_t p)
204 gf_mul(a, p->x, p->y);
205 gf_mul(b, p->z, p->t);
211 gf_mulw(c, b, TWISTED_D);
215 out &= ~gf_eq(p->z, ZERO);
216 return mask_to_bool(out);
219 static ossl_inline void constant_time_lookup_niels(niels_s * RESTRICT ni,
220 const niels_t * table,
223 constant_time_lookup(ni, table, sizeof(niels_s), nelts, idx);
226 void curve448_precomputed_scalarmul(curve448_point_t out,
227 const curve448_precomputed_s * table,
228 const curve448_scalar_t scalar)
230 unsigned int i, j, k;
231 const unsigned int n = COMBS_N, t = COMBS_T, s = COMBS_S;
233 curve448_scalar_t scalar1x;
235 curve448_scalar_add(scalar1x, scalar, precomputed_scalarmul_adjustment);
236 curve448_scalar_halve(scalar1x, scalar1x);
238 for (i = s; i > 0; i--) {
240 point_double_internal(out, out, 0);
242 for (j = 0; j < n; j++) {
246 for (k = 0; k < t; k++) {
247 unsigned int bit = (i - 1) + s * (k + j * t);
249 if (bit < C448_SCALAR_BITS)
251 (scalar1x->limb[bit / WBITS] >> (bit % WBITS) & 1) << k;
254 invert = (tab >> (t - 1)) - 1;
256 tab &= (1 << (t - 1)) - 1;
258 constant_time_lookup_niels(ni, &table->table[j << (t - 1)],
261 cond_neg_niels(ni, invert);
262 if ((i != s) || j != 0)
263 add_niels_to_pt(out, ni, j == n - 1 && i != 1);
265 niels_to_pt(out, ni);
269 OPENSSL_cleanse(ni, sizeof(ni));
270 OPENSSL_cleanse(scalar1x, sizeof(scalar1x));
273 void curve448_point_mul_by_ratio_and_encode_like_eddsa(
274 uint8_t enc[EDDSA_448_PUBLIC_BYTES],
275 const curve448_point_t p)
280 /* The point is now on the twisted curve. Move it to untwisted. */
281 curve448_point_copy(q, p);
284 /* 4-isogeny: 2xy/(y^+x^2), (y^2-x^2)/(2z^2-y^2+x^2) */
290 gf_add(z, q->y, q->x);
300 OPENSSL_cleanse(u, sizeof(u));
309 enc[EDDSA_448_PRIVATE_BYTES - 1] = 0;
310 gf_serialize(enc, x, 1);
311 enc[EDDSA_448_PRIVATE_BYTES - 1] |= 0x80 & gf_lobit(t);
313 OPENSSL_cleanse(x, sizeof(x));
314 OPENSSL_cleanse(y, sizeof(y));
315 OPENSSL_cleanse(z, sizeof(z));
316 OPENSSL_cleanse(t, sizeof(t));
317 curve448_point_destroy(q);
320 c448_error_t curve448_point_decode_like_eddsa_and_mul_by_ratio(
322 const uint8_t enc[EDDSA_448_PUBLIC_BYTES])
324 uint8_t enc2[EDDSA_448_PUBLIC_BYTES];
328 memcpy(enc2, enc, sizeof(enc2));
330 low = ~word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1] & 0x80);
331 enc2[EDDSA_448_PRIVATE_BYTES - 1] &= ~0x80;
333 succ = gf_deserialize(p->y, enc2, 1, 0);
334 succ &= word_is_zero(enc2[EDDSA_448_PRIVATE_BYTES - 1]);
