2 * Copyright 2014-2019 The OpenSSL Project Authors. All Rights Reserved.
3 * Copyright (c) 2014, Intel Corporation. All Rights Reserved.
4 * Copyright (c) 2015, CloudFlare, Inc.
6 * Licensed under the OpenSSL license (the "License"). You may not use
7 * this file except in compliance with the License. You can obtain a copy
8 * in the file LICENSE in the source distribution or at
9 * https://www.openssl.org/source/license.html
11 * Originally written by Shay Gueron (1, 2), and Vlad Krasnov (1, 3)
12 * (1) Intel Corporation, Israel Development Center, Haifa, Israel
13 * (2) University of Haifa, Israel
14 * (3) CloudFlare, Inc.
17 * S.Gueron and V.Krasnov, "Fast Prime Field Elliptic Curve Cryptography with
23 #include "internal/cryptlib.h"
24 #include "crypto/bn.h"
26 #include "internal/refcount.h"
29 # define TOBN(hi,lo) lo,hi
31 # define TOBN(hi,lo) ((BN_ULONG)hi<<32|lo)
35 # define ALIGN32 __attribute((aligned(32)))
36 #elif defined(_MSC_VER)
37 # define ALIGN32 __declspec(align(32))
42 #define ALIGNPTR(p,N) ((unsigned char *)p+N-(size_t)p%N)
43 #define P256_LIMBS (256/BN_BITS2)
45 typedef unsigned short u16
;
48 BN_ULONG X
[P256_LIMBS
];
49 BN_ULONG Y
[P256_LIMBS
];
50 BN_ULONG Z
[P256_LIMBS
];
54 BN_ULONG X
[P256_LIMBS
];
55 BN_ULONG Y
[P256_LIMBS
];
58 typedef P256_POINT_AFFINE PRECOMP256_ROW
[64];
60 /* structure for precomputed multiples of the generator */
61 struct nistz256_pre_comp_st
{
62 const EC_GROUP
*group
; /* Parent EC_GROUP object */
63 size_t w
; /* Window size */
65 * Constant time access to the X and Y coordinates of the pre-computed,
66 * generator multiplies, in the Montgomery domain. Pre-calculated
67 * multiplies are stored in affine form.
69 PRECOMP256_ROW
*precomp
;
70 void *precomp_storage
;
71 CRYPTO_REF_COUNT references
;
75 /* Functions implemented in assembly */
77 * Most of below mentioned functions *preserve* the property of inputs
78 * being fully reduced, i.e. being in [0, modulus) range. Simply put if
79 * inputs are fully reduced, then output is too. Note that reverse is
80 * not true, in sense that given partially reduced inputs output can be
81 * either, not unlikely reduced. And "most" in first sentence refers to
82 * the fact that given the calculations flow one can tolerate that
83 * addition, 1st function below, produces partially reduced result *if*
84 * multiplications by 2 and 3, which customarily use addition, fully
85 * reduce it. This effectively gives two options: a) addition produces
86 * fully reduced result [as long as inputs are, just like remaining
87 * functions]; b) addition is allowed to produce partially reduced
88 * result, but multiplications by 2 and 3 perform additional reduction
89 * step. Choice between the two can be platform-specific, but it was a)
90 * in all cases so far...
92 /* Modular add: res = a+b mod P */
93 void ecp_nistz256_add(BN_ULONG res
[P256_LIMBS
],
94 const BN_ULONG a
[P256_LIMBS
],
95 const BN_ULONG b
[P256_LIMBS
]);
96 /* Modular mul by 2: res = 2*a mod P */
97 void ecp_nistz256_mul_by_2(BN_ULONG res
[P256_LIMBS
],
98 const BN_ULONG a
[P256_LIMBS
]);
99 /* Modular mul by 3: res = 3*a mod P */
100 void ecp_nistz256_mul_by_3(BN_ULONG res
[P256_LIMBS
],
101 const BN_ULONG a
[P256_LIMBS
]);
103 /* Modular div by 2: res = a/2 mod P */
104 void ecp_nistz256_div_by_2(BN_ULONG res
[P256_LIMBS
],
105 const BN_ULONG a
[P256_LIMBS
]);
106 /* Modular sub: res = a-b mod P */
107 void ecp_nistz256_sub(BN_ULONG res
[P256_LIMBS
],
108 const BN_ULONG a
[P256_LIMBS
],
109 const BN_ULONG b
[P256_LIMBS
]);
110 /* Modular neg: res = -a mod P */
111 void ecp_nistz256_neg(BN_ULONG res
[P256_LIMBS
], const BN_ULONG a
[P256_LIMBS
]);
112 /* Montgomery mul: res = a*b*2^-256 mod P */
113 void ecp_nistz256_mul_mont(BN_ULONG res
[P256_LIMBS
],
114 const BN_ULONG a
[P256_LIMBS
],
115 const BN_ULONG b
[P256_LIMBS
]);
116 /* Montgomery sqr: res = a*a*2^-256 mod P */
117 void ecp_nistz256_sqr_mont(BN_ULONG res
[P256_LIMBS
],
118 const BN_ULONG a
[P256_LIMBS
]);
119 /* Convert a number from Montgomery domain, by multiplying with 1 */
120 void ecp_nistz256_from_mont(BN_ULONG res
[P256_LIMBS
],
121 const BN_ULONG in
[P256_LIMBS
]);
122 /* Convert a number to Montgomery domain, by multiplying with 2^512 mod P*/
123 void ecp_nistz256_to_mont(BN_ULONG res
[P256_LIMBS
],
124 const BN_ULONG in
[P256_LIMBS
]);
125 /* Functions that perform constant time access to the precomputed tables */
126 void ecp_nistz256_scatter_w5(P256_POINT
*val
,
127 const P256_POINT
*in_t
, int idx
);
128 void ecp_nistz256_gather_w5(P256_POINT
*val
,
129 const P256_POINT
*in_t
, int idx
);
130 void ecp_nistz256_scatter_w7(P256_POINT_AFFINE
*val
,
131 const P256_POINT_AFFINE
*in_t
, int idx
);
132 void ecp_nistz256_gather_w7(P256_POINT_AFFINE
*val
,
133 const P256_POINT_AFFINE
*in_t
, int idx
);
135 /* One converted into the Montgomery domain */
136 static const BN_ULONG ONE
[P256_LIMBS
] = {
137 TOBN(0x00000000, 0x00000001), TOBN(0xffffffff, 0x00000000),
138 TOBN(0xffffffff, 0xffffffff), TOBN(0x00000000, 0xfffffffe)
141 static NISTZ256_PRE_COMP
*ecp_nistz256_pre_comp_new(const EC_GROUP
*group
);
143 /* Precomputed tables for the default generator */
144 extern const PRECOMP256_ROW ecp_nistz256_precomputed
[37];
146 /* Recode window to a signed digit, see ecp_nistputil.c for details */
147 static unsigned int _booth_recode_w5(unsigned int in
)
151 s
= ~((in
>> 5) - 1);
152 d
= (1 << 6) - in
- 1;
153 d
= (d
& s
) | (in
& ~s
);
154 d
= (d
>> 1) + (d
& 1);
156 return (d
<< 1) + (s
& 1);
159 static unsigned int _booth_recode_w7(unsigned int in
)
163 s
= ~((in
>> 7) - 1);
164 d
= (1 << 8) - in
- 1;
165 d
= (d
& s
) | (in
& ~s
);
166 d
= (d
>> 1) + (d
& 1);
168 return (d
<< 1) + (s
& 1);
171 static void copy_conditional(BN_ULONG dst
[P256_LIMBS
],
172 const BN_ULONG src
[P256_LIMBS
], BN_ULONG move
)
174 BN_ULONG mask1
= 0-move
;
175 BN_ULONG mask2
= ~mask1
;
177 dst
[0] = (src
[0] & mask1
) ^ (dst
[0] & mask2
);
178 dst
[1] = (src
[1] & mask1
) ^ (dst
[1] & mask2
);
179 dst
[2] = (src
[2] & mask1
) ^ (dst
[2] & mask2
);
180 dst
[3] = (src
[3] & mask1
) ^ (dst
[3] & mask2
);
181 if (P256_LIMBS
== 8) {
182 dst
[4] = (src
[4] & mask1
) ^ (dst
[4] & mask2
);
183 dst
[5] = (src
[5] & mask1
) ^ (dst
[5] & mask2
);
184 dst
[6] = (src
[6] & mask1
) ^ (dst
[6] & mask2
);
185 dst
[7] = (src
[7] & mask1
) ^ (dst
[7] & mask2
);
189 static BN_ULONG
is_zero(BN_ULONG in
)
197 static BN_ULONG
is_equal(const BN_ULONG a
[P256_LIMBS
],
198 const BN_ULONG b
[P256_LIMBS
])
206 if (P256_LIMBS
== 8) {
216 static BN_ULONG
is_one(const BIGNUM
*z
)
219 BN_ULONG
*a
= bn_get_words(z
);
221 if (bn_get_top(z
) == (P256_LIMBS
- P256_LIMBS
/ 8)) {
223 res
|= a
[1] ^ ONE
[1];
224 res
|= a
[2] ^ ONE
[2];
225 res
|= a
[3] ^ ONE
[3];
226 if (P256_LIMBS
== 8) {
227 res
|= a
[4] ^ ONE
[4];
228 res
|= a
[5] ^ ONE
[5];
229 res
|= a
[6] ^ ONE
[6];
231 * no check for a[7] (being zero) on 32-bit platforms,
232 * because value of "one" takes only 7 limbs.
