2 * Copyright 2014-2020 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 Apache License 2.0 (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
22 * ECDSA low level APIs are deprecated for public use, but still ok for
25 #include "internal/deprecated.h"
29 #include "internal/cryptlib.h"
30 #include "crypto/bn.h"
32 #include "internal/refcount.h"
35 # define TOBN(hi,lo) lo,hi
37 # define TOBN(hi,lo) ((BN_ULONG)hi<<32|lo)
41 # define ALIGN32 __attribute((aligned(32)))
42 #elif defined(_MSC_VER)
43 # define ALIGN32 __declspec(align(32))
48 #define ALIGNPTR(p,N) ((unsigned char *)p+N-(size_t)p%N)
49 #define P256_LIMBS (256/BN_BITS2)
51 typedef unsigned short u16;
54 BN_ULONG X[P256_LIMBS];
55 BN_ULONG Y[P256_LIMBS];
56 BN_ULONG Z[P256_LIMBS];
60 BN_ULONG X[P256_LIMBS];
61 BN_ULONG Y[P256_LIMBS];
64 typedef P256_POINT_AFFINE PRECOMP256_ROW[64];
66 /* structure for precomputed multiples of the generator */
67 struct nistz256_pre_comp_st {
68 const EC_GROUP *group; /* Parent EC_GROUP object */
69 size_t w; /* Window size */
71 * Constant time access to the X and Y coordinates of the pre-computed,
72 * generator multiplies, in the Montgomery domain. Pre-calculated
73 * multiplies are stored in affine form.
75 PRECOMP256_ROW *precomp;
76 void *precomp_storage;
77 CRYPTO_REF_COUNT references;
81 /* Functions implemented in assembly */
83 * Most of below mentioned functions *preserve* the property of inputs
84 * being fully reduced, i.e. being in [0, modulus) range. Simply put if
85 * inputs are fully reduced, then output is too. Note that reverse is
86 * not true, in sense that given partially reduced inputs output can be
87 * either, not unlikely reduced. And "most" in first sentence refers to
88 * the fact that given the calculations flow one can tolerate that
89 * addition, 1st function below, produces partially reduced result *if*
90 * multiplications by 2 and 3, which customarily use addition, fully
91 * reduce it. This effectively gives two options: a) addition produces
92 * fully reduced result [as long as inputs are, just like remaining
93 * functions]; b) addition is allowed to produce partially reduced
94 * result, but multiplications by 2 and 3 perform additional reduction
95 * step. Choice between the two can be platform-specific, but it was a)
96 * in all cases so far...
98 /* Modular add: res = a+b mod P */
99 void ecp_nistz256_add(BN_ULONG res[P256_LIMBS],
100 const BN_ULONG a[P256_LIMBS],
101 const BN_ULONG b[P256_LIMBS]);
102 /* Modular mul by 2: res = 2*a mod P */
103 void ecp_nistz256_mul_by_2(BN_ULONG res[P256_LIMBS],
104 const BN_ULONG a[P256_LIMBS]);
105 /* Modular mul by 3: res = 3*a mod P */
106 void ecp_nistz256_mul_by_3(BN_ULONG res[P256_LIMBS],
107 const BN_ULONG a[P256_LIMBS]);
109 /* Modular div by 2: res = a/2 mod P */
110 void ecp_nistz256_div_by_2(BN_ULONG res[P256_LIMBS],
111 const BN_ULONG a[P256_LIMBS]);
112 /* Modular sub: res = a-b mod P */
113 void ecp_nistz256_sub(BN_ULONG res[P256_LIMBS],
114 const BN_ULONG a[P256_LIMBS],
115 const BN_ULONG b[P256_LIMBS]);
116 /* Modular neg: res = -a mod P */
117 void ecp_nistz256_neg(BN_ULONG res[P256_LIMBS], const BN_ULONG a[P256_LIMBS]);
118 /* Montgomery mul: res = a*b*2^-256 mod P */
119 void ecp_nistz256_mul_mont(BN_ULONG res[P256_LIMBS],
120 const BN_ULONG a[P256_LIMBS],
121 const BN_ULONG b[P256_LIMBS]);
122 /* Montgomery sqr: res = a*a*2^-256 mod P */
123 void ecp_nistz256_sqr_mont(BN_ULONG res[P256_LIMBS],
124 const BN_ULONG a[P256_LIMBS]);
125 /* Convert a number from Montgomery domain, by multiplying with 1 */
126 void ecp_nistz256_from_mont(BN_ULONG res[P256_LIMBS],
127 const BN_ULONG in[P256_LIMBS]);
128 /* Convert a number to Montgomery domain, by multiplying with 2^512 mod P*/
129 void ecp_nistz256_to_mont(BN_ULONG res[P256_LIMBS],
130 const BN_ULONG in[P256_LIMBS]);
131 /* Functions that perform constant time access to the precomputed tables */
132 void ecp_nistz256_scatter_w5(P256_POINT *val,
133 const P256_POINT *in_t, int idx);
134 void ecp_nistz256_gather_w5(P256_POINT *val,
135 const P256_POINT *in_t, int idx);
136 void ecp_nistz256_scatter_w7(P256_POINT_AFFINE *val,
137 const P256_POINT_AFFINE *in_t, int idx);
138 void ecp_nistz256_gather_w7(P256_POINT_AFFINE *val,
139 const P256_POINT_AFFINE *in_t, int idx);
141 /* One converted into the Montgomery domain */
142 static const BN_ULONG ONE[P256_LIMBS] = {
143 TOBN(0x00000000, 0x00000001), TOBN(0xffffffff, 0x00000000),
144 TOBN(0xffffffff, 0xffffffff), TOBN(0x00000000, 0xfffffffe)
147 static NISTZ256_PRE_COMP *ecp_nistz256_pre_comp_new(const EC_GROUP *group);
149 /* Precomputed tables for the default generator */
150 extern const PRECOMP256_ROW ecp_nistz256_precomputed[37];
152 /* Recode window to a signed digit, see ecp_nistputil.c for details */
153 static unsigned int _booth_recode_w5(unsigned int in)
157 s = ~((in >> 5) - 1);
158 d = (1 << 6) - in - 1;
159 d = (d & s) | (in & ~s);
160 d = (d >> 1) + (d & 1);
162 return (d << 1) + (s & 1);
165 static unsigned int _booth_recode_w7(unsigned int in)
169 s = ~((in >> 7) - 1);
170 d = (1 << 8) - in - 1;
171 d = (d & s) | (in & ~s);
172 d = (d >> 1) + (d & 1);
174 return (d << 1) + (s & 1);
177 static void copy_conditional(BN_ULONG dst[P256_LIMBS],
178 const BN_ULONG src[P256_LIMBS], BN_ULONG move)
180 BN_ULONG mask1 = 0-move;
181 BN_ULONG mask2 = ~mask1;
183 dst[0] = (src[0] & mask1) ^ (dst[0] & mask2);
184 dst[1] = (src[1] & mask1) ^ (dst[1] & mask2);
185 dst[2] = (src[2] & mask1) ^ (dst[2] & mask2);
186 dst[3] = (src[3] & mask1) ^ (dst[3] & mask2);
187 if (P256_LIMBS == 8) {
188 dst[4] = (src[4] & mask1) ^ (dst[4] & mask2);
189 dst[5] = (src[5] & mask1) ^ (dst[5] & mask2);
190 dst[6] = (src[6] & mask1) ^ (dst[6] & mask2);
191 dst[7] = (src[7] & mask1) ^ (dst[7] & mask2);
195 static BN_ULONG is_zero(BN_ULONG in)
203 static BN_ULONG is_equal(const BN_ULONG a[P256_LIMBS],
204 const BN_ULONG b[P256_LIMBS])
212 if (P256_LIMBS == 8) {
222 static BN_ULONG is_one(const BIGNUM *z)
225 BN_ULONG *a = bn_get_words(z);
227 if (bn_get_top(z) == (P256_LIMBS - P256_LIMBS / 8)) {
229 res |= a[1] ^ ONE[1];
230 res |= a[2] ^ ONE[2];
231 res |= a[3] ^ ONE[3];
232 if (P256_LIMBS == 8) {
233 res |= a[4] ^ ONE[4];
234 res |= a[5] ^ ONE[5];
235 res |= a[6] ^ ONE[6];
237 * no check for a[7] (being zero) on 32-bit platforms,
238 * because value of "one" takes only 7 limbs.
