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__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ > 3))
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
);