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 /* 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
,
171 gf_sub_nr(b
, d
->y
, d
->x
); /* 3+e */
173 gf_add_nr(b
, d
->x
, d
->y
); /* 2+e */
174 gf_mul(d
->y
, e
->a
, b
);
175 gf_mul(d
->x
, e
->c
, d
->t
);
176 gf_add_nr(c
, a
, d
->y
); /* 2+e */
177 gf_sub_nr(b
, d
->y
, a
); /* 3+e */
178 gf_add_nr(d
->y
, d
->z
, d
->x
); /* 2+e */
179 gf_sub_nr(a
, d
->z
, d
->x
); /* 3+e */
180 gf_mul(d
->z
, a
, d
->y
);
181 gf_mul(d
->x
, d
->y
, b
);
187 static void add_pniels_to_pt(curve448_point_t p
, const pniels_t pn
,
192 gf_mul(L0
, p
->z
, pn
->z
);
194 add_niels_to_pt(p
, pn
->n
, before_double
);
197 static void sub_pniels_from_pt(curve448_point_t p
, const pniels_t pn
,
202 gf_mul(L0
, p
->z
, pn
->z
);
204 sub_niels_from_pt(p
, pn
->n
, before_double
);
207 c448_bool_t
curve448_point_eq(const curve448_point_t p
,
208 const curve448_point_t q
)
213 /* equality mod 2-torsion compares x/y */
214 gf_mul(a
, p
->y
, q
->x
);
215 gf_mul(b
, q
->y
, p
->x
);
218 return mask_to_bool(succ
);
221 c448_bool_t
curve448_point_valid(const curve448_point_t p
)
226 gf_mul(a
, p
->x
, p
->y
);
227 gf_mul(b
, p
->z
, p
->t
);
233 gf_mulw(c
, b
, TWISTED_D
);
237 out
&= ~gf_eq(p
->z
, ZERO
);
238 return mask_to_bool(out
);
241 static ossl_inline
void constant_time_lookup_niels(niels_s
* RESTRICT ni
,
242 const niels_t
* table
,
245 constant_time_lookup(ni
, table
, sizeof(niels_s
), nelts
, idx
);
248 void curve448_precomputed_scalarmul(curve448_point_t out
,
249 const curve448_precomputed_s
* table
,
250 const curve448_scalar_t scalar
)
252 unsigned int i
, j
, k
;
253 const unsigned int n
= COMBS_N
, t
= COMBS_T
, s
= COMBS_S
;
255 curve448_scalar_t scalar1x
;
257 curve448_scalar_add(scalar1x
, scalar
, precomputed_scalarmul_adjustment
);
258 curve448_scalar_halve(scalar1x
, scalar1x
);
260 for (i
= s
; 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
- 1) + s
* (k
+ j
* t
);
271 if (bit
< C448_SCALAR_BITS
) {
273 (scalar1x
->limb
[bit
/ WBITS
] >> (bit
% WBITS
) & 1) << k
;
277 invert
= (tab
>> (t
- 1)) - 1;
279 tab
&= (1 << (t
- 1)) - 1;
281 constant_time_lookup_niels(ni
, &table
->table
[j
<< (t
- 1)],
284 cond_neg_niels(ni
, invert
);
285 if ((i
!= s
) || j
!= 0) {
286 add_niels_to_pt(out
, ni
, j
== n
- 1 && i
!= 1);
288 niels_to_pt(out
, ni
);
293 OPENSSL_cleanse(ni
, sizeof(ni
));
294 OPENSSL_cleanse(scalar1x
, sizeof(scalar1x
));
297 void curve448_point_mul_by_ratio_and_encode_like_eddsa(
298 uint8_t enc
[EDDSA_448_PUBLIC_BYTES
],
299 const curve448_point_t p
)
304 /* The point is now on the twisted curve. Move it to untwisted. */
305 curve448_point_copy(q
, p
);
308 /* 4-isogeny: 2xy/(y^+x^2), (y^2-x^2)/(2z^2-y^2+x^2) */
314 gf_add(z
, q
->y
, q
->x
);
324 OPENSSL_cleanse(u
, sizeof(u
));
333 enc
[EDDSA_448_PRIVATE_BYTES
- 1] = 0;
334 gf_serialize(enc
, x
, 1);
335 enc
[EDDSA_448_PRIVATE_BYTES
- 1] |= 0x80 & gf_lobit(t
);
337 OPENSSL_cleanse(x
, sizeof(x
));
338 OPENSSL_cleanse(y
, sizeof(y
));
339 OPENSSL_cleanse(z
, sizeof(z
));