248 * For reference, this macro is used only when new ecp_nistz256 assembly
249 * module is being developed. For example, configure with
250 * -DECP_NISTZ256_REFERENCE_IMPLEMENTATION and implement only functions
251 * performing simplest arithmetic operations on 256-bit vectors. Then
252 * work on implementation of higher-level functions performing point
253 * operations. Then remove ECP_NISTZ256_REFERENCE_IMPLEMENTATION
254 * and never define it again. (The correct macro denoting presence of
255 * ecp_nistz256 module is ECP_NISTZ256_ASM.)
257 #ifndef ECP_NISTZ256_REFERENCE_IMPLEMENTATION
258 void ecp_nistz256_point_double(P256_POINT *r, const P256_POINT *a);
259 void ecp_nistz256_point_add(P256_POINT *r,
260 const P256_POINT *a, const P256_POINT *b);
261 void ecp_nistz256_point_add_affine(P256_POINT *r,
263 const P256_POINT_AFFINE *b);
265 /* Point double: r = 2*a */
266 static void ecp_nistz256_point_double(P256_POINT *r, const P256_POINT *a)
268 BN_ULONG S[P256_LIMBS];
269 BN_ULONG M[P256_LIMBS];
270 BN_ULONG Zsqr[P256_LIMBS];
271 BN_ULONG tmp0[P256_LIMBS];
273 const BN_ULONG *in_x = a->X;
274 const BN_ULONG *in_y = a->Y;
275 const BN_ULONG *in_z = a->Z;
277 BN_ULONG *res_x = r->X;
278 BN_ULONG *res_y = r->Y;
279 BN_ULONG *res_z = r->Z;
281 ecp_nistz256_mul_by_2(S, in_y);
283 ecp_nistz256_sqr_mont(Zsqr, in_z);
285 ecp_nistz256_sqr_mont(S, S);
287 ecp_nistz256_mul_mont(res_z, in_z, in_y);
288 ecp_nistz256_mul_by_2(res_z, res_z);
290 ecp_nistz256_add(M, in_x, Zsqr);
291 ecp_nistz256_sub(Zsqr, in_x, Zsqr);
293 ecp_nistz256_sqr_mont(res_y, S);
294 ecp_nistz256_div_by_2(res_y, res_y);
296 ecp_nistz256_mul_mont(M, M, Zsqr);
297 ecp_nistz256_mul_by_3(M, M);
299 ecp_nistz256_mul_mont(S, S, in_x);
300 ecp_nistz256_mul_by_2(tmp0, S);
302 ecp_nistz256_sqr_mont(res_x, M);
304 ecp_nistz256_sub(res_x, res_x, tmp0);
305 ecp_nistz256_sub(S, S, res_x);
307 ecp_nistz256_mul_mont(S, S, M);
308 ecp_nistz256_sub(res_y, S, res_y);
311 /* Point addition: r = a+b */
312 static void ecp_nistz256_point_add(P256_POINT *r,
313 const P256_POINT *a, const P256_POINT *b)
315 BN_ULONG U2[P256_LIMBS], S2[P256_LIMBS];
316 BN_ULONG U1[P256_LIMBS], S1[P256_LIMBS];
317 BN_ULONG Z1sqr[P256_LIMBS];
318 BN_ULONG Z2sqr[P256_LIMBS];
319 BN_ULONG H[P256_LIMBS], R[P256_LIMBS];
320 BN_ULONG Hsqr[P256_LIMBS];
321 BN_ULONG Rsqr[P256_LIMBS];
322 BN_ULONG Hcub[P256_LIMBS];
324 BN_ULONG res_x[P256_LIMBS];
325 BN_ULONG res_y[P256_LIMBS];
326 BN_ULONG res_z[P256_LIMBS];
328 BN_ULONG in1infty, in2infty;
330 const BN_ULONG *in1_x = a->X;
331 const BN_ULONG *in1_y = a->Y;
332 const BN_ULONG *in1_z = a->Z;
334 const BN_ULONG *in2_x = b->X;
335 const BN_ULONG *in2_y = b->Y;
336 const BN_ULONG *in2_z = b->Z;
339 * Infinity in encoded as (,,0)
341 in1infty = (in1_z[0] | in1_z[1] | in1_z[2] | in1_z[3]);
343 in1infty |= (in1_z[4] | in1_z[5] | in1_z[6] | in1_z[7]);
345 in2infty = (in2_z[0] | in2_z[1] | in2_z[2] | in2_z[3]);
347 in2infty |= (in2_z[4] | in2_z[5] | in2_z[6] | in2_z[7]);
349 in1infty = is_zero(in1infty);
350 in2infty = is_zero(in2infty);
352 ecp_nistz256_sqr_mont(Z2sqr, in2_z); /* Z2^2 */
353 ecp_nistz256_sqr_mont(Z1sqr, in1_z); /* Z1^2 */
355 ecp_nistz256_mul_mont(S1, Z2sqr, in2_z); /* S1 = Z2^3 */
356 ecp_nistz256_mul_mont(S2, Z1sqr, in1_z); /* S2 = Z1^3 */
358 ecp_nistz256_mul_mont(S1, S1, in1_y); /* S1 = Y1*Z2^3 */
359 ecp_nistz256_mul_mont(S2, S2, in2_y); /* S2 = Y2*Z1^3 */
360 ecp_nistz256_sub(R, S2, S1); /* R = S2 - S1 */
362 ecp_nistz256_mul_mont(U1, in1_x, Z2sqr); /* U1 = X1*Z2^2 */
363 ecp_nistz256_mul_mont(U2, in2_x, Z1sqr); /* U2 = X2*Z1^2 */
364 ecp_nistz256_sub(H, U2, U1); /* H = U2 - U1 */
367 * The formulae are incorrect if the points are equal so we check for
368 * this and do doubling if this happens.