337 gf_sub(p->z, ONE, p->x); /* num = 1-y^2 */
338 gf_mulw(p->t, p->x, EDWARDS_D); /* dy^2 */
339 gf_sub(p->t, ONE, p->t); /* denom = 1-dy^2 or 1-d + dy^2 */
341 gf_mul(p->x, p->z, p->t);
342 succ &= gf_isr(p->t, p->x); /* 1/sqrt(num * denom) */
344 gf_mul(p->x, p->t, p->z); /* sqrt(num / denom) */
345 gf_cond_neg(p->x, gf_lobit(p->x) ^ low);
351 /* 4-isogeny 2xy/(y^2-ax^2), (y^2+ax^2)/(2-y^2-ax^2) */
355 gf_add(p->t, p->y, p->x);
360 gf_add(p->z, p->x, p->x);
363 gf_mul(p->z, p->t, a);
364 gf_mul(p->y, p->t, d);
366 OPENSSL_cleanse(a, sizeof(a));
367 OPENSSL_cleanse(b, sizeof(b));
368 OPENSSL_cleanse(c, sizeof(c));
369 OPENSSL_cleanse(d, sizeof(d));
372 OPENSSL_cleanse(enc2, sizeof(enc2));
373 assert(curve448_point_valid(p) || ~succ);
375 return c448_succeed_if(mask_to_bool(succ));
378 c448_error_t x448_int(uint8_t out[X_PUBLIC_BYTES],
379 const uint8_t base[X_PUBLIC_BYTES],
380 const uint8_t scalar[X_PRIVATE_BYTES])
382 gf x1, x2, z2, x3, z3, t1, t2;
387 ignore_result(gf_deserialize(x1, base, 1, 0));
393 for (t = X_PRIVATE_BITS - 1; t >= 0; t--) {
394 uint8_t sb = scalar[t / 8];
397 /* Scalar conditioning */
399 sb &= -(uint8_t)COFACTOR;
400 else if (t == X_PRIVATE_BITS - 1)
403 k_t = (sb >> (t % 8)) & 1;
404 k_t = 0 - k_t; /* set to all 0s or all 1s */
407 gf_cond_swap(x2, x3, swap);
408 gf_cond_swap(z2, z3, swap);
411 gf_add_nr(t1, x2, z2); /* A = x2 + z2 *//* 2+e */
412 gf_sub_nr(t2, x2, z2); /* B = x2 - z2 *//* 3+e */
413 gf_sub_nr(z2, x3, z3); /* D = x3 - z3 *//* 3+e */
414 gf_mul(x2, t1, z2); /* DA */
415 gf_add_nr(z2, z3, x3); /* C = x3 + z3 *//* 2+e */
416 gf_mul(x3, t2, z2); /* CB */
417 gf_sub_nr(z3, x2, x3); /* DA-CB *//* 3+e */
418 gf_sqr(z2, z3); /* (DA-CB)^2 */
419 gf_mul(z3, x1, z2); /* z3 = x1(DA-CB)^2 */
420 gf_add_nr(z2, x2, x3); /* (DA+CB) *//* 2+e */
421 gf_sqr(x3, z2); /* x3 = (DA+CB)^2 */
423 gf_sqr(z2, t1); /* AA = A^2 */
424 gf_sqr(t1, t2); /* BB = B^2 */
425 gf_mul(x2, z2, t1); /* x2 = AA*BB */
426 gf_sub_nr(t2, z2, t1); /* E = AA-BB *//* 3+e */
428 gf_mulw(t1, t2, -EDWARDS_D); /* E*-d = a24*E */
429 gf_add_nr(t1, t1, z2); /* AA + a24*E *//* 2+e */
430 gf_mul(z2, t2, t1); /* z2 = E(AA+a24*E) */
434 gf_cond_swap(x2, x3, swap);
435 gf_cond_swap(z2, z3, swap);
436 gf_invert(z2, z2, 0);
438 gf_serialize(out, x1, 1);
439 nz = ~gf_eq(x1, ZERO);
441 OPENSSL_cleanse(x1, sizeof(x1));
442 OPENSSL_cleanse(x2, sizeof(x2));