242 * For reference, this macro is used only when new ecp_nistz256 assembly
243 * module is being developed. For example, configure with
244 * -DECP_NISTZ256_REFERENCE_IMPLEMENTATION and implement only functions
245 * performing simplest arithmetic operations on 256-bit vectors. Then
246 * work on implementation of higher-level functions performing point
247 * operations. Then remove ECP_NISTZ256_REFERENCE_IMPLEMENTATION
248 * and never define it again. (The correct macro denoting presence of
249 * ecp_nistz256 module is ECP_NISTZ256_ASM.)
251 #ifndef ECP_NISTZ256_REFERENCE_IMPLEMENTATION
252 void ecp_nistz256_point_double(P256_POINT
*r
, const P256_POINT
*a
);
253 void ecp_nistz256_point_add(P256_POINT
*r
,
254 const P256_POINT
*a
, const P256_POINT
*b
);
255 void ecp_nistz256_point_add_affine(P256_POINT
*r
,
257 const P256_POINT_AFFINE
*b
);
259 /* Point double: r = 2*a */
260 static void ecp_nistz256_point_double(P256_POINT
*r
, const P256_POINT
*a
)
262 BN_ULONG S
[P256_LIMBS
];
263 BN_ULONG M
[P256_LIMBS
];
264 BN_ULONG Zsqr
[P256_LIMBS
];
265 BN_ULONG tmp0
[P256_LIMBS
];
267 const BN_ULONG
*in_x
= a
->X
;
268 const BN_ULONG
*in_y
= a
->Y
;
269 const BN_ULONG
*in_z
= a
->Z
;
271 BN_ULONG
*res_x
= r
->X
;
272 BN_ULONG
*res_y
= r
->Y
;
273 BN_ULONG
*res_z
= r
->Z
;
275 ecp_nistz256_mul_by_2(S
, in_y
);
277 ecp_nistz256_sqr_mont(Zsqr
, in_z
);
279 ecp_nistz256_sqr_mont(S
, S
);
281 ecp_nistz256_mul_mont(res_z
, in_z
, in_y
);
282 ecp_nistz256_mul_by_2(res_z
, res_z
);
284 ecp_nistz256_add(M
, in_x
, Zsqr
);
285 ecp_nistz256_sub(Zsqr
, in_x
, Zsqr
);
287 ecp_nistz256_sqr_mont(res_y
, S
);
288 ecp_nistz256_div_by_2(res_y
, res_y
);
290 ecp_nistz256_mul_mont(M
, M
, Zsqr
);
291 ecp_nistz256_mul_by_3(M
, M
);
293 ecp_nistz256_mul_mont(S
, S
, in_x
);
294 ecp_nistz256_mul_by_2(tmp0
, S
);
296 ecp_nistz256_sqr_mont(res_x
, M
);
298 ecp_nistz256_sub(res_x
, res_x
, tmp0
);
299 ecp_nistz256_sub(S
, S
, res_x
);
301 ecp_nistz256_mul_mont(S
, S
, M
);
302 ecp_nistz256_sub(res_y
, S
, res_y
);
305 /* Point addition: r = a+b */
306 static void ecp_nistz256_point_add(P256_POINT
*r
,
307 const P256_POINT
*a
, const P256_POINT
*b
)
309 BN_ULONG U2
[P256_LIMBS
], S2
[P256_LIMBS
];
310 BN_ULONG U1
[P256_LIMBS
], S1
[P256_LIMBS
];
311 BN_ULONG Z1sqr
[P256_LIMBS
];
312 BN_ULONG Z2sqr
[P256_LIMBS
];
313 BN_ULONG H
[P256_LIMBS
], R
[P256_LIMBS
];
314 BN_ULONG Hsqr
[P256_LIMBS
];
315 BN_ULONG Rsqr
[P256_LIMBS
];
316 BN_ULONG Hcub
[P256_LIMBS
];
318 BN_ULONG res_x
[P256_LIMBS
];
319 BN_ULONG res_y
[P256_LIMBS
];
320 BN_ULONG res_z
[P256_LIMBS
];
322 BN_ULONG in1infty
, in2infty
;
324 const BN_ULONG
*in1_x
= a
->X
;
325 const BN_ULONG
*in1_y
= a
->Y
;
326 const BN_ULONG
*in1_z
= a
->Z
;
328 const BN_ULONG
*in2_x
= b
->X
;
329 const BN_ULONG
*in2_y
= b
->Y
;
330 const BN_ULONG
*in2_z
= b
->Z
;
333 * Infinity in encoded as (,,0)
335 in1infty
= (in1_z
[0] | in1_z
[1] | in1_z
[2] | in1_z
[3]);
337 in1infty
|= (in1_z
[4] | in1_z
[5] | in1_z
[6] | in1_z
[7]);
339 in2infty
= (in2_z
[0] | in2_z
[1] | in2_z
[2] | in2_z
[3]);
341 in2infty
|= (in2_z
[4] | in2_z
[5] | in2_z
[6] | in2_z
[7]);
343 in1infty
= is_zero(in1infty
);
344 in2infty
= is_zero(in2infty
);
346 ecp_nistz256_sqr_mont(Z2sqr
, in2_z
); /* Z2^2 */
347 ecp_nistz256_sqr_mont(Z1sqr
, in1_z
); /* Z1^2 */
349 ecp_nistz256_mul_mont(S1
, Z2sqr
, in2_z
); /* S1 = Z2^3 */
350 ecp_nistz256_mul_mont(S2
, Z1sqr
, in1_z
); /* S2 = Z1^3 */
352 ecp_nistz256_mul_mont(S1
, S1
, in1_y
); /* S1 = Y1*Z2^3 */
353 ecp_nistz256_mul_mont(S2
, S2
, in2_y
); /* S2 = Y2*Z1^3 */
354 ecp_nistz256_sub(R
, S2
, S1
); /* R = S2 - S1 */
356 ecp_nistz256_mul_mont(U1
, in1_x
, Z2sqr
); /* U1 = X1*Z2^2 */
357 ecp_nistz256_mul_mont(U2
, in2_x
, Z1sqr
); /* U2 = X2*Z1^2 */
358 ecp_nistz256_sub(H
, U2
, U1
); /* H = U2 - U1 */
361 * The formulae are incorrect if the points are equal so we check for
362 * this and do doubling if this happens.
364 * Points here are in Jacobian projective coordinates (Xi, Yi, Zi)
365 * that are bound to the affine coordinates (xi, yi) by the following
370 * For the sake of optimization, the algorithm operates over
371 * intermediate variables U1, U2 and S1, S2 that are derived from
372 * the projective coordinates:
373 * - U1 = X1 * (Z2)^2 ; U2 = X2 * (Z1)^2
374 * - S1 = Y1 * (Z2)^3 ; S2 = Y2 * (Z1)^3
376 * It is easy to prove that is_equal(U1, U2) implies that the affine
377 * x-coordinates are equal, or either point is at infinity.
378 * Likewise is_equal(S1, S2) implies that the affine y-coordinates are
379 * equal, or either point is at infinity.
381 * The special case of either point being the point at infinity (Z1 or Z2
382 * is zero), is handled separately later on in this function, so we avoid
383 * jumping to point_double here in those special cases.
385 * When both points are inverse of each other, we know that the affine
386 * x-coordinates are equal, and the y-coordinates have different sign.
387 * Therefore since U1 = U2, we know H = 0, and therefore Z3 = H*Z1*Z2
388 * will equal 0, thus the result is infinity, if we simply let this
389 * function continue normally.
391 * We use bitwise operations to avoid potential side-channels introduced by
392 * the short-circuiting behaviour of boolean operators.