370 * Points here are in Jacobian projective coordinates (Xi, Yi, Zi)
371 * that are bound to the affine coordinates (xi, yi) by the following
376 * For the sake of optimization, the algorithm operates over
377 * intermediate variables U1, U2 and S1, S2 that are derived from
378 * the projective coordinates:
379 * - U1 = X1 * (Z2)^2 ; U2 = X2 * (Z1)^2
380 * - S1 = Y1 * (Z2)^3 ; S2 = Y2 * (Z1)^3
382 * It is easy to prove that is_equal(U1, U2) implies that the affine
383 * x-coordinates are equal, or either point is at infinity.
384 * Likewise is_equal(S1, S2) implies that the affine y-coordinates are
385 * equal, or either point is at infinity.
387 * The special case of either point being the point at infinity (Z1 or Z2
388 * is zero), is handled separately later on in this function, so we avoid
389 * jumping to point_double here in those special cases.
391 * When both points are inverse of each other, we know that the affine
392 * x-coordinates are equal, and the y-coordinates have different sign.
393 * Therefore since U1 = U2, we know H = 0, and therefore Z3 = H*Z1*Z2
394 * will equal 0, thus the result is infinity, if we simply let this
395 * function continue normally.
397 * We use bitwise operations to avoid potential side-channels introduced by
398 * the short-circuiting behaviour of boolean operators.
400 if (is_equal(U1, U2) & ~in1infty & ~in2infty & is_equal(S1, S2)) {
402 * This is obviously not constant-time but it should never happen during
403 * single point multiplication, so there is no timing leak for ECDH or
406 ecp_nistz256_point_double(r, a);
410 ecp_nistz256_sqr_mont(Rsqr, R); /* R^2 */
411 ecp_nistz256_mul_mont(res_z, H, in1_z); /* Z3 = H*Z1*Z2 */
412 ecp_nistz256_sqr_mont(Hsqr, H); /* H^2 */
413 ecp_nistz256_mul_mont(res_z, res_z, in2_z); /* Z3 = H*Z1*Z2 */
414 ecp_nistz256_mul_mont(Hcub, Hsqr, H); /* H^3 */
416 ecp_nistz256_mul_mont(U2, U1, Hsqr); /* U1*H^2 */
417 ecp_nistz256_mul_by_2(Hsqr, U2); /* 2*U1*H^2 */
419 ecp_nistz256_sub(res_x, Rsqr, Hsqr);
420 ecp_nistz256_sub(res_x, res_x, Hcub);
422 ecp_nistz256_sub(res_y, U2, res_x);
424 ecp_nistz256_mul_mont(S2, S1, Hcub);
425 ecp_nistz256_mul_mont(res_y, R, res_y);
426 ecp_nistz256_sub(res_y, res_y, S2);
428 copy_conditional(res_x, in2_x, in1infty);
429 copy_conditional(res_y, in2_y, in1infty);
430 copy_conditional(res_z, in2_z, in1infty);
432 copy_conditional(res_x, in1_x, in2infty);
433 copy_conditional(res_y, in1_y, in2infty);
434 copy_conditional(res_z, in1_z, in2infty);
436 memcpy(r->X, res_x, sizeof(res_x));
437 memcpy(r->Y, res_y, sizeof(res_y));
438 memcpy(r->Z, res_z, sizeof(res_z));
441 /* Point addition when b is known to be affine: r = a+b */
442 static void ecp_nistz256_point_add_affine(P256_POINT *r,
444 const P256_POINT_AFFINE *b)
446 BN_ULONG U2[P256_LIMBS], S2[P256_LIMBS];
447 BN_ULONG Z1sqr[P256_LIMBS];
448 BN_ULONG H[P256_LIMBS], R[P256_LIMBS];
449 BN_ULONG Hsqr[P256_LIMBS];
450 BN_ULONG Rsqr[P256_LIMBS];
451 BN_ULONG Hcub[P256_LIMBS];
453 BN_ULONG res_x[P256_LIMBS];
454 BN_ULONG res_y[P256_LIMBS];
455 BN_ULONG res_z[P256_LIMBS];
457 BN_ULONG in1infty, in2infty;
459 const BN_ULONG *in1_x = a->X;
460 const BN_ULONG *in1_y = a->Y;
461 const BN_ULONG *in1_z = a->Z;
463 const BN_ULONG *in2_x = b->X;
464 const BN_ULONG *in2_y = b->Y;
467 * Infinity in encoded as (,,0)
469 in1infty = (in1_z[0] | in1_z[1] | in1_z[2] | in1_z[3]);
471 in1infty |= (in1_z[4] | in1_z[5] | in1_z[6] | in1_z[7]);
474 * In affine representation we encode infinity as (0,0), which is
475 * not on the curve, so it is OK
477 in2infty = (in2_x[0] | in2_x[1] | in2_x[2] | in2_x[3] |
478 in2_y[0] | in2_y[1] | in2_y[2] | in2_y[3]);
480 in2infty |= (in2_x[4] | in2_x[5] | in2_x[6] | in2_x[7] |
481 in2_y[4] | in2_y[5] | in2_y[6] | in2_y[7]);
483 in1infty = is_zero(in1infty);
484 in2infty = is_zero(in2infty);
486 ecp_nistz256_sqr_mont(Z1sqr, in1_z); /* Z1^2 */
488 ecp_nistz256_mul_mont(U2, in2_x, Z1sqr); /* U2 = X2*Z1^2 */
489 ecp_nistz256_sub(H, U2, in1_x); /* H = U2 - U1 */
491 ecp_nistz256_mul_mont(S2, Z1sqr, in1_z); /* S2 = Z1^3 */
493 ecp_nistz256_mul_mont(res_z, H, in1_z); /* Z3 = H*Z1*Z2 */
495 ecp_nistz256_mul_mont(S2, S2, in2_y); /* S2 = Y2*Z1^3 */
496 ecp_nistz256_sub(R, S2, in1_y); /* R = S2 - S1 */
498 ecp_nistz256_sqr_mont(Hsqr, H); /* H^2 */
499 ecp_nistz256_sqr_mont(Rsqr, R); /* R^2 */
500 ecp_nistz256_mul_mont(Hcub, Hsqr, H); /* H^3 */
502 ecp_nistz256_mul_mont(U2, in1_x, Hsqr); /* U1*H^2 */
503 ecp_nistz256_mul_by_2(Hsqr, U2); /* 2*U1*H^2 */
505 ecp_nistz256_sub(res_x, Rsqr, Hsqr);
506 ecp_nistz256_sub(res_x, res_x, Hcub);
507 ecp_nistz256_sub(H, U2, res_x);
509 ecp_nistz256_mul_mont(S2, in1_y, Hcub);
510 ecp_nistz256_mul_mont(H, H, R);
511 ecp_nistz256_sub(res_y, H, S2);
513 copy_conditional(res_x, in2_x, in1infty);
514 copy_conditional(res_x, in1_x, in2infty);
516 copy_conditional(res_y, in2_y, in1infty);