443 OPENSSL_cleanse(z2, sizeof(z2));
444 OPENSSL_cleanse(x3, sizeof(x3));
445 OPENSSL_cleanse(z3, sizeof(z3));
446 OPENSSL_cleanse(t1, sizeof(t1));
447 OPENSSL_cleanse(t2, sizeof(t2));
449 return c448_succeed_if(mask_to_bool(nz));
452 void curve448_point_mul_by_ratio_and_encode_like_x448(uint8_t
454 const curve448_point_t p)
458 curve448_point_copy(q, p);
459 gf_invert(q->t, q->x, 0); /* 1/x */
460 gf_mul(q->z, q->t, q->y); /* y/x */
461 gf_sqr(q->y, q->z); /* (y/x)^2 */
462 gf_serialize(out, q->y, 1);
463 curve448_point_destroy(q);
466 void x448_derive_public_key(uint8_t out[X_PUBLIC_BYTES],
467 const uint8_t scalar[X_PRIVATE_BYTES])
469 /* Scalar conditioning */
470 uint8_t scalar2[X_PRIVATE_BYTES];
471 curve448_scalar_t the_scalar;
475 memcpy(scalar2, scalar, sizeof(scalar2));
476 scalar2[0] &= -(uint8_t)COFACTOR;
478 scalar2[X_PRIVATE_BYTES - 1] &= ~((0u - 1u) << ((X_PRIVATE_BITS + 7) % 8));
479 scalar2[X_PRIVATE_BYTES - 1] |= 1 << ((X_PRIVATE_BITS + 7) % 8);
481 curve448_scalar_decode_long(the_scalar, scalar2, sizeof(scalar2));
483 /* Compensate for the encoding ratio */
484 for (i = 1; i < X448_ENCODE_RATIO; i <<= 1)
485 curve448_scalar_halve(the_scalar, the_scalar);
487 curve448_precomputed_scalarmul(p, curve448_precomputed_base, the_scalar);
488 curve448_point_mul_by_ratio_and_encode_like_x448(out, p);
489 curve448_point_destroy(p);
492 /* Control for variable-time scalar multiply algorithms. */
493 struct smvt_control {
497 #if defined(__GNUC__) || defined(__clang__)
498 # define NUMTRAILINGZEROS __builtin_ctz
500 # define NUMTRAILINGZEROS numtrailingzeros
501 static uint32_t numtrailingzeros(uint32_t i)
537 static int recode_wnaf(struct smvt_control *control,
538 /* [nbits/(table_bits + 1) + 3] */
539 const curve448_scalar_t scalar,
540 unsigned int table_bits)
542 unsigned int table_size = C448_SCALAR_BITS / (table_bits + 1) + 3;
543 int position = table_size - 1; /* at the end */
544 uint64_t current = scalar->limb[0] & 0xFFFF;
545 uint32_t mask = (1 << (table_bits + 1)) - 1;
547 const unsigned int B_OVER_16 = sizeof(scalar->limb[0]) / 2;
550 /* place the end marker */
551 control[position].power = -1;
552 control[position].addend = 0;
556 * PERF: Could negate scalar if it's large. But then would need more cases
557 * in the actual code that uses it, all for an expected reduction of like
558 * 1/5 op. Probably not worth it.