394 if (is_equal(U1
, U2
) & ~in1infty
& ~in2infty
& is_equal(S1
, S2
)) {
396 * This is obviously not constant-time but it should never happen during
397 * single point multiplication, so there is no timing leak for ECDH or
400 ecp_nistz256_point_double(r
, a
);
404 ecp_nistz256_sqr_mont(Rsqr
, R
); /* R^2 */
405 ecp_nistz256_mul_mont(res_z
, H
, in1_z
); /* Z3 = H*Z1*Z2 */
406 ecp_nistz256_sqr_mont(Hsqr
, H
); /* H^2 */
407 ecp_nistz256_mul_mont(res_z
, res_z
, in2_z
); /* Z3 = H*Z1*Z2 */
408 ecp_nistz256_mul_mont(Hcub
, Hsqr
, H
); /* H^3 */
410 ecp_nistz256_mul_mont(U2
, U1
, Hsqr
); /* U1*H^2 */
411 ecp_nistz256_mul_by_2(Hsqr
, U2
); /* 2*U1*H^2 */
413 ecp_nistz256_sub(res_x
, Rsqr
, Hsqr
);
414 ecp_nistz256_sub(res_x
, res_x
, Hcub
);
416 ecp_nistz256_sub(res_y
, U2
, res_x
);
418 ecp_nistz256_mul_mont(S2
, S1
, Hcub
);
419 ecp_nistz256_mul_mont(res_y
, R
, res_y
);
420 ecp_nistz256_sub(res_y
, res_y
, S2
);
422 copy_conditional(res_x
, in2_x
, in1infty
);
423 copy_conditional(res_y
, in2_y
, in1infty
);
424 copy_conditional(res_z
, in2_z
, in1infty
);
426 copy_conditional(res_x
, in1_x
, in2infty
);
427 copy_conditional(res_y
, in1_y
, in2infty
);
428 copy_conditional(res_z
, in1_z
, in2infty
);
430 memcpy(r
->X
, res_x
, sizeof(res_x
));
431 memcpy(r
->Y
, res_y
, sizeof(res_y
));
432 memcpy(r
->Z
, res_z
, sizeof(res_z
));
435 /* Point addition when b is known to be affine: r = a+b */
436 static void ecp_nistz256_point_add_affine(P256_POINT
*r
,
438 const P256_POINT_AFFINE
*b
)
440 BN_ULONG U2
[P256_LIMBS
], S2
[P256_LIMBS
];
441 BN_ULONG Z1sqr
[P256_LIMBS
];
442 BN_ULONG H
[P256_LIMBS
], R
[P256_LIMBS
];
443 BN_ULONG Hsqr
[P256_LIMBS
];
444 BN_ULONG Rsqr
[P256_LIMBS
];
445 BN_ULONG Hcub
[P256_LIMBS
];
447 BN_ULONG res_x
[P256_LIMBS
];
448 BN_ULONG res_y
[P256_LIMBS
];
449 BN_ULONG res_z
[P256_LIMBS
];
451 BN_ULONG in1infty
, in2infty
;
453 const BN_ULONG
*in1_x
= a
->X
;
454 const BN_ULONG
*in1_y
= a
->Y
;
455 const BN_ULONG
*in1_z
= a
->Z
;
457 const BN_ULONG
*in2_x
= b
->X
;
458 const BN_ULONG
*in2_y
= b
->Y
;
461 * Infinity in encoded as (,,0)
463 in1infty
= (in1_z
[0] | in1_z
[1] | in1_z
[2] | in1_z
[3]);
465 in1infty
|= (in1_z
[4] | in1_z
[5] | in1_z
[6] | in1_z
[7]);
468 * In affine representation we encode infinity as (0,0), which is
469 * not on the curve, so it is OK
471 in2infty
= (in2_x
[0] | in2_x
[1] | in2_x
[2] | in2_x
[3] |
472 in2_y
[0] | in2_y
[1] | in2_y
[2] | in2_y
[3]);
474 in2infty
|= (in2_x
[4] | in2_x
[5] | in2_x
[6] | in2_x
[7] |
475 in2_y
[4] | in2_y
[5] | in2_y
[6] | in2_y
[7]);
477 in1infty
= is_zero(in1infty
);
478 in2infty
= is_zero(in2infty
);
480 ecp_nistz256_sqr_mont(Z1sqr
, in1_z
); /* Z1^2 */
482 ecp_nistz256_mul_mont(U2
, in2_x
, Z1sqr
); /* U2 = X2*Z1^2 */
483 ecp_nistz256_sub(H
, U2
, in1_x
); /* H = U2 - U1 */
485 ecp_nistz256_mul_mont(S2
, Z1sqr
, in1_z
); /* S2 = Z1^3 */
487 ecp_nistz256_mul_mont(res_z
, H
, in1_z
); /* Z3 = H*Z1*Z2 */
489 ecp_nistz256_mul_mont(S2
, S2
, in2_y
); /* S2 = Y2*Z1^3 */
490 ecp_nistz256_sub(R
, S2
, in1_y
); /* R = S2 - S1 */
492 ecp_nistz256_sqr_mont(Hsqr
, H
); /* H^2 */
493 ecp_nistz256_sqr_mont(Rsqr
, R
); /* R^2 */
494 ecp_nistz256_mul_mont(Hcub
, Hsqr
, H
); /* H^3 */
496 ecp_nistz256_mul_mont(U2
, in1_x
, Hsqr
); /* U1*H^2 */
497 ecp_nistz256_mul_by_2(Hsqr
, U2
); /* 2*U1*H^2 */
499 ecp_nistz256_sub(res_x
, Rsqr
, Hsqr
);
500 ecp_nistz256_sub(res_x
, res_x
, Hcub
);
501 ecp_nistz256_sub(H
, U2
, res_x
);
503 ecp_nistz256_mul_mont(S2
, in1_y
, Hcub
);
504 ecp_nistz256_mul_mont(H
, H
, R
);
505 ecp_nistz256_sub(res_y
, H
, S2
);
507 copy_conditional(res_x
, in2_x
, in1infty
);
508 copy_conditional(res_x
, in1_x
, in2infty
);
510 copy_conditional(res_y
, in2_y
, in1infty
);
511 copy_conditional(res_y
, in1_y
, in2infty
);
513 copy_conditional(res_z
, ONE
, in1infty
);
514 copy_conditional(res_z
, in1_z
, in2infty
);
516 memcpy(r
->X
, res_x
, sizeof(res_x
));
517 memcpy(r
->Y
, res_y
, sizeof(res_y
));
518 memcpy(r
->Z
, res_z
, sizeof(res_z
));
522 /* r = in^-1 mod p */
523 static void ecp_nistz256_mod_inverse(BN_ULONG r
[P256_LIMBS
],
524 const BN_ULONG in
[P256_LIMBS
])
527 * The poly is ffffffff 00000001 00000000 00000000 00000000 ffffffff
528 * ffffffff ffffffff We use FLT and used poly-2 as exponent
530 BN_ULONG p2
[P256_LIMBS
];
531 BN_ULONG p4
[P256_LIMBS
];
532 BN_ULONG p8
[P256_LIMBS
];
533 BN_ULONG p16
[P256_LIMBS
];
534 BN_ULONG p32
[P256_LIMBS
];
535 BN_ULONG res
[P256_LIMBS
];
538 ecp_nistz256_sqr_mont(res
, in
);
539 ecp_nistz256_mul_mont(p2
, res
, in
); /* 3*p */
541 ecp_nistz256_sqr_mont(res
, p2
);
542 ecp_nistz256_sqr_mont(res
, res
);
543 ecp_nistz256_mul_mont(p4
, res
, p2
); /* f*p */
545 ecp_nistz256_sqr_mont(res
, p4
);
546 ecp_nistz256_sqr_mont(res
, res
);
547 ecp_nistz256_sqr_mont(res
, res
);
548 ecp_nistz256_sqr_mont(res
, res
);
549 ecp_nistz256_mul_mont(p8
, res
, p4
); /* ff*p */
551 ecp_nistz256_sqr_mont(res
, p8
);
552 for (i
= 0; i
< 7; i
++)
553 ecp_nistz256_sqr_mont(res
, res
);
554 ecp_nistz256_mul_mont(p16
, res
, p8
); /* ffff*p */
556 ecp_nistz256_sqr_mont(res
, p16
);
557 for (i
= 0; i
< 15; i
++)
558 ecp_nistz256_sqr_mont(res
, res
);
559 ecp_nistz256_mul_mont(p32
, res
, p16
); /* ffffffff*p */
561 ecp_nistz256_sqr_mont(res
, p32
);
562 for (i
= 0; i
< 31; i
++)
563 ecp_nistz256_sqr_mont(res
, res
);
564 ecp_nistz256_mul_mont(res
, res
, in
);
566 for (i
= 0; i
< 32 * 4; i
++)
567 ecp_nistz256_sqr_mont(res
, res
);
568 ecp_nistz256_mul_mont(res
, res
, p32
);
570 for (i
= 0; i
< 32; i
++)
571 ecp_nistz256_sqr_mont(res
, res
);
572 ecp_nistz256_mul_mont(res
, res
, p32
);
574 for (i
= 0; i
< 16; i
++)
575 ecp_nistz256_sqr_mont(res
, res
);
576 ecp_nistz256_mul_mont(res
, res
, p16
);
578 for (i
= 0; i
< 8; i
++)
579 ecp_nistz256_sqr_mont(res
, res
);
580 ecp_nistz256_mul_mont(res
, res
, p8
);
582 ecp_nistz256_sqr_mont(res
, res
);
583 ecp_nistz256_sqr_mont(res
, res
);
584 ecp_nistz256_sqr_mont(res
, res
);
585 ecp_nistz256_sqr_mont(res
, res
);
586 ecp_nistz256_mul_mont(res
, res
, p4
);
588 ecp_nistz256_sqr_mont(res
, res
);
589 ecp_nistz256_sqr_mont(res
, res
);
590 ecp_nistz256_mul_mont(res
, res
, p2
);
592 ecp_nistz256_sqr_mont(res
, res
);
593 ecp_nistz256_sqr_mont(res
, res
);
594 ecp_nistz256_mul_mont(res
, res
, in
);
596 memcpy(r
, res
, sizeof(res
));
600 * ecp_nistz256_bignum_to_field_elem copies the contents of |in| to |out| and
601 * returns one if it fits. Otherwise it returns zero.