517 copy_conditional(res_y, in1_y, in2infty);
519 copy_conditional(res_z, ONE, in1infty);
520 copy_conditional(res_z, in1_z, in2infty);
522 memcpy(r->X, res_x, sizeof(res_x));
523 memcpy(r->Y, res_y, sizeof(res_y));
524 memcpy(r->Z, res_z, sizeof(res_z));
528 /* r = in^-1 mod p */
529 static void ecp_nistz256_mod_inverse(BN_ULONG r[P256_LIMBS],
530 const BN_ULONG in[P256_LIMBS])
533 * The poly is ffffffff 00000001 00000000 00000000 00000000 ffffffff
534 * ffffffff ffffffff We use FLT and used poly-2 as exponent
536 BN_ULONG p2[P256_LIMBS];
537 BN_ULONG p4[P256_LIMBS];
538 BN_ULONG p8[P256_LIMBS];
539 BN_ULONG p16[P256_LIMBS];
540 BN_ULONG p32[P256_LIMBS];
541 BN_ULONG res[P256_LIMBS];
544 ecp_nistz256_sqr_mont(res, in);
545 ecp_nistz256_mul_mont(p2, res, in); /* 3*p */
547 ecp_nistz256_sqr_mont(res, p2);
548 ecp_nistz256_sqr_mont(res, res);
549 ecp_nistz256_mul_mont(p4, res, p2); /* f*p */
551 ecp_nistz256_sqr_mont(res, p4);
552 ecp_nistz256_sqr_mont(res, res);
553 ecp_nistz256_sqr_mont(res, res);
554 ecp_nistz256_sqr_mont(res, res);
555 ecp_nistz256_mul_mont(p8, res, p4); /* ff*p */
557 ecp_nistz256_sqr_mont(res, p8);
558 for (i = 0; i < 7; i++)
559 ecp_nistz256_sqr_mont(res, res);
560 ecp_nistz256_mul_mont(p16, res, p8); /* ffff*p */
562 ecp_nistz256_sqr_mont(res, p16);
563 for (i = 0; i < 15; i++)
564 ecp_nistz256_sqr_mont(res, res);
565 ecp_nistz256_mul_mont(p32, res, p16); /* ffffffff*p */
567 ecp_nistz256_sqr_mont(res, p32);
568 for (i = 0; i < 31; i++)
569 ecp_nistz256_sqr_mont(res, res);
570 ecp_nistz256_mul_mont(res, res, in);
572 for (i = 0; i < 32 * 4; i++)
573 ecp_nistz256_sqr_mont(res, res);
574 ecp_nistz256_mul_mont(res, res, p32);
576 for (i = 0; i < 32; i++)
577 ecp_nistz256_sqr_mont(res, res);
578 ecp_nistz256_mul_mont(res, res, p32);
580 for (i = 0; i < 16; i++)
581 ecp_nistz256_sqr_mont(res, res);
582 ecp_nistz256_mul_mont(res, res, p16);
584 for (i = 0; i < 8; i++)
585 ecp_nistz256_sqr_mont(res, res);
586 ecp_nistz256_mul_mont(res, res, p8);
588 ecp_nistz256_sqr_mont(res, res);
589 ecp_nistz256_sqr_mont(res, res);
590 ecp_nistz256_sqr_mont(res, res);
591 ecp_nistz256_sqr_mont(res, res);
592 ecp_nistz256_mul_mont(res, res, p4);
594 ecp_nistz256_sqr_mont(res, res);
595 ecp_nistz256_sqr_mont(res, res);
596 ecp_nistz256_mul_mont(res, res, p2);
598 ecp_nistz256_sqr_mont(res, res);
599 ecp_nistz256_sqr_mont(res, res);
600 ecp_nistz256_mul_mont(res, res, in);
602 memcpy(r, res, sizeof(res));
606 * ecp_nistz256_bignum_to_field_elem copies the contents of |in| to |out| and
607 * returns one if it fits. Otherwise it returns zero.
609 __owur static int ecp_nistz256_bignum_to_field_elem(BN_ULONG out[P256_LIMBS],
612 return bn_copy_words(out, in, P256_LIMBS);
615 /* r = sum(scalar[i]*point[i]) */
616 __owur static int ecp_nistz256_windowed_mul(const EC_GROUP *group,
618 const BIGNUM **scalar,
619 const EC_POINT **point,
620 size_t num, BN_CTX *ctx)
625 unsigned char (*p_str)[33] = NULL;
626 const unsigned int window_size = 5;
627 const unsigned int mask = (1 << (window_size + 1)) - 1;
629 P256_POINT *temp; /* place for 5 temporary points */
630 const BIGNUM **scalars = NULL;
631 P256_POINT (*table)[16] = NULL;
632 void *table_storage = NULL;
634 if ((num * 16 + 6) > OPENSSL_MALLOC_MAX_NELEMS(P256_POINT)
636 OPENSSL_malloc((num * 16 + 5) * sizeof(P256_POINT) + 64)) == NULL
638 OPENSSL_malloc(num * 33 * sizeof(unsigned char))) == NULL
639 || (scalars = OPENSSL_malloc(num * sizeof(BIGNUM *))) == NULL) {
640 ECerr(EC_F_ECP_NISTZ256_WINDOWED_MUL, ERR_R_MALLOC_FAILURE);
644 table = (void *)ALIGNPTR(table_storage, 64);
645 temp = (P256_POINT *)(table + num);
647 for (i = 0; i < num; i++) {
648 P256_POINT *row = table[i];
650 /* This is an unusual input, we don't guarantee constant-timeness. */
651 if ((BN_num_bits(scalar[i]) > 256) || BN_is_negative(scalar[i])) {
654 if ((mod = BN_CTX_get(ctx)) == NULL)
656 if (!BN_nnmod(mod, scalar[i], group->order, ctx)) {
657 ECerr(EC_F_ECP_NISTZ256_WINDOWED_MUL, ERR_R_BN_LIB);
662 scalars[i] = scalar[i];
664 for (j = 0; j < bn_get_top(scalars[i]) * BN_BYTES; j += BN_BYTES) {
665 BN_ULONG d = bn_get_words(scalars[i])[j / BN_BYTES];
667 p_str[i][j + 0] = (unsigned char)d;
668 p_str[i][j + 1] = (unsigned char)(d >> 8);
669 p_str[i][j + 2] = (unsigned char)(d >> 16);
670 p_str[i][j + 3] = (unsigned char)(d >>= 24);
673 p_str[i][j + 4] = (unsigned char)d;
674 p_str[i][j + 5] = (unsigned char)(d >> 8);
675 p_str[i][j + 6] = (unsigned char)(d >> 16);
676 p_str[i][j + 7] = (unsigned char)(d >> 24);
682 if (!ecp_nistz256_bignum_to_field_elem(temp[0].X, point[i]->X)
683 || !ecp_nistz256_bignum_to_field_elem(temp[0].Y, point[i]->Y)
684 || !ecp_nistz256_bignum_to_field_elem(temp[0].Z, point[i]->Z)) {
685 ECerr(EC_F_ECP_NISTZ256_WINDOWED_MUL,
686 EC_R_COORDINATES_OUT_OF_RANGE);
691 * row[0] is implicitly (0,0,0) (the point at infinity), therefore it