340 OPENSSL_cleanse(t
, sizeof(t
));
341 curve448_point_destroy(q
);
344 c448_error_t
curve448_point_decode_like_eddsa_and_mul_by_ratio(
346 const uint8_t enc
[EDDSA_448_PUBLIC_BYTES
])
348 uint8_t enc2
[EDDSA_448_PUBLIC_BYTES
];
352 memcpy(enc2
, enc
, sizeof(enc2
));
354 low
= ~word_is_zero(enc2
[EDDSA_448_PRIVATE_BYTES
- 1] & 0x80);
355 enc2
[EDDSA_448_PRIVATE_BYTES
- 1] &= ~0x80;
357 succ
= gf_deserialize(p
->y
, enc2
, 1, 0);
358 succ
&= word_is_zero(enc2
[EDDSA_448_PRIVATE_BYTES
- 1]);
361 gf_sub(p
->z
, ONE
, p
->x
); /* num = 1-y^2 */
362 gf_mulw(p
->t
, p
->x
, EDWARDS_D
); /* dy^2 */
363 gf_sub(p
->t
, ONE
, p
->t
); /* denom = 1-dy^2 or 1-d + dy^2 */
365 gf_mul(p
->x
, p
->z
, p
->t
);
366 succ
&= gf_isr(p
->t
, p
->x
); /* 1/sqrt(num * denom) */
368 gf_mul(p
->x
, p
->t
, p
->z
); /* sqrt(num / denom) */
369 gf_cond_neg(p
->x
, gf_lobit(p
->x
) ^ low
);
375 /* 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
)
482 curve448_point_copy(q
, p
);
483 gf_invert(q
->t
, q
->x
, 0); /* 1/x */
484 gf_mul(q
->z
, q
->t
, q
->y
); /* y/x */
485 gf_sqr(q
->y
, q
->z
); /* (y/x)^2 */
486 gf_serialize(out
, q
->y
, 1);
487 curve448_point_destroy(q
);
490 void x448_derive_public_key(uint8_t out
[X_PUBLIC_BYTES
],
491 const uint8_t scalar
[X_PRIVATE_BYTES
])
493 /* Scalar conditioning */
494 uint8_t scalar2
[X_PRIVATE_BYTES
];
495 curve448_scalar_t the_scalar
;
499 memcpy(scalar2
, scalar
, sizeof(scalar2
));
500 scalar2
[0] &= -(uint8_t)COFACTOR
;
502 scalar2
[X_PRIVATE_BYTES
- 1] &= ~((0u - 1u) << ((X_PRIVATE_BITS
+ 7) % 8));
503 scalar2
[X_PRIVATE_BYTES
- 1] |= 1 << ((X_PRIVATE_BITS
+ 7) % 8);
505 curve448_scalar_decode_long(the_scalar
, scalar2
, sizeof(scalar2
));
507 /* Compensate for the encoding ratio */
508 for (i
= 1; i
< X448_ENCODE_RATIO
; i
<<= 1) {
509 curve448_scalar_halve(the_scalar
, the_scalar
);
511 curve448_precomputed_scalarmul(p
, curve448_precomputed_base
, the_scalar
);
512 curve448_point_mul_by_ratio_and_encode_like_x448(out
, p
);
513 curve448_point_destroy(p
);
516 /* Control for variable-time scalar multiply algorithms. */
517 struct smvt_control
{
521 #if defined(__GNUC__) || defined(__clang__)
522 # define NUMTRAILINGZEROS __builtin_ctz
524 # define NUMTRAILINGZEROS numtrailingzeros
525 static uint32_t numtrailingzeros(uint32_t i
)
561 static int recode_wnaf(struct smvt_control
*control
,
562 /* [nbits/(table_bits + 1) + 3] */
563 const curve448_scalar_t scalar
,
564 unsigned int table_bits
)
566 unsigned int table_size
= C448_SCALAR_BITS
/ (table_bits
+ 1) + 3;
567 int position
= table_size
- 1; /* at the end */
568 uint64_t current
= scalar
->limb
[0] & 0xFFFF;
569 uint32_t mask
= (1 << (table_bits
+ 1)) - 1;
571 const unsigned int B_OVER_16
= sizeof(scalar
->limb
[0]) / 2;
574 /* place the end marker */
575 control
[position
].power
= -1;
576 control
[position
].addend
= 0;
580 * PERF: Could negate scalar if it's large. But then would need more cases
581 * in the actual code that uses it, all for an expected reduction of like
582 * 1/5 op. Probably not worth it.