603 __owur
static int ecp_nistz256_bignum_to_field_elem(BN_ULONG out
[P256_LIMBS
],
606 return bn_copy_words(out
, in
, P256_LIMBS
);
609 /* r = sum(scalar[i]*point[i]) */
610 __owur
static int ecp_nistz256_windowed_mul(const EC_GROUP
*group
,
612 const BIGNUM
**scalar
,
613 const EC_POINT
**point
,
614 size_t num
, BN_CTX
*ctx
)
619 unsigned char (*p_str
)[33] = NULL
;
620 const unsigned int window_size
= 5;
621 const unsigned int mask
= (1 << (window_size
+ 1)) - 1;
623 P256_POINT
*temp
; /* place for 5 temporary points */
624 const BIGNUM
**scalars
= NULL
;
625 P256_POINT (*table
)[16] = NULL
;
626 void *table_storage
= NULL
;
628 if ((num
* 16 + 6) > OPENSSL_MALLOC_MAX_NELEMS(P256_POINT
)
630 OPENSSL_malloc((num
* 16 + 5) * sizeof(P256_POINT
) + 64)) == NULL
632 OPENSSL_malloc(num
* 33 * sizeof(unsigned char))) == NULL
633 || (scalars
= OPENSSL_malloc(num
* sizeof(BIGNUM
*))) == NULL
) {
634 ECerr(EC_F_ECP_NISTZ256_WINDOWED_MUL
, ERR_R_MALLOC_FAILURE
);
638 table
= (void *)ALIGNPTR(table_storage
, 64);
639 temp
= (P256_POINT
*)(table
+ num
);
641 for (i
= 0; i
< num
; i
++) {
642 P256_POINT
*row
= table
[i
];
644 /* This is an unusual input, we don't guarantee constant-timeness. */
645 if ((BN_num_bits(scalar
[i
]) > 256) || BN_is_negative(scalar
[i
])) {
648 if ((mod
= BN_CTX_get(ctx
)) == NULL
)
650 if (!BN_nnmod(mod
, scalar
[i
], group
->order
, ctx
)) {
651 ECerr(EC_F_ECP_NISTZ256_WINDOWED_MUL
, ERR_R_BN_LIB
);
656 scalars
[i
] = scalar
[i
];
658 for (j
= 0; j
< bn_get_top(scalars
[i
]) * BN_BYTES
; j
+= BN_BYTES
) {
659 BN_ULONG d
= bn_get_words(scalars
[i
])[j
/ BN_BYTES
];
661 p_str
[i
][j
+ 0] = (unsigned char)d
;
662 p_str
[i
][j
+ 1] = (unsigned char)(d
>> 8);
663 p_str
[i
][j
+ 2] = (unsigned char)(d
>> 16);
664 p_str
[i
][j
+ 3] = (unsigned char)(d
>>= 24);
667 p_str
[i
][j
+ 4] = (unsigned char)d
;
668 p_str
[i
][j
+ 5] = (unsigned char)(d
>> 8);
669 p_str
[i
][j
+ 6] = (unsigned char)(d
>> 16);
670 p_str
[i
][j
+ 7] = (unsigned char)(d
>> 24);
676 if (!ecp_nistz256_bignum_to_field_elem(temp
[0].X
, point
[i
]->X
)
677 || !ecp_nistz256_bignum_to_field_elem(temp
[0].Y
, point
[i
]->Y
)
678 || !ecp_nistz256_bignum_to_field_elem(temp
[0].Z
, point
[i
]->Z
)) {
679 ECerr(EC_F_ECP_NISTZ256_WINDOWED_MUL
,
680 EC_R_COORDINATES_OUT_OF_RANGE
);
685 * row[0] is implicitly (0,0,0) (the point at infinity), therefore it
686 * is not stored. All other values are actually stored with an offset
690 ecp_nistz256_scatter_w5 (row
, &temp
[0], 1);
691 ecp_nistz256_point_double(&temp
[1], &temp
[0]); /*1+1=2 */
692 ecp_nistz256_scatter_w5 (row
, &temp
[1], 2);
693 ecp_nistz256_point_add (&temp
[2], &temp
[1], &temp
[0]); /*2+1=3 */
694 ecp_nistz256_scatter_w5 (row
, &temp
[2], 3);
695 ecp_nistz256_point_double(&temp
[1], &temp
[1]); /*2*2=4 */
696 ecp_nistz256_scatter_w5 (row
, &temp
[1], 4);
697 ecp_nistz256_point_double(&temp
[2], &temp
[2]); /*2*3=6 */
698 ecp_nistz256_scatter_w5 (row
, &temp
[2], 6);
699 ecp_nistz256_point_add (&temp
[3], &temp
[1], &temp
[0]); /*4+1=5 */
700 ecp_nistz256_scatter_w5 (row
, &temp
[3], 5);
701 ecp_nistz256_point_add (&temp
[4], &temp
[2], &temp
[0]); /*6+1=7 */
702 ecp_nistz256_scatter_w5 (row
, &temp
[4], 7);
703 ecp_nistz256_point_double(&temp
[1], &temp
[1]); /*2*4=8 */
704 ecp_nistz256_scatter_w5 (row
, &temp
[1], 8);
705 ecp_nistz256_point_double(&temp
[2], &temp
[2]); /*2*6=12 */
706 ecp_nistz256_scatter_w5 (row
, &temp
[2], 12);
707 ecp_nistz256_point_double(&temp
[3], &temp
[3]); /*2*5=10 */
708 ecp_nistz256_scatter_w5 (row
, &temp
[3], 10);
709 ecp_nistz256_point_double(&temp
[4], &temp
[4]); /*2*7=14 */
710 ecp_nistz256_scatter_w5 (row
, &temp
[4], 14);
711 ecp_nistz256_point_add (&temp
[2], &temp
[2], &temp
[0]); /*12+1=13*/
712 ecp_nistz256_scatter_w5 (row
, &temp
[2], 13);
713 ecp_nistz256_point_add (&temp
[3], &temp
[3], &temp
[0]); /*10+1=11*/
714 ecp_nistz256_scatter_w5 (row
, &temp
[3], 11);
715 ecp_nistz256_point_add (&temp
[4], &temp
[4], &temp
[0]); /*14+1=15*/
716 ecp_nistz256_scatter_w5 (row
, &temp
[4], 15);
717 ecp_nistz256_point_add (&temp
[2], &temp
[1], &temp
[0]); /*8+1=9 */
718 ecp_nistz256_scatter_w5 (row
, &temp
[2], 9);
719 ecp_nistz256_point_double(&temp
[1], &temp
[1]); /*2*8=16 */
720 ecp_nistz256_scatter_w5 (row
, &temp
[1], 16);
725 wvalue
= p_str
[0][(idx
- 1) / 8];
726 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
729 * We gather to temp[0], because we know it's position relative
732 ecp_nistz256_gather_w5(&temp
[0], table
[0], _booth_recode_w5(wvalue
) >> 1);
733 memcpy(r
, &temp
[0], sizeof(temp
[0]));
736 for (i
= (idx
== 255 ? 1 : 0); i
< num
; i
++) {
737 unsigned int off
= (idx
- 1) / 8;
739 wvalue
= p_str
[i
][off
] | p_str
[i
][off
+ 1] << 8;
740 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
742 wvalue
= _booth_recode_w5(wvalue
);
744 ecp_nistz256_gather_w5(&temp
[0], table
[i
], wvalue
>> 1);
746 ecp_nistz256_neg(temp
[1].Y
, temp
[0].Y
);
747 copy_conditional(temp
[0].Y
, temp
[1].Y
, (wvalue
& 1));
749 ecp_nistz256_point_add(r
, r
, &temp
[0]);
754 ecp_nistz256_point_double(r
, r
);
755 ecp_nistz256_point_double(r
, r
);
756 ecp_nistz256_point_double(r
, r
);
757 ecp_nistz256_point_double(r
, r
);
758 ecp_nistz256_point_double(r
, r
);
762 for (i
= 0; i
< num
; i
++) {
763 wvalue
= p_str
[i
][0];
764 wvalue
= (wvalue
<< 1) & mask
;
766 wvalue
= _booth_recode_w5(wvalue
);
768 ecp_nistz256_gather_w5(&temp
[0], table
[i
], wvalue
>> 1);
770 ecp_nistz256_neg(temp
[1].Y
, temp
[0].Y
);
771 copy_conditional(temp
[0].Y
, temp
[1].Y
, wvalue
& 1);
773 ecp_nistz256_point_add(r
, r
, &temp
[0]);
778 OPENSSL_free(table_storage
);
780 OPENSSL_free(scalars
);
784 /* Coordinates of G, for which we have precomputed tables */
785 static const BN_ULONG def_xG
[P256_LIMBS
] = {
786 TOBN(0x79e730d4, 0x18a9143c), TOBN(0x75ba95fc, 0x5fedb601),
787 TOBN(0x79fb732b, 0x77622510), TOBN(0x18905f76, 0xa53755c6)
790 static const BN_ULONG def_yG
[P256_LIMBS
] = {
791 TOBN(0xddf25357, 0xce95560a), TOBN(0x8b4ab8e4, 0xba19e45c),
792 TOBN(0xd2e88688, 0xdd21f325), TOBN(0x8571ff18, 0x25885d85)
796 * ecp_nistz256_is_affine_G returns one if |generator| is the standard, P-256
799 static int ecp_nistz256_is_affine_G(const EC_POINT
*generator
)
801 return (bn_get_top(generator
->X
) == P256_LIMBS
) &&
802 (bn_get_top(generator
->Y
) == P256_LIMBS
) &&
803 is_equal(bn_get_words(generator
->X
), def_xG
) &&
804 is_equal(bn_get_words(generator
->Y
), def_yG
) &&
805 is_one(generator
->Z
);
808 __owur
static int ecp_nistz256_mult_precompute(EC_GROUP
*group
, BN_CTX
*ctx
)
811 * We precompute a table for a Booth encoded exponent (wNAF) based
812 * computation. Each table holds 64 values for safe access, with an
813 * implicit value of infinity at index zero. We use window of size 7, and
814 * therefore require ceil(256/7) = 37 tables.