692 * is not stored. All other values are actually stored with an offset
696 ecp_nistz256_scatter_w5 (row, &temp[0], 1);
697 ecp_nistz256_point_double(&temp[1], &temp[0]); /*1+1=2 */
698 ecp_nistz256_scatter_w5 (row, &temp[1], 2);
699 ecp_nistz256_point_add (&temp[2], &temp[1], &temp[0]); /*2+1=3 */
700 ecp_nistz256_scatter_w5 (row, &temp[2], 3);
701 ecp_nistz256_point_double(&temp[1], &temp[1]); /*2*2=4 */
702 ecp_nistz256_scatter_w5 (row, &temp[1], 4);
703 ecp_nistz256_point_double(&temp[2], &temp[2]); /*2*3=6 */
704 ecp_nistz256_scatter_w5 (row, &temp[2], 6);
705 ecp_nistz256_point_add (&temp[3], &temp[1], &temp[0]); /*4+1=5 */
706 ecp_nistz256_scatter_w5 (row, &temp[3], 5);
707 ecp_nistz256_point_add (&temp[4], &temp[2], &temp[0]); /*6+1=7 */
708 ecp_nistz256_scatter_w5 (row, &temp[4], 7);
709 ecp_nistz256_point_double(&temp[1], &temp[1]); /*2*4=8 */
710 ecp_nistz256_scatter_w5 (row, &temp[1], 8);
711 ecp_nistz256_point_double(&temp[2], &temp[2]); /*2*6=12 */
712 ecp_nistz256_scatter_w5 (row, &temp[2], 12);
713 ecp_nistz256_point_double(&temp[3], &temp[3]); /*2*5=10 */
714 ecp_nistz256_scatter_w5 (row, &temp[3], 10);
715 ecp_nistz256_point_double(&temp[4], &temp[4]); /*2*7=14 */
716 ecp_nistz256_scatter_w5 (row, &temp[4], 14);
717 ecp_nistz256_point_add (&temp[2], &temp[2], &temp[0]); /*12+1=13*/
718 ecp_nistz256_scatter_w5 (row, &temp[2], 13);
719 ecp_nistz256_point_add (&temp[3], &temp[3], &temp[0]); /*10+1=11*/
720 ecp_nistz256_scatter_w5 (row, &temp[3], 11);
721 ecp_nistz256_point_add (&temp[4], &temp[4], &temp[0]); /*14+1=15*/
722 ecp_nistz256_scatter_w5 (row, &temp[4], 15);
723 ecp_nistz256_point_add (&temp[2], &temp[1], &temp[0]); /*8+1=9 */
724 ecp_nistz256_scatter_w5 (row, &temp[2], 9);
725 ecp_nistz256_point_double(&temp[1], &temp[1]); /*2*8=16 */
726 ecp_nistz256_scatter_w5 (row, &temp[1], 16);
731 wvalue = p_str[0][(idx - 1) / 8];
732 wvalue = (wvalue >> ((idx - 1) % 8)) & mask;
735 * We gather to temp[0], because we know it's position relative
738 ecp_nistz256_gather_w5(&temp[0], table[0], _booth_recode_w5(wvalue) >> 1);
739 memcpy(r, &temp[0], sizeof(temp[0]));
742 for (i = (idx == 255 ? 1 : 0); i < num; i++) {
743 unsigned int off = (idx - 1) / 8;
745 wvalue = p_str[i][off] | p_str[i][off + 1] << 8;
746 wvalue = (wvalue >> ((idx - 1) % 8)) & mask;
748 wvalue = _booth_recode_w5(wvalue);
750 ecp_nistz256_gather_w5(&temp[0], table[i], wvalue >> 1);
752 ecp_nistz256_neg(temp[1].Y, temp[0].Y);
753 copy_conditional(temp[0].Y, temp[1].Y, (wvalue & 1));
755 ecp_nistz256_point_add(r, r, &temp[0]);
760 ecp_nistz256_point_double(r, r);
761 ecp_nistz256_point_double(r, r);
762 ecp_nistz256_point_double(r, r);
763 ecp_nistz256_point_double(r, r);
764 ecp_nistz256_point_double(r, r);
768 for (i = 0; i < num; i++) {
769 wvalue = p_str[i][0];
770 wvalue = (wvalue << 1) & mask;
772 wvalue = _booth_recode_w5(wvalue);
774 ecp_nistz256_gather_w5(&temp[0], table[i], wvalue >> 1);
776 ecp_nistz256_neg(temp[1].Y, temp[0].Y);
777 copy_conditional(temp[0].Y, temp[1].Y, wvalue & 1);
779 ecp_nistz256_point_add(r, r, &temp[0]);
784 OPENSSL_free(table_storage);
786 OPENSSL_free(scalars);
790 /* Coordinates of G, for which we have precomputed tables */
791 static const BN_ULONG def_xG[P256_LIMBS] = {
792 TOBN(0x79e730d4, 0x18a9143c), TOBN(0x75ba95fc, 0x5fedb601),
793 TOBN(0x79fb732b, 0x77622510), TOBN(0x18905f76, 0xa53755c6)
796 static const BN_ULONG def_yG[P256_LIMBS] = {
797 TOBN(0xddf25357, 0xce95560a), TOBN(0x8b4ab8e4, 0xba19e45c),
798 TOBN(0xd2e88688, 0xdd21f325), TOBN(0x8571ff18, 0x25885d85)
802 * ecp_nistz256_is_affine_G returns one if |generator| is the standard, P-256
805 static int ecp_nistz256_is_affine_G(const EC_POINT *generator)
807 return (bn_get_top(generator->X) == P256_LIMBS) &&
808 (bn_get_top(generator->Y) == P256_LIMBS) &&
809 is_equal(bn_get_words(generator->X), def_xG) &&
810 is_equal(bn_get_words(generator->Y), def_yG) &&
811 is_one(generator->Z);
814 __owur static int ecp_nistz256_mult_precompute(EC_GROUP *group, BN_CTX *ctx)
817 * We precompute a table for a Booth encoded exponent (wNAF) based
818 * computation. Each table holds 64 values for safe access, with an
819 * implicit value of infinity at index zero. We use window of size 7, and
820 * therefore require ceil(256/7) = 37 tables.
823 EC_POINT *P = NULL, *T = NULL;
824 const EC_POINT *generator;
825 NISTZ256_PRE_COMP *pre_comp;
826 BN_CTX *new_ctx = NULL;
827 int i, j, k, ret = 0;
830 PRECOMP256_ROW *preComputedTable = NULL;
831 unsigned char *precomp_storage = NULL;
833 /* if there is an old NISTZ256_PRE_COMP object, throw it away */
834 EC_pre_comp_free(group);
835 generator = EC_GROUP_get0_generator(group);
836 if (generator == NULL) {
837 ECerr(EC_F_ECP_NISTZ256_MULT_PRECOMPUTE, EC_R_UNDEFINED_GENERATOR);
841 if (ecp_nistz256_is_affine_G(generator)) {
843 * No need to calculate tables for the standard generator because we
844 * have them statically.