585 for (w
= 1; w
< (C448_SCALAR_BITS
- 1) / 16 + 3; w
++) {
586 if (w
< (C448_SCALAR_BITS
- 1) / 16 + 1) {
587 /* Refill the 16 high bits of current */
588 current
+= (uint32_t)((scalar
->limb
[w
/ B_OVER_16
]
589 >> (16 * (w
% B_OVER_16
))) << 16);
592 while (current
& 0xFFFF) {
593 uint32_t pos
= NUMTRAILINGZEROS((uint32_t)current
);
594 uint32_t odd
= (uint32_t)current
>> pos
;
595 int32_t delta
= odd
& mask
;
597 assert(position
>= 0);
598 if (odd
& (1 << (table_bits
+ 1)))
599 delta
-= (1 << (table_bits
+ 1));
600 current
-= delta
<< pos
;
601 control
[position
].power
= pos
+ 16 * (w
- 1);
602 control
[position
].addend
= delta
;
607 assert(current
== 0);
610 n
= table_size
- position
;
611 for (i
= 0; i
< n
; i
++)
612 control
[i
] = control
[i
+ position
];
617 static void prepare_wnaf_table(pniels_t
* output
,
618 const curve448_point_t working
,
621 curve448_point_t tmp
;
625 pt_to_pniels(output
[0], working
);
630 curve448_point_double(tmp
, working
);
631 pt_to_pniels(twop
, tmp
);
633 add_pniels_to_pt(tmp
, output
[0], 0);
634 pt_to_pniels(output
[1], tmp
);
636 for (i
= 2; i
< 1 << tbits
; i
++) {
637 add_pniels_to_pt(tmp
, twop
, 0);
638 pt_to_pniels(output
[i
], tmp
);
641 curve448_point_destroy(tmp
);
642 OPENSSL_cleanse(twop
, sizeof(twop
));
645 extern const gf curve448_precomputed_wnaf_as_fe
[];
646 static const niels_t
*curve448_wnaf_base
=
647 (const niels_t
*)curve448_precomputed_wnaf_as_fe
;
649 void curve448_base_double_scalarmul_non_secret(curve448_point_t combo
,
650 const curve448_scalar_t scalar1
,
651 const curve448_point_t base2
,
652 const curve448_scalar_t scalar2
)
654 const int table_bits_var
= C448_WNAF_VAR_TABLE_BITS
;
655 const int table_bits_pre
= C448_WNAF_FIXED_TABLE_BITS
;
656 struct smvt_control control_var
[C448_SCALAR_BITS
/
657 (C448_WNAF_VAR_TABLE_BITS
+ 1) + 3];
658 struct smvt_control control_pre
[C448_SCALAR_BITS
/
659 (C448_WNAF_FIXED_TABLE_BITS
+ 1) + 3];
660 int ncb_pre
= recode_wnaf(control_pre
, scalar1
, table_bits_pre
);
661 int ncb_var
= recode_wnaf(control_var
, scalar2
, table_bits_var
);
662 pniels_t precmp_var
[1 << C448_WNAF_VAR_TABLE_BITS
];
663 int contp
= 0, contv
= 0, i
;
665 prepare_wnaf_table(precmp_var
, base2
, table_bits_var
);
666 i
= control_var
[0].power
;
669 curve448_point_copy(combo
, curve448_point_identity
);
671 } else if (i
> control_pre
[0].power
) {
672 pniels_to_pt(combo
, precmp_var
[control_var
[0].addend
>> 1]);
674 } else if (i
== control_pre
[0].power
&& i
>= 0) {
675 pniels_to_pt(combo
, precmp_var
[control_var
[0].addend
>> 1]);
676 add_niels_to_pt(combo
, curve448_wnaf_base
[control_pre
[0].addend
>> 1],
681 i
= control_pre
[0].power
;
682 niels_to_pt(combo
, curve448_wnaf_base
[control_pre
[0].addend
>> 1]);
686 for (i
--; i
>= 0; i
--) {
687 int cv
= (i
== control_var
[contv
].power
);
688 int cp
= (i
== control_pre
[contp
].power
);
690 point_double_internal(combo
, combo
, i
&& !(cv
|| cp
));
693 assert(control_var
[contv
].addend
);
695 if (control_var
[contv
].addend
> 0)
696 add_pniels_to_pt(combo
,
697 precmp_var
[control_var
[contv
].addend
>> 1],
700 sub_pniels_from_pt(combo
,
701 precmp_var
[(-control_var
[contv
].addend
)
707 assert(control_pre
[contp
].addend
);
709 if (control_pre
[contp
].addend
> 0)
710 add_niels_to_pt(combo
,
711 curve448_wnaf_base
[control_pre
[contp
].addend
714 sub_niels_from_pt(combo
,
715 curve448_wnaf_base
[(-control_pre
716 [contp
].addend
) >> 1], i
);
721 /* This function is non-secret, but whatever this is cheap. */
722 OPENSSL_cleanse(control_var
, sizeof(control_var
));
723 OPENSSL_cleanse(control_pre
, sizeof(control_pre
));
724 OPENSSL_cleanse(precmp_var
, sizeof(precmp_var
));
726 assert(contv
== ncb_var
);
728 assert(contp
== ncb_pre
);
732 void curve448_point_destroy(curve448_point_t point
)
734 OPENSSL_cleanse(point
, sizeof(curve448_point_t
));
737 int X448(uint8_t out_shared_key
[56], const uint8_t private_key
[56],
738 const uint8_t peer_public_value
[56])
740 return x448_int(out_shared_key
, peer_public_value
, private_key
)
744 void X448_public_from_private(uint8_t out_public_value
[56],
745 const uint8_t private_key
[56])
747 x448_derive_public_key(out_public_value
, private_key
);