817 EC_POINT
*P
= NULL
, *T
= NULL
;
818 const EC_POINT
*generator
;
819 NISTZ256_PRE_COMP
*pre_comp
;
820 BN_CTX
*new_ctx
= NULL
;
821 int i
, j
, k
, ret
= 0;
824 PRECOMP256_ROW
*preComputedTable
= NULL
;
825 unsigned char *precomp_storage
= NULL
;
827 /* if there is an old NISTZ256_PRE_COMP object, throw it away */
828 EC_pre_comp_free(group
);
829 generator
= EC_GROUP_get0_generator(group
);
830 if (generator
== NULL
) {
831 ECerr(EC_F_ECP_NISTZ256_MULT_PRECOMPUTE
, EC_R_UNDEFINED_GENERATOR
);
835 if (ecp_nistz256_is_affine_G(generator
)) {
837 * No need to calculate tables for the standard generator because we
838 * have them statically.
843 if ((pre_comp
= ecp_nistz256_pre_comp_new(group
)) == NULL
)
847 ctx
= new_ctx
= BN_CTX_new();
854 order
= EC_GROUP_get0_order(group
);
858 if (BN_is_zero(order
)) {
859 ECerr(EC_F_ECP_NISTZ256_MULT_PRECOMPUTE
, EC_R_UNKNOWN_ORDER
);
865 if ((precomp_storage
=
866 OPENSSL_malloc(37 * 64 * sizeof(P256_POINT_AFFINE
) + 64)) == NULL
) {
867 ECerr(EC_F_ECP_NISTZ256_MULT_PRECOMPUTE
, ERR_R_MALLOC_FAILURE
);
871 preComputedTable
= (void *)ALIGNPTR(precomp_storage
, 64);
873 P
= EC_POINT_new(group
);
874 T
= EC_POINT_new(group
);
875 if (P
== NULL
|| T
== NULL
)
879 * The zero entry is implicitly infinity, and we skip it, storing other
880 * values with -1 offset.
882 if (!EC_POINT_copy(T
, generator
))
885 for (k
= 0; k
< 64; k
++) {
886 if (!EC_POINT_copy(P
, T
))
888 for (j
= 0; j
< 37; j
++) {
889 P256_POINT_AFFINE temp
;
891 * It would be faster to use EC_POINTs_make_affine and
892 * make multiple points affine at the same time.
894 if (!EC_POINT_make_affine(group
, P
, ctx
))
896 if (!ecp_nistz256_bignum_to_field_elem(temp
.X
, P
->X
) ||
897 !ecp_nistz256_bignum_to_field_elem(temp
.Y
, P
->Y
)) {
898 ECerr(EC_F_ECP_NISTZ256_MULT_PRECOMPUTE
,
899 EC_R_COORDINATES_OUT_OF_RANGE
);
902 ecp_nistz256_scatter_w7(preComputedTable
[j
], &temp
, k
);
903 for (i
= 0; i
< 7; i
++) {
904 if (!EC_POINT_dbl(group
, P
, P
, ctx
))
908 if (!EC_POINT_add(group
, T
, T
, generator
, ctx
))
912 pre_comp
->group
= group
;
914 pre_comp
->precomp
= preComputedTable
;
915 pre_comp
->precomp_storage
= precomp_storage
;
916 precomp_storage
= NULL
;
917 SETPRECOMP(group
, nistz256
, pre_comp
);
923 BN_CTX_free(new_ctx
);
925 EC_nistz256_pre_comp_free(pre_comp
);
926 OPENSSL_free(precomp_storage
);
933 * Note that by default ECP_NISTZ256_AVX2 is undefined. While it's great
934 * code processing 4 points in parallel, corresponding serial operation
935 * is several times slower, because it uses 29x29=58-bit multiplication
936 * as opposite to 64x64=128-bit in integer-only scalar case. As result
937 * it doesn't provide *significant* performance improvement. Note that
938 * just defining ECP_NISTZ256_AVX2 is not sufficient to make it work,
939 * you'd need to compile even asm/ecp_nistz256-avx.pl module.
941 #if defined(ECP_NISTZ256_AVX2)
942 # if !(defined(__x86_64) || defined(__x86_64__) || \
943 defined(_M_AMD64) || defined(_M_X64)) || \
944 !(defined(__GNUC__) || defined(_MSC_VER)) /* this is for ALIGN32 */
945 # undef ECP_NISTZ256_AVX2
947 /* Constant time access, loading four values, from four consecutive tables */
948 void ecp_nistz256_avx2_multi_gather_w7(void *result
, const void *in
,
949 int index0
, int index1
, int index2
,
951 void ecp_nistz256_avx2_transpose_convert(void *RESULTx4
, const void *in
);
952 void ecp_nistz256_avx2_convert_transpose_back(void *result
, const void *Ax4
);
953 void ecp_nistz256_avx2_point_add_affine_x4(void *RESULTx4
, const void *Ax4
,
955 void ecp_nistz256_avx2_point_add_affines_x4(void *RESULTx4
, const void *Ax4
,
957 void ecp_nistz256_avx2_to_mont(void *RESULTx4
, const void *Ax4
);
958 void ecp_nistz256_avx2_from_mont(void *RESULTx4
, const void *Ax4
);
959 void ecp_nistz256_avx2_set1(void *RESULTx4
);
960 int ecp_nistz_avx2_eligible(void);
962 static void booth_recode_w7(unsigned char *sign
,
963 unsigned char *digit
, unsigned char in
)
967 s
= ~((in
>> 7) - 1);
968 d
= (1 << 8) - in
- 1;
969 d
= (d
& s
) | (in
& ~s
);
970 d
= (d
>> 1) + (d
& 1);
977 * ecp_nistz256_avx2_mul_g performs multiplication by G, using only the
978 * precomputed table. It does 4 affine point additions in parallel,
979 * significantly speeding up point multiplication for a fixed value.
981 static void ecp_nistz256_avx2_mul_g(P256_POINT
*r
,
982 unsigned char p_str
[33],
983 const P256_POINT_AFFINE(*preComputedTable
)[64])
985 const unsigned int window_size
= 7;
986 const unsigned int mask
= (1 << (window_size
+ 1)) - 1;
988 /* Using 4 windows at a time */
989 unsigned char sign0
, digit0
;
990 unsigned char sign1
, digit1
;
991 unsigned char sign2
, digit2
;
992 unsigned char sign3
, digit3
;
993 unsigned int idx
= 0;
994 BN_ULONG tmp
[P256_LIMBS
];
997 ALIGN32 BN_ULONG aX4
[4 * 9 * 3] = { 0 };
998 ALIGN32 BN_ULONG bX4
[4 * 9 * 2] = { 0 };
999 ALIGN32 P256_POINT_AFFINE point_arr
[4];
1000 ALIGN32 P256_POINT res_point_arr
[4];
1002 /* Initial four windows */
1003 wvalue
= *((u16
*) & p_str
[0]);
1004 wvalue
= (wvalue
<< 1) & mask
;
1006 booth_recode_w7(&sign0
, &digit0
, wvalue
);
1007 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1008 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1010 booth_recode_w7(&sign1
, &digit1
, wvalue
);
1011 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1012 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1014 booth_recode_w7(&sign2
, &digit2
, wvalue
);
1015 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1016 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1018 booth_recode_w7(&sign3
, &digit3
, wvalue
);
1020 ecp_nistz256_avx2_multi_gather_w7(point_arr
, preComputedTable
[0],
1021 digit0
, digit1
, digit2
, digit3
);
1023 ecp_nistz256_neg(tmp
, point_arr
[0].Y
);
1024 copy_conditional(point_arr
[0].Y
, tmp
, sign0
);
1025 ecp_nistz256_neg(tmp
, point_arr
[1].Y
);
1026 copy_conditional(point_arr
[1].Y
, tmp
, sign1
);
1027 ecp_nistz256_neg(tmp
, point_arr
[2].Y
);
1028 copy_conditional(point_arr
[2].Y
, tmp
, sign2
);
1029 ecp_nistz256_neg(tmp
, point_arr
[3].Y
);
1030 copy_conditional(point_arr
[3].Y
, tmp
, sign3
);
1032 ecp_nistz256_avx2_transpose_convert(aX4
, point_arr
);
1033 ecp_nistz256_avx2_to_mont(aX4
, aX4
);
1034 ecp_nistz256_avx2_to_mont(&aX4
[4 * 9], &aX4
[4 * 9]);
1035 ecp_nistz256_avx2_set1(&aX4
[4 * 9 * 2]);
1037 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1038 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1040 booth_recode_w7(&sign0
, &digit0
, wvalue
);
1041 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1042 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1044 booth_recode_w7(&sign1
, &digit1
, wvalue
);
1045 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1046 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1048 booth_recode_w7(&sign2
, &digit2
, wvalue
);
1049 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1050 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1052 booth_recode_w7(&sign3
, &digit3
, wvalue
);