849 if ((pre_comp = ecp_nistz256_pre_comp_new(group)) == NULL)
853 ctx = new_ctx = BN_CTX_new_ex(group->libctx);
860 order = EC_GROUP_get0_order(group);
864 if (BN_is_zero(order)) {
865 ECerr(EC_F_ECP_NISTZ256_MULT_PRECOMPUTE, EC_R_UNKNOWN_ORDER);
871 if ((precomp_storage =
872 OPENSSL_malloc(37 * 64 * sizeof(P256_POINT_AFFINE) + 64)) == NULL) {
873 ECerr(EC_F_ECP_NISTZ256_MULT_PRECOMPUTE, ERR_R_MALLOC_FAILURE);
877 preComputedTable = (void *)ALIGNPTR(precomp_storage, 64);
879 P = EC_POINT_new(group);
880 T = EC_POINT_new(group);
881 if (P == NULL || T == NULL)
885 * The zero entry is implicitly infinity, and we skip it, storing other
886 * values with -1 offset.
888 if (!EC_POINT_copy(T, generator))
891 for (k = 0; k < 64; k++) {
892 if (!EC_POINT_copy(P, T))
894 for (j = 0; j < 37; j++) {
895 P256_POINT_AFFINE temp;
897 * It would be faster to use EC_POINTs_make_affine and
898 * make multiple points affine at the same time.
900 if (group->meth->make_affine == NULL
901 || !group->meth->make_affine(group, P, ctx))
903 if (!ecp_nistz256_bignum_to_field_elem(temp.X, P->X) ||
904 !ecp_nistz256_bignum_to_field_elem(temp.Y, P->Y)) {
905 ECerr(EC_F_ECP_NISTZ256_MULT_PRECOMPUTE,
906 EC_R_COORDINATES_OUT_OF_RANGE);
909 ecp_nistz256_scatter_w7(preComputedTable[j], &temp, k);
910 for (i = 0; i < 7; i++) {
911 if (!EC_POINT_dbl(group, P, P, ctx))
915 if (!EC_POINT_add(group, T, T, generator, ctx))
919 pre_comp->group = group;
921 pre_comp->precomp = preComputedTable;
922 pre_comp->precomp_storage = precomp_storage;
923 precomp_storage = NULL;
924 SETPRECOMP(group, nistz256, pre_comp);
930 BN_CTX_free(new_ctx);
932 EC_nistz256_pre_comp_free(pre_comp);
933 OPENSSL_free(precomp_storage);
939 __owur static int ecp_nistz256_set_from_affine(EC_POINT *out, const EC_GROUP *group,
940 const P256_POINT_AFFINE *in,
945 if ((ret = bn_set_words(out->X, in->X, P256_LIMBS))
946 && (ret = bn_set_words(out->Y, in->Y, P256_LIMBS))
947 && (ret = bn_set_words(out->Z, ONE, P256_LIMBS)))
953 /* r = scalar*G + sum(scalars[i]*points[i]) */
954 __owur static int ecp_nistz256_points_mul(const EC_GROUP *group,
956 const BIGNUM *scalar,
958 const EC_POINT *points[],
959 const BIGNUM *scalars[], BN_CTX *ctx)
961 int i = 0, ret = 0, no_precomp_for_generator = 0, p_is_infinity = 0;
962 unsigned char p_str[33] = { 0 };
963 const PRECOMP256_ROW *preComputedTable = NULL;
964 const NISTZ256_PRE_COMP *pre_comp = NULL;
965 const EC_POINT *generator = NULL;
966 const BIGNUM **new_scalars = NULL;
967 const EC_POINT **new_points = NULL;
968 unsigned int idx = 0;
969 const unsigned int window_size = 7;
970 const unsigned int mask = (1 << (window_size + 1)) - 1;
978 if ((num + 1) == 0 || (num + 1) > OPENSSL_MALLOC_MAX_NELEMS(void *)) {
979 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL, ERR_R_MALLOC_FAILURE);
986 generator = EC_GROUP_get0_generator(group);
987 if (generator == NULL) {
988 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL, EC_R_UNDEFINED_GENERATOR);
992 /* look if we can use precomputed multiples of generator */
993 pre_comp = group->pre_comp.nistz256;
997 * If there is a precomputed table for the generator, check that
998 * it was generated with the same generator.
1000 EC_POINT *pre_comp_generator = EC_POINT_new(group);
1001 if (pre_comp_generator == NULL)
1004 ecp_nistz256_gather_w7(&p.a, pre_comp->precomp[0], 1);
1005 if (!ecp_nistz256_set_from_affine(pre_comp_generator,
1006 group, &p.a, ctx)) {
1007 EC_POINT_free(pre_comp_generator);
1011 if (0 == EC_POINT_cmp(group, generator, pre_comp_generator, ctx))
1012 preComputedTable = (const PRECOMP256_ROW *)pre_comp->precomp;
1014 EC_POINT_free(pre_comp_generator);
1017 if (preComputedTable == NULL && ecp_nistz256_is_affine_G(generator)) {
1019 * If there is no precomputed data, but the generator is the
1020 * default, a hardcoded table of precomputed data is used. This
1021 * is because applications, such as Apache, do not use
1022 * EC_KEY_precompute_mult.
1024 preComputedTable = ecp_nistz256_precomputed;
1027 if (preComputedTable) {
1030 if ((BN_num_bits(scalar) > 256)
1031 || BN_is_negative(scalar)) {
1032 if ((tmp_scalar = BN_CTX_get(ctx)) == NULL)
1035 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
1036 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL, ERR_R_BN_LIB);
1039 scalar = tmp_scalar;
1042 for (i = 0; i < bn_get_top(scalar) * BN_BYTES; i += BN_BYTES) {
1043 BN_ULONG d = bn_get_words(scalar)[i / BN_BYTES];
1045 p_str[i + 0] = (unsigned char)d;
1046 p_str[i + 1] = (unsigned char)(d >> 8);
1047 p_str[i + 2] = (unsigned char)(d >> 16);
1048 p_str[i + 3] = (unsigned char)(d >>= 24);
1049 if (BN_BYTES == 8) {
1051 p_str[i + 4] = (unsigned char)d;
1052 p_str[i + 5] = (unsigned char)(d >> 8);
1053 p_str[i + 6] = (unsigned char)(d >> 16);
1054 p_str[i + 7] = (unsigned char)(d >> 24);
1062 wvalue = (p_str[0] << 1) & mask;
1065 wvalue = _booth_recode_w7(wvalue);
1067 ecp_nistz256_gather_w7(&p.a, preComputedTable[0],
1070 ecp_nistz256_neg(p.p.Z, p.p.Y);
1071 copy_conditional(p.p.Y, p.p.Z, wvalue & 1);
1074 * Since affine infinity is encoded as (0,0) and
1075 * Jacobian is (,,0), we need to harmonize them
1076 * by assigning "one" or zero to Z.