561 for (w = 1; w < (C448_SCALAR_BITS - 1) / 16 + 3; w++) {
562 if (w < (C448_SCALAR_BITS - 1) / 16 + 1) {
563 /* Refill the 16 high bits of current */
564 current += (uint32_t)((scalar->limb[w / B_OVER_16]
565 >> (16 * (w % B_OVER_16))) << 16);
568 while (current & 0xFFFF) {
569 uint32_t pos = NUMTRAILINGZEROS((uint32_t)current);
570 uint32_t odd = (uint32_t)current >> pos;
571 int32_t delta = odd & mask;
573 assert(position >= 0);
574 if (odd & (1 << (table_bits + 1)))
575 delta -= (1 << (table_bits + 1));
576 current -= delta << pos;
577 control[position].power = pos + 16 * (w - 1);
578 control[position].addend = delta;
583 assert(current == 0);
586 n = table_size - position;
587 for (i = 0; i < n; i++)
588 control[i] = control[i + position];
593 static void prepare_wnaf_table(pniels_t * output,
594 const curve448_point_t working,
597 curve448_point_t tmp;
601 pt_to_pniels(output[0], working);
606 curve448_point_double(tmp, working);
607 pt_to_pniels(twop, tmp);
609 add_pniels_to_pt(tmp, output[0], 0);
610 pt_to_pniels(output[1], tmp);
612 for (i = 2; i < 1 << tbits; i++) {
613 add_pniels_to_pt(tmp, twop, 0);
614 pt_to_pniels(output[i], tmp);
617 curve448_point_destroy(tmp);
618 OPENSSL_cleanse(twop, sizeof(twop));
621 void curve448_base_double_scalarmul_non_secret(curve448_point_t combo,
622 const curve448_scalar_t scalar1,
623 const curve448_point_t base2,
624 const curve448_scalar_t scalar2)
626 const int table_bits_var = C448_WNAF_VAR_TABLE_BITS;
627 const int table_bits_pre = C448_WNAF_FIXED_TABLE_BITS;
628 struct smvt_control control_var[C448_SCALAR_BITS /
629 (C448_WNAF_VAR_TABLE_BITS + 1) + 3];
630 struct smvt_control control_pre[C448_SCALAR_BITS /
631 (C448_WNAF_FIXED_TABLE_BITS + 1) + 3];
632 int ncb_pre = recode_wnaf(control_pre, scalar1, table_bits_pre);
633 int ncb_var = recode_wnaf(control_var, scalar2, table_bits_var);
634 pniels_t precmp_var[1 << C448_WNAF_VAR_TABLE_BITS];
635 int contp = 0, contv = 0, i;
637 prepare_wnaf_table(precmp_var, base2, table_bits_var);
638 i = control_var[0].power;
641 curve448_point_copy(combo, curve448_point_identity);
644 if (i > control_pre[0].power) {
645 pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
647 } else if (i == control_pre[0].power && i >= 0) {
648 pniels_to_pt(combo, precmp_var[control_var[0].addend >> 1]);
649 add_niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1],
654 i = control_pre[0].power;
655 niels_to_pt(combo, curve448_wnaf_base[control_pre[0].addend >> 1]);
659 for (i--; i >= 0; i--) {
660 int cv = (i == control_var[contv].power);
661 int cp = (i == control_pre[contp].power);
663 point_double_internal(combo, combo, i && !(cv || cp));
666 assert(control_var[contv].addend);
668 if (control_var[contv].addend > 0)
669 add_pniels_to_pt(combo,
670 precmp_var[control_var[contv].addend >> 1],
673 sub_pniels_from_pt(combo,
674 precmp_var[(-control_var[contv].addend)
680 assert(control_pre[contp].addend);
682 if (control_pre[contp].addend > 0)
683 add_niels_to_pt(combo,
684 curve448_wnaf_base[control_pre[contp].addend
687 sub_niels_from_pt(combo,
688 curve448_wnaf_base[(-control_pre
689 [contp].addend) >> 1], i);
694 /* This function is non-secret, but whatever this is cheap. */
695 OPENSSL_cleanse(control_var, sizeof(control_var));
696 OPENSSL_cleanse(control_pre, sizeof(control_pre));
697 OPENSSL_cleanse(precmp_var, sizeof(precmp_var));
699 assert(contv == ncb_var);
701 assert(contp == ncb_pre);
705 void curve448_point_destroy(curve448_point_t point)
707 OPENSSL_cleanse(point, sizeof(curve448_point_t));
710 int X448(uint8_t out_shared_key[56], const uint8_t private_key[56],
711 const uint8_t peer_public_value[56])
713 return x448_int(out_shared_key, peer_public_value, private_key)
717 void X448_public_from_private(uint8_t out_public_value[56],
718 const uint8_t private_key[56])
720 x448_derive_public_key(out_public_value, private_key);