1054 ecp_nistz256_avx2_multi_gather_w7(point_arr
, preComputedTable
[4 * 1],
1055 digit0
, digit1
, digit2
, digit3
);
1057 ecp_nistz256_neg(tmp
, point_arr
[0].Y
);
1058 copy_conditional(point_arr
[0].Y
, tmp
, sign0
);
1059 ecp_nistz256_neg(tmp
, point_arr
[1].Y
);
1060 copy_conditional(point_arr
[1].Y
, tmp
, sign1
);
1061 ecp_nistz256_neg(tmp
, point_arr
[2].Y
);
1062 copy_conditional(point_arr
[2].Y
, tmp
, sign2
);
1063 ecp_nistz256_neg(tmp
, point_arr
[3].Y
);
1064 copy_conditional(point_arr
[3].Y
, tmp
, sign3
);
1066 ecp_nistz256_avx2_transpose_convert(bX4
, point_arr
);
1067 ecp_nistz256_avx2_to_mont(bX4
, bX4
);
1068 ecp_nistz256_avx2_to_mont(&bX4
[4 * 9], &bX4
[4 * 9]);
1069 /* Optimized when both inputs are affine */
1070 ecp_nistz256_avx2_point_add_affines_x4(aX4
, aX4
, bX4
);
1072 for (i
= 2; i
< 9; i
++) {
1073 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1074 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1076 booth_recode_w7(&sign0
, &digit0
, wvalue
);
1077 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1078 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1080 booth_recode_w7(&sign1
, &digit1
, wvalue
);
1081 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1082 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1084 booth_recode_w7(&sign2
, &digit2
, wvalue
);
1085 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1086 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1088 booth_recode_w7(&sign3
, &digit3
, wvalue
);
1090 ecp_nistz256_avx2_multi_gather_w7(point_arr
,
1091 preComputedTable
[4 * i
],
1092 digit0
, digit1
, digit2
, digit3
);
1094 ecp_nistz256_neg(tmp
, point_arr
[0].Y
);
1095 copy_conditional(point_arr
[0].Y
, tmp
, sign0
);
1096 ecp_nistz256_neg(tmp
, point_arr
[1].Y
);
1097 copy_conditional(point_arr
[1].Y
, tmp
, sign1
);
1098 ecp_nistz256_neg(tmp
, point_arr
[2].Y
);
1099 copy_conditional(point_arr
[2].Y
, tmp
, sign2
);
1100 ecp_nistz256_neg(tmp
, point_arr
[3].Y
);
1101 copy_conditional(point_arr
[3].Y
, tmp
, sign3
);
1103 ecp_nistz256_avx2_transpose_convert(bX4
, point_arr
);
1104 ecp_nistz256_avx2_to_mont(bX4
, bX4
);
1105 ecp_nistz256_avx2_to_mont(&bX4
[4 * 9], &bX4
[4 * 9]);
1107 ecp_nistz256_avx2_point_add_affine_x4(aX4
, aX4
, bX4
);
1110 ecp_nistz256_avx2_from_mont(&aX4
[4 * 9 * 0], &aX4
[4 * 9 * 0]);
1111 ecp_nistz256_avx2_from_mont(&aX4
[4 * 9 * 1], &aX4
[4 * 9 * 1]);
1112 ecp_nistz256_avx2_from_mont(&aX4
[4 * 9 * 2], &aX4
[4 * 9 * 2]);
1114 ecp_nistz256_avx2_convert_transpose_back(res_point_arr
, aX4
);
1115 /* Last window is performed serially */
1116 wvalue
= *((u16
*) & p_str
[(idx
- 1) / 8]);
1117 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1118 booth_recode_w7(&sign0
, &digit0
, wvalue
);
1119 ecp_nistz256_gather_w7((P256_POINT_AFFINE
*)r
,
1120 preComputedTable
[36], digit0
);
1121 ecp_nistz256_neg(tmp
, r
->Y
);
1122 copy_conditional(r
->Y
, tmp
, sign0
);
1123 memcpy(r
->Z
, ONE
, sizeof(ONE
));
1124 /* Sum the four windows */
1125 ecp_nistz256_point_add(r
, r
, &res_point_arr
[0]);
1126 ecp_nistz256_point_add(r
, r
, &res_point_arr
[1]);
1127 ecp_nistz256_point_add(r
, r
, &res_point_arr
[2]);
1128 ecp_nistz256_point_add(r
, r
, &res_point_arr
[3]);
1133 __owur
static int ecp_nistz256_set_from_affine(EC_POINT
*out
, const EC_GROUP
*group
,
1134 const P256_POINT_AFFINE
*in
,
1139 if ((ret
= bn_set_words(out
->X
, in
->X
, P256_LIMBS
))
1140 && (ret
= bn_set_words(out
->Y
, in
->Y
, P256_LIMBS
))
1141 && (ret
= bn_set_words(out
->Z
, ONE
, P256_LIMBS
)))
1147 /* r = scalar*G + sum(scalars[i]*points[i]) */
1148 __owur
static int ecp_nistz256_points_mul(const EC_GROUP
*group
,
1150 const BIGNUM
*scalar
,
1152 const EC_POINT
*points
[],
1153 const BIGNUM
*scalars
[], BN_CTX
*ctx
)
1155 int i
= 0, ret
= 0, no_precomp_for_generator
= 0, p_is_infinity
= 0;
1156 unsigned char p_str
[33] = { 0 };
1157 const PRECOMP256_ROW
*preComputedTable
= NULL
;
1158 const NISTZ256_PRE_COMP
*pre_comp
= NULL
;
1159 const EC_POINT
*generator
= NULL
;
1160 const BIGNUM
**new_scalars
= NULL
;
1161 const EC_POINT
**new_points
= NULL
;
1162 unsigned int idx
= 0;
1163 const unsigned int window_size
= 7;
1164 const unsigned int mask
= (1 << (window_size
+ 1)) - 1;
1165 unsigned int wvalue
;
1168 P256_POINT_AFFINE a
;
1172 if ((num
+ 1) == 0 || (num
+ 1) > OPENSSL_MALLOC_MAX_NELEMS(void *)) {
1173 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL
, ERR_R_MALLOC_FAILURE
);
1180 generator
= EC_GROUP_get0_generator(group
);
1181 if (generator
== NULL
) {
1182 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL
, EC_R_UNDEFINED_GENERATOR
);
1186 /* look if we can use precomputed multiples of generator */
1187 pre_comp
= group
->pre_comp
.nistz256
;
1191 * If there is a precomputed table for the generator, check that
1192 * it was generated with the same generator.
1194 EC_POINT
*pre_comp_generator
= EC_POINT_new(group
);
1195 if (pre_comp_generator
== NULL
)
1198 ecp_nistz256_gather_w7(&p
.a
, pre_comp
->precomp
[0], 1);
1199 if (!ecp_nistz256_set_from_affine(pre_comp_generator
,
1200 group
, &p
.a
, ctx
)) {
1201 EC_POINT_free(pre_comp_generator
);
1205 if (0 == EC_POINT_cmp(group
, generator
, pre_comp_generator
, ctx
))
1206 preComputedTable
= (const PRECOMP256_ROW
*)pre_comp
->precomp
;
1208 EC_POINT_free(pre_comp_generator
);
1211 if (preComputedTable
== NULL
&& ecp_nistz256_is_affine_G(generator
)) {
1213 * If there is no precomputed data, but the generator is the
1214 * default, a hardcoded table of precomputed data is used. This
1215 * is because applications, such as Apache, do not use
1216 * EC_KEY_precompute_mult.
1218 preComputedTable
= ecp_nistz256_precomputed
;
1221 if (preComputedTable
) {
1222 if ((BN_num_bits(scalar
) > 256)
1223 || BN_is_negative(scalar
)) {
1224 if ((tmp_scalar
= BN_CTX_get(ctx
)) == NULL
)
1227 if (!BN_nnmod(tmp_scalar
, scalar
, group
->order
, ctx
)) {
1228 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL
, ERR_R_BN_LIB
);
1231 scalar
= tmp_scalar
;
1234 for (i
= 0; i
< bn_get_top(scalar
) * BN_BYTES
; i
+= BN_BYTES
) {
1235 BN_ULONG d
= bn_get_words(scalar
)[i
/ BN_BYTES
];
1237 p_str
[i
+ 0] = (unsigned char)d
;
1238 p_str
[i
+ 1] = (unsigned char)(d
>> 8);
1239 p_str
[i
+ 2] = (unsigned char)(d
>> 16);
1240 p_str
[i
+ 3] = (unsigned char)(d
>>= 24);
1241 if (BN_BYTES
== 8) {
1243 p_str
[i
+ 4] = (unsigned char)d
;
1244 p_str
[i
+ 5] = (unsigned char)(d
>> 8);
1245 p_str
[i
+ 6] = (unsigned char)(d
>> 16);
1246 p_str
[i
+ 7] = (unsigned char)(d
>> 24);
1253 #if defined(ECP_NISTZ256_AVX2)
1254 if (ecp_nistz_avx2_eligible()) {
1255 ecp_nistz256_avx2_mul_g(&p
.p
, p_str
, preComputedTable
);
1262 wvalue
= (p_str
[0] << 1) & mask
;
1265 wvalue
= _booth_recode_w7(wvalue
);
1267 ecp_nistz256_gather_w7(&p
.a
, preComputedTable
[0],
1270 ecp_nistz256_neg(p
.p
.Z
, p
.p
.Y
);
1271 copy_conditional(p
.p
.Y
, p
.p
.Z
, wvalue
& 1);
1274 * Since affine infinity is encoded as (0,0) and
1275 * Jacobian ias (,,0), we need to harmonize them
1276 * by assigning "one" or zero to Z.