1078 infty = (p.p.X[0] | p.p.X[1] | p.p.X[2] | p.p.X[3] |
1079 p.p.Y[0] | p.p.Y[1] | p.p.Y[2] | p.p.Y[3]);
1080 if (P256_LIMBS == 8)
1081 infty |= (p.p.X[4] | p.p.X[5] | p.p.X[6] | p.p.X[7] |
1082 p.p.Y[4] | p.p.Y[5] | p.p.Y[6] | p.p.Y[7]);
1084 infty = 0 - is_zero(infty);
1087 p.p.Z[0] = ONE[0] & infty;
1088 p.p.Z[1] = ONE[1] & infty;
1089 p.p.Z[2] = ONE[2] & infty;
1090 p.p.Z[3] = ONE[3] & infty;
1091 if (P256_LIMBS == 8) {
1092 p.p.Z[4] = ONE[4] & infty;
1093 p.p.Z[5] = ONE[5] & infty;
1094 p.p.Z[6] = ONE[6] & infty;
1095 p.p.Z[7] = ONE[7] & infty;
1098 for (i = 1; i < 37; i++) {
1099 unsigned int off = (idx - 1) / 8;
1100 wvalue = p_str[off] | p_str[off + 1] << 8;
1101 wvalue = (wvalue >> ((idx - 1) % 8)) & mask;
1104 wvalue = _booth_recode_w7(wvalue);
1106 ecp_nistz256_gather_w7(&t.a,
1107 preComputedTable[i], wvalue >> 1);
1109 ecp_nistz256_neg(t.p.Z, t.a.Y);
1110 copy_conditional(t.a.Y, t.p.Z, wvalue & 1);
1112 ecp_nistz256_point_add_affine(&p.p, &p.p, &t.a);
1116 no_precomp_for_generator = 1;
1121 if (no_precomp_for_generator) {
1123 * Without a precomputed table for the generator, it has to be
1124 * handled like a normal point.
1126 new_scalars = OPENSSL_malloc((num + 1) * sizeof(BIGNUM *));
1127 if (new_scalars == NULL) {
1128 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1132 new_points = OPENSSL_malloc((num + 1) * sizeof(EC_POINT *));
1133 if (new_points == NULL) {
1134 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1138 memcpy(new_scalars, scalars, num * sizeof(BIGNUM *));
1139 new_scalars[num] = scalar;
1140 memcpy(new_points, points, num * sizeof(EC_POINT *));
1141 new_points[num] = generator;
1143 scalars = new_scalars;
1144 points = new_points;
1149 P256_POINT *out = &t.p;
1153 if (!ecp_nistz256_windowed_mul(group, out, scalars, points, num, ctx))
1157 ecp_nistz256_point_add(&p.p, &p.p, out);
1160 /* Not constant-time, but we're only operating on the public output. */
1161 if (!bn_set_words(r->X, p.p.X, P256_LIMBS) ||
1162 !bn_set_words(r->Y, p.p.Y, P256_LIMBS) ||
1163 !bn_set_words(r->Z, p.p.Z, P256_LIMBS)) {
1166 r->Z_is_one = is_one(r->Z) & 1;
1172 OPENSSL_free(new_points);
1173 OPENSSL_free(new_scalars);
1177 __owur static int ecp_nistz256_get_affine(const EC_GROUP *group,
1178 const EC_POINT *point,
1179 BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
1181 BN_ULONG z_inv2[P256_LIMBS];
1182 BN_ULONG z_inv3[P256_LIMBS];
1183 BN_ULONG x_aff[P256_LIMBS];
1184 BN_ULONG y_aff[P256_LIMBS];
1185 BN_ULONG point_x[P256_LIMBS], point_y[P256_LIMBS], point_z[P256_LIMBS];
1186 BN_ULONG x_ret[P256_LIMBS], y_ret[P256_LIMBS];
1188 if (EC_POINT_is_at_infinity(group, point)) {
1189 ECerr(EC_F_ECP_NISTZ256_GET_AFFINE, EC_R_POINT_AT_INFINITY);
1193 if (!ecp_nistz256_bignum_to_field_elem(point_x, point->X) ||
1194 !ecp_nistz256_bignum_to_field_elem(point_y, point->Y) ||
1195 !ecp_nistz256_bignum_to_field_elem(point_z, point->Z)) {
1196 ECerr(EC_F_ECP_NISTZ256_GET_AFFINE, EC_R_COORDINATES_OUT_OF_RANGE);
1200 ecp_nistz256_mod_inverse(z_inv3, point_z);
1201 ecp_nistz256_sqr_mont(z_inv2, z_inv3);
1202 ecp_nistz256_mul_mont(x_aff, z_inv2, point_x);
1205 ecp_nistz256_from_mont(x_ret, x_aff);
1206 if (!bn_set_words(x, x_ret, P256_LIMBS))
1211 ecp_nistz256_mul_mont(z_inv3, z_inv3, z_inv2);
1212 ecp_nistz256_mul_mont(y_aff, z_inv3, point_y);
1213 ecp_nistz256_from_mont(y_ret, y_aff);
1214 if (!bn_set_words(y, y_ret, P256_LIMBS))
1221 static NISTZ256_PRE_COMP *ecp_nistz256_pre_comp_new(const EC_GROUP *group)
1223 NISTZ256_PRE_COMP *ret = NULL;
1228 ret = OPENSSL_zalloc(sizeof(*ret));
1231 ECerr(EC_F_ECP_NISTZ256_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1236 ret->w = 6; /* default */
1237 ret->references = 1;
1239 ret->lock = CRYPTO_THREAD_lock_new();
1240 if (ret->lock == NULL) {
1241 ECerr(EC_F_ECP_NISTZ256_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1248 NISTZ256_PRE_COMP *EC_nistz256_pre_comp_dup(NISTZ256_PRE_COMP *p)
1252 CRYPTO_UP_REF(&p->references, &i, p->lock);
1256 void EC_nistz256_pre_comp_free(NISTZ256_PRE_COMP *pre)
1263 CRYPTO_DOWN_REF(&pre->references, &i, pre->lock);
1264 REF_PRINT_COUNT("EC_nistz256", pre);
1267 REF_ASSERT_ISNT(i < 0);
1269 OPENSSL_free(pre->precomp_storage);
1270 CRYPTO_THREAD_lock_free(pre->lock);
1275 static int ecp_nistz256_window_have_precompute_mult(const EC_GROUP *group)
1277 /* There is a hard-coded table for the default generator. */
1278 const EC_POINT *generator = EC_GROUP_get0_generator(group);
1280 if (generator != NULL && ecp_nistz256_is_affine_G(generator)) {
1281 /* There is a hard-coded table for the default generator. */
1285 return HAVEPRECOMP(group, nistz256);
1288 #if defined(__x86_64) || defined(__x86_64__) || \
1289 defined(_M_AMD64) || defined(_M_X64) || \
1290 defined(__powerpc64__) || defined(_ARCH_PP64) || \
1291 defined(__aarch64__)
1293 * Montgomery mul modulo Order(P): res = a*b*2^-256 mod Order(P)
1295 void ecp_nistz256_ord_mul_mont(BN_ULONG res[P256_LIMBS],
1296 const BN_ULONG a[P256_LIMBS],
1297 const BN_ULONG b[P256_LIMBS]);
1298 void ecp_nistz256_ord_sqr_mont(BN_ULONG res[P256_LIMBS],
1299 const BN_ULONG a[P256_LIMBS],
1302 static int ecp_nistz256_inv_mod_ord(const EC_GROUP *group, BIGNUM *r,
1303 const BIGNUM *x, BN_CTX *ctx)
1305 /* RR = 2^512 mod ord(p256) */
1306 static const BN_ULONG RR[P256_LIMBS] = {
1307 TOBN(0x83244c95,0xbe79eea2), TOBN(0x4699799c,0x49bd6fa6),
1308 TOBN(0x2845b239,0x2b6bec59), TOBN(0x66e12d94,0xf3d95620)
1310 /* The constant 1 (unlike ONE that is one in Montgomery representation) */
1311 static const BN_ULONG one[P256_LIMBS] = {
1312 TOBN(0,1), TOBN(0,0), TOBN(0,0), TOBN(0,0)
1315 * We don't use entry 0 in the table, so we omit it and address
1318 BN_ULONG table[15][P256_LIMBS];
1319 BN_ULONG out[P256_LIMBS], t[P256_LIMBS];
1322 i_1 = 0, i_10, i_11, i_101, i_111, i_1010, i_1111,
1323 i_10101, i_101010, i_101111, i_x6, i_x8, i_x16, i_x32
1327 * Catch allocation failure early.