1278 infty
= (p
.p
.X
[0] | p
.p
.X
[1] | p
.p
.X
[2] | p
.p
.X
[3] |
1279 p
.p
.Y
[0] | p
.p
.Y
[1] | p
.p
.Y
[2] | p
.p
.Y
[3]);
1280 if (P256_LIMBS
== 8)
1281 infty
|= (p
.p
.X
[4] | p
.p
.X
[5] | p
.p
.X
[6] | p
.p
.X
[7] |
1282 p
.p
.Y
[4] | p
.p
.Y
[5] | p
.p
.Y
[6] | p
.p
.Y
[7]);
1284 infty
= 0 - is_zero(infty
);
1287 p
.p
.Z
[0] = ONE
[0] & infty
;
1288 p
.p
.Z
[1] = ONE
[1] & infty
;
1289 p
.p
.Z
[2] = ONE
[2] & infty
;
1290 p
.p
.Z
[3] = ONE
[3] & infty
;
1291 if (P256_LIMBS
== 8) {
1292 p
.p
.Z
[4] = ONE
[4] & infty
;
1293 p
.p
.Z
[5] = ONE
[5] & infty
;
1294 p
.p
.Z
[6] = ONE
[6] & infty
;
1295 p
.p
.Z
[7] = ONE
[7] & infty
;
1298 for (i
= 1; i
< 37; i
++) {
1299 unsigned int off
= (idx
- 1) / 8;
1300 wvalue
= p_str
[off
] | p_str
[off
+ 1] << 8;
1301 wvalue
= (wvalue
>> ((idx
- 1) % 8)) & mask
;
1304 wvalue
= _booth_recode_w7(wvalue
);
1306 ecp_nistz256_gather_w7(&t
.a
,
1307 preComputedTable
[i
], wvalue
>> 1);
1309 ecp_nistz256_neg(t
.p
.Z
, t
.a
.Y
);
1310 copy_conditional(t
.a
.Y
, t
.p
.Z
, wvalue
& 1);
1312 ecp_nistz256_point_add_affine(&p
.p
, &p
.p
, &t
.a
);
1317 no_precomp_for_generator
= 1;
1322 if (no_precomp_for_generator
) {
1324 * Without a precomputed table for the generator, it has to be
1325 * handled like a normal point.
1327 new_scalars
= OPENSSL_malloc((num
+ 1) * sizeof(BIGNUM
*));
1328 if (new_scalars
== NULL
) {
1329 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL
, ERR_R_MALLOC_FAILURE
);
1333 new_points
= OPENSSL_malloc((num
+ 1) * sizeof(EC_POINT
*));
1334 if (new_points
== NULL
) {
1335 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL
, ERR_R_MALLOC_FAILURE
);
1339 memcpy(new_scalars
, scalars
, num
* sizeof(BIGNUM
*));
1340 new_scalars
[num
] = scalar
;
1341 memcpy(new_points
, points
, num
* sizeof(EC_POINT
*));
1342 new_points
[num
] = generator
;
1344 scalars
= new_scalars
;
1345 points
= new_points
;
1350 P256_POINT
*out
= &t
.p
;
1354 if (!ecp_nistz256_windowed_mul(group
, out
, scalars
, points
, num
, ctx
))
1358 ecp_nistz256_point_add(&p
.p
, &p
.p
, out
);
1361 /* Not constant-time, but we're only operating on the public output. */
1362 if (!bn_set_words(r
->X
, p
.p
.X
, P256_LIMBS
) ||
1363 !bn_set_words(r
->Y
, p
.p
.Y
, P256_LIMBS
) ||
1364 !bn_set_words(r
->Z
, p
.p
.Z
, P256_LIMBS
)) {
1367 r
->Z_is_one
= is_one(r
->Z
) & 1;
1373 OPENSSL_free(new_points
);
1374 OPENSSL_free(new_scalars
);
1378 __owur
static int ecp_nistz256_get_affine(const EC_GROUP
*group
,
1379 const EC_POINT
*point
,
1380 BIGNUM
*x
, BIGNUM
*y
, BN_CTX
*ctx
)
1382 BN_ULONG z_inv2
[P256_LIMBS
];
1383 BN_ULONG z_inv3
[P256_LIMBS
];
1384 BN_ULONG x_aff
[P256_LIMBS
];
1385 BN_ULONG y_aff
[P256_LIMBS
];
1386 BN_ULONG point_x
[P256_LIMBS
], point_y
[P256_LIMBS
], point_z
[P256_LIMBS
];
1387 BN_ULONG x_ret
[P256_LIMBS
], y_ret
[P256_LIMBS
];
1389 if (EC_POINT_is_at_infinity(group
, point
)) {
1390 ECerr(EC_F_ECP_NISTZ256_GET_AFFINE
, EC_R_POINT_AT_INFINITY
);
1394 if (!ecp_nistz256_bignum_to_field_elem(point_x
, point
->X
) ||
1395 !ecp_nistz256_bignum_to_field_elem(point_y
, point
->Y
) ||
1396 !ecp_nistz256_bignum_to_field_elem(point_z
, point
->Z
)) {
1397 ECerr(EC_F_ECP_NISTZ256_GET_AFFINE
, EC_R_COORDINATES_OUT_OF_RANGE
);
1401 ecp_nistz256_mod_inverse(z_inv3
, point_z
);
1402 ecp_nistz256_sqr_mont(z_inv2
, z_inv3
);
1403 ecp_nistz256_mul_mont(x_aff
, z_inv2
, point_x
);
1406 ecp_nistz256_from_mont(x_ret
, x_aff
);
1407 if (!bn_set_words(x
, x_ret
, P256_LIMBS
))
1412 ecp_nistz256_mul_mont(z_inv3
, z_inv3
, z_inv2
);
1413 ecp_nistz256_mul_mont(y_aff
, z_inv3
, point_y
);
1414 ecp_nistz256_from_mont(y_ret
, y_aff
);
1415 if (!bn_set_words(y
, y_ret
, P256_LIMBS
))
1422 static NISTZ256_PRE_COMP
*ecp_nistz256_pre_comp_new(const EC_GROUP
*group
)
1424 NISTZ256_PRE_COMP
*ret
= NULL
;
1429 ret
= OPENSSL_zalloc(sizeof(*ret
));
1432 ECerr(EC_F_ECP_NISTZ256_PRE_COMP_NEW
, ERR_R_MALLOC_FAILURE
);
1437 ret
->w
= 6; /* default */
1438 ret
->references
= 1;
1440 ret
->lock
= CRYPTO_THREAD_lock_new();
1441 if (ret
->lock
== NULL
) {
1442 ECerr(EC_F_ECP_NISTZ256_PRE_COMP_NEW
, ERR_R_MALLOC_FAILURE
);
1449 NISTZ256_PRE_COMP
*EC_nistz256_pre_comp_dup(NISTZ256_PRE_COMP
*p
)
1453 CRYPTO_UP_REF(&p
->references
, &i
, p
->lock
);
1457 void EC_nistz256_pre_comp_free(NISTZ256_PRE_COMP
*pre
)
1464 CRYPTO_DOWN_REF(&pre
->references
, &i
, pre
->lock
);
1465 REF_PRINT_COUNT("EC_nistz256", x
);
1468 REF_ASSERT_ISNT(i
< 0);
1470 OPENSSL_free(pre
->precomp_storage
);
1471 CRYPTO_THREAD_lock_free(pre
->lock
);
1476 static int ecp_nistz256_window_have_precompute_mult(const EC_GROUP
*group
)
1478 /* There is a hard-coded table for the default generator. */
1479 const EC_POINT
*generator
= EC_GROUP_get0_generator(group
);
1481 if (generator
!= NULL
&& ecp_nistz256_is_affine_G(generator
)) {
1482 /* There is a hard-coded table for the default generator. */
1486 return HAVEPRECOMP(group
, nistz256
);
1489 #if defined(__x86_64) || defined(__x86_64__) || \
1490 defined(_M_AMD64) || defined(_M_X64) || \
1491 defined(__powerpc64__) || defined(_ARCH_PP64) || \
1492 defined(__aarch64__)
1494 * Montgomery mul modulo Order(P): res = a*b*2^-256 mod Order(P)
1496 void ecp_nistz256_ord_mul_mont(BN_ULONG res
[P256_LIMBS
],
1497 const BN_ULONG a
[P256_LIMBS
],
1498 const BN_ULONG b
[P256_LIMBS
]);
1499 void ecp_nistz256_ord_sqr_mont(BN_ULONG res
[P256_LIMBS
],
1500 const BN_ULONG a
[P256_LIMBS
],
1503 static int ecp_nistz256_inv_mod_ord(const EC_GROUP
*group
, BIGNUM
*r
,
1504 const BIGNUM
*x
, BN_CTX
*ctx
)
1506 /* RR = 2^512 mod ord(p256) */
1507 static const BN_ULONG RR
[P256_LIMBS
] = {
1508 TOBN(0x83244c95,0xbe79eea2), TOBN(0x4699799c,0x49bd6fa6),
1509 TOBN(0x2845b239,0x2b6bec59), TOBN(0x66e12d94,0xf3d95620)
1511 /* The constant 1 (unlike ONE that is one in Montgomery representation) */
1512 static const BN_ULONG one
[P256_LIMBS
] = {
1513 TOBN(0,1), TOBN(0,0), TOBN(0,0), TOBN(0,0)
1516 * We don't use entry 0 in the table, so we omit it and address
1519 BN_ULONG table
[15][P256_LIMBS
];
1520 BN_ULONG out
[P256_LIMBS
], t
[P256_LIMBS
];
1523 i_1
= 0, i_10
, i_11
, i_101
, i_111
, i_1010
, i_1111
,
1524 i_10101
, i_101010
, i_101111
, i_x6
, i_x8
, i_x16
, i_x32
1528 * Catch allocation failure early.
1530 if (bn_wexpand(r
, P256_LIMBS
) == NULL
) {
1531 ECerr(EC_F_ECP_NISTZ256_INV_MOD_ORD
, ERR_R_BN_LIB
);
1535 if ((BN_num_bits(x
) > 256) || BN_is_negative(x
)) {
1538 if ((tmp
= BN_CTX_get(ctx
)) == NULL
1539 || !BN_nnmod(tmp
, x
, group
->order
, ctx
)) {
1540 ECerr(EC_F_ECP_NISTZ256_INV_MOD_ORD
, ERR_R_BN_LIB
);
1546 if (!ecp_nistz256_bignum_to_field_elem(t
, x
)) {
1547 ECerr(EC_F_ECP_NISTZ256_INV_MOD_ORD
, EC_R_COORDINATES_OUT_OF_RANGE
);
1551 ecp_nistz256_ord_mul_mont(table
[0], t
, RR
);
1554 * Original sparse-then-fixed-window algorithm, retained for reference.
1556 for (i
= 2; i
< 16; i
+= 2) {
1557 ecp_nistz256_ord_sqr_mont(table
[i
-1], table
[i
/2-1], 1);
1558 ecp_nistz256_ord_mul_mont(table
[i
], table
[i
-1], table
[0]);
1562 * The top 128bit of the exponent are highly redudndant, so we
1563 * perform an optimized flow
1565 ecp_nistz256_ord_sqr_mont(t
, table
[15-1], 4); /* f0 */
1566 ecp_nistz256_ord_mul_mont(t
, t
, table
[15-1]); /* ff */
1568 ecp_nistz256_ord_sqr_mont(out
, t
, 8); /* ff00 */
1569 ecp_nistz256_ord_mul_mont(out
, out
, t
); /* ffff */
1571 ecp_nistz256_ord_sqr_mont(t
, out
, 16); /* ffff0000 */
1572 ecp_nistz256_ord_mul_mont(t
, t
, out
); /* ffffffff */
1574 ecp_nistz256_ord_sqr_mont(out
, t
, 64); /* ffffffff0000000000000000 */
1575 ecp_nistz256_ord_mul_mont(out
, out
, t
); /* ffffffff00000000ffffffff */
1577 ecp_nistz256_ord_sqr_mont(out
, out
, 32); /* ffffffff00000000ffffffff00000000 */
1578 ecp_nistz256_ord_mul_mont(out
, out
, t
); /* ffffffff00000000ffffffffffffffff */
1581 * The bottom 128 bit of the exponent are processed with fixed 4-bit window
1583 for(i
= 0; i
< 32; i
++) {
1584 /* expLo - the low 128 bits of the exponent we use (ord(p256) - 2),
1585 * split into nibbles */
1586 static const unsigned char expLo
[32] = {
1587 0xb,0xc,0xe,0x6,0xf,0xa,0xa,0xd,0xa,0x7,0x1,0x7,0x9,0xe,0x8,0x4,
1588 0xf,0x3,0xb,0x9,0xc,0xa,0xc,0x2,0xf,0xc,0x6,0x3,0x2,0x5,0x4,0xf
1591 ecp_nistz256_ord_sqr_mont(out
, out
, 4);
1592 /* The exponent is public, no need in constant-time access */
1593 ecp_nistz256_ord_mul_mont(out
, out
, table
[expLo
[i
]-1]);
1597 * https://briansmith.org/ecc-inversion-addition-chains-01#p256_scalar_inversion
1599 * Even though this code path spares 12 squarings, 4.5%, and 13
1600 * multiplications, 25%, on grand scale sign operation is not that
1601 * much faster, not more that 2%...
1604 /* pre-calculate powers */
1605 ecp_nistz256_ord_sqr_mont(table
[i_10
], table
[i_1
], 1);
1607 ecp_nistz256_ord_mul_mont(table
[i_11
], table
[i_1
], table
[i_10
]);
1609 ecp_nistz256_ord_mul_mont(table
[i_101
], table
[i_11
], table
[i_10
]);
1611 ecp_nistz256_ord_mul_mont(table
[i_111
], table
[i_101
], table
[i_10
]);
1613 ecp_nistz256_ord_sqr_mont(table
[i_1010
], table
[i_101
], 1);
1615 ecp_nistz256_ord_mul_mont(table
[i_1111
], table
[i_1010
], table
[i_101
]);
1617 ecp_nistz256_ord_sqr_mont(table
[i_10101
], table
[i_1010
], 1);
1618 ecp_nistz256_ord_mul_mont(table
[i_10101
], table
[i_10101
], table
[i_1
]);
1620 ecp_nistz256_ord_sqr_mont(table
[i_101010
], table
[i_10101
], 1);
1622 ecp_nistz256_ord_mul_mont(table
[i_101111
], table
[i_101010
], table
[i_101
]);
1624 ecp_nistz256_ord_mul_mont(table
[i_x6
], table
[i_101010
], table
[i_10101
]);
1626 ecp_nistz256_ord_sqr_mont(table
[i_x8
], table
[i_x6
], 2);
1627 ecp_nistz256_ord_mul_mont(table
[i_x8
], table
[i_x8
], table
[i_11
]);
1629 ecp_nistz256_ord_sqr_mont(table
[i_x16
], table
[i_x8
], 8);
1630 ecp_nistz256_ord_mul_mont(table
[i_x16
], table
[i_x16
], table
[i_x8
]);
1632 ecp_nistz256_ord_sqr_mont(table
[i_x32
], table
[i_x16
], 16);
1633 ecp_nistz256_ord_mul_mont(table
[i_x32
], table
[i_x32
], table
[i_x16
]);
1636 ecp_nistz256_ord_sqr_mont(out
, table
[i_x32
], 64);
1637 ecp_nistz256_ord_mul_mont(out
, out
, table
[i_x32
]);
1639 for (i
= 0; i
< 27; i
++) {
1640 static const struct { unsigned char p
, i
; } chain
[27] = {
1641 { 32, i_x32
}, { 6, i_101111
}, { 5, i_111
},
1642 { 4, i_11
}, { 5, i_1111
}, { 5, i_10101
},
1643 { 4, i_101
}, { 3, i_101
}, { 3, i_101
},
1644 { 5, i_111
}, { 9, i_101111
}, { 6, i_1111
},
1645 { 2, i_1
}, { 5, i_1
}, { 6, i_1111
},
1646 { 5, i_111
}, { 4, i_111
}, { 5, i_111
},
1647 { 5, i_101
}, { 3, i_11
}, { 10, i_101111
},
1648 { 2, i_11
}, { 5, i_11
}, { 5, i_11
},
1649 { 3, i_1
}, { 7, i_10101
}, { 6, i_1111
}
1652 ecp_nistz256_ord_sqr_mont(out
, out
, chain
[i
].p
);
1653 ecp_nistz256_ord_mul_mont(out
, out
, table
[chain
[i
].i
]);
1656 ecp_nistz256_ord_mul_mont(out
, out
, one
);
1659 * Can't fail, but check return code to be consistent anyway.
1661 if (!bn_set_words(r
, out
, P256_LIMBS
))
1669 # define ecp_nistz256_inv_mod_ord NULL
1672 const EC_METHOD
*EC_GFp_nistz256_method(void)
1674 static const EC_METHOD ret
= {
1675 EC_FLAGS_DEFAULT_OCT
,
1676 NID_X9_62_prime_field
,
1677 ec_GFp_mont_group_init
,
1678 ec_GFp_mont_group_finish
,
1679 ec_GFp_mont_group_clear_finish
,
1680 ec_GFp_mont_group_copy
,
1681 ec_GFp_mont_group_set_curve
,
1682 ec_GFp_simple_group_get_curve
,
1683 ec_GFp_simple_group_get_degree
,
1684 ec_group_simple_order_bits
,
1685 ec_GFp_simple_group_check_discriminant
,
1686 ec_GFp_simple_point_init
,
1687 ec_GFp_simple_point_finish
,
1688 ec_GFp_simple_point_clear_finish
,
1689 ec_GFp_simple_point_copy
,
1690 ec_GFp_simple_point_set_to_infinity
,
1691 ec_GFp_simple_set_Jprojective_coordinates_GFp
,
1692 ec_GFp_simple_get_Jprojective_coordinates_GFp
,
1693 ec_GFp_simple_point_set_affine_coordinates
,
1694 ecp_nistz256_get_affine
,
1698 ec_GFp_simple_invert
,
1699 ec_GFp_simple_is_at_infinity
,
1700 ec_GFp_simple_is_on_curve
,
1702 ec_GFp_simple_make_affine
,
1703 ec_GFp_simple_points_make_affine
,
1704 ecp_nistz256_points_mul
, /* mul */
1705 ecp_nistz256_mult_precompute
, /* precompute_mult */
1706 ecp_nistz256_window_have_precompute_mult
, /* have_precompute_mult */
1707 ec_GFp_mont_field_mul
,
1708 ec_GFp_mont_field_sqr
,
1710 ec_GFp_mont_field_inv
,
1711 ec_GFp_mont_field_encode
,
1712 ec_GFp_mont_field_decode
,
1713 ec_GFp_mont_field_set_to_one
,
1714 ec_key_simple_priv2oct
,
1715 ec_key_simple_oct2priv
,
1716 0, /* set private */
1717 ec_key_simple_generate_key
,
1718 ec_key_simple_check_key
,
1719 ec_key_simple_generate_public_key
,
1722 ecdh_simple_compute_key
,
1723 ecp_nistz256_inv_mod_ord
, /* can be #define-d NULL */
1724 0, /* blind_coordinates */
1726 0, /* ladder_step */