1329 if (bn_wexpand(r, P256_LIMBS) == NULL) {
1330 ECerr(EC_F_ECP_NISTZ256_INV_MOD_ORD, ERR_R_BN_LIB);
1334 if ((BN_num_bits(x) > 256) || BN_is_negative(x)) {
1337 if ((tmp = BN_CTX_get(ctx)) == NULL
1338 || !BN_nnmod(tmp, x, group->order, ctx)) {
1339 ECerr(EC_F_ECP_NISTZ256_INV_MOD_ORD, ERR_R_BN_LIB);
1345 if (!ecp_nistz256_bignum_to_field_elem(t, x)) {
1346 ECerr(EC_F_ECP_NISTZ256_INV_MOD_ORD, EC_R_COORDINATES_OUT_OF_RANGE);
1350 ecp_nistz256_ord_mul_mont(table[0], t, RR);
1353 * Original sparse-then-fixed-window algorithm, retained for reference.
1355 for (i = 2; i < 16; i += 2) {
1356 ecp_nistz256_ord_sqr_mont(table[i-1], table[i/2-1], 1);
1357 ecp_nistz256_ord_mul_mont(table[i], table[i-1], table[0]);
1361 * The top 128bit of the exponent are highly redudndant, so we
1362 * perform an optimized flow
1364 ecp_nistz256_ord_sqr_mont(t, table[15-1], 4); /* f0 */
1365 ecp_nistz256_ord_mul_mont(t, t, table[15-1]); /* ff */
1367 ecp_nistz256_ord_sqr_mont(out, t, 8); /* ff00 */
1368 ecp_nistz256_ord_mul_mont(out, out, t); /* ffff */
1370 ecp_nistz256_ord_sqr_mont(t, out, 16); /* ffff0000 */
1371 ecp_nistz256_ord_mul_mont(t, t, out); /* ffffffff */
1373 ecp_nistz256_ord_sqr_mont(out, t, 64); /* ffffffff0000000000000000 */
1374 ecp_nistz256_ord_mul_mont(out, out, t); /* ffffffff00000000ffffffff */
1376 ecp_nistz256_ord_sqr_mont(out, out, 32); /* ffffffff00000000ffffffff00000000 */
1377 ecp_nistz256_ord_mul_mont(out, out, t); /* ffffffff00000000ffffffffffffffff */
1380 * The bottom 128 bit of the exponent are processed with fixed 4-bit window
1382 for(i = 0; i < 32; i++) {
1383 /* expLo - the low 128 bits of the exponent we use (ord(p256) - 2),
1384 * split into nibbles */
1385 static const unsigned char expLo[32] = {
1386 0xb,0xc,0xe,0x6,0xf,0xa,0xa,0xd,0xa,0x7,0x1,0x7,0x9,0xe,0x8,0x4,
1387 0xf,0x3,0xb,0x9,0xc,0xa,0xc,0x2,0xf,0xc,0x6,0x3,0x2,0x5,0x4,0xf
1390 ecp_nistz256_ord_sqr_mont(out, out, 4);
1391 /* The exponent is public, no need in constant-time access */
1392 ecp_nistz256_ord_mul_mont(out, out, table[expLo[i]-1]);
1396 * https://briansmith.org/ecc-inversion-addition-chains-01#p256_scalar_inversion
1398 * Even though this code path spares 12 squarings, 4.5%, and 13
1399 * multiplications, 25%, on grand scale sign operation is not that
1400 * much faster, not more that 2%...
1403 /* pre-calculate powers */
1404 ecp_nistz256_ord_sqr_mont(table[i_10], table[i_1], 1);
1406 ecp_nistz256_ord_mul_mont(table[i_11], table[i_1], table[i_10]);
1408 ecp_nistz256_ord_mul_mont(table[i_101], table[i_11], table[i_10]);
1410 ecp_nistz256_ord_mul_mont(table[i_111], table[i_101], table[i_10]);
1412 ecp_nistz256_ord_sqr_mont(table[i_1010], table[i_101], 1);
1414 ecp_nistz256_ord_mul_mont(table[i_1111], table[i_1010], table[i_101]);
1416 ecp_nistz256_ord_sqr_mont(table[i_10101], table[i_1010], 1);
1417 ecp_nistz256_ord_mul_mont(table[i_10101], table[i_10101], table[i_1]);
1419 ecp_nistz256_ord_sqr_mont(table[i_101010], table[i_10101], 1);
1421 ecp_nistz256_ord_mul_mont(table[i_101111], table[i_101010], table[i_101]);
1423 ecp_nistz256_ord_mul_mont(table[i_x6], table[i_101010], table[i_10101]);
1425 ecp_nistz256_ord_sqr_mont(table[i_x8], table[i_x6], 2);
1426 ecp_nistz256_ord_mul_mont(table[i_x8], table[i_x8], table[i_11]);
1428 ecp_nistz256_ord_sqr_mont(table[i_x16], table[i_x8], 8);
1429 ecp_nistz256_ord_mul_mont(table[i_x16], table[i_x16], table[i_x8]);
1431 ecp_nistz256_ord_sqr_mont(table[i_x32], table[i_x16], 16);
1432 ecp_nistz256_ord_mul_mont(table[i_x32], table[i_x32], table[i_x16]);
1435 ecp_nistz256_ord_sqr_mont(out, table[i_x32], 64);
1436 ecp_nistz256_ord_mul_mont(out, out, table[i_x32]);
1438 for (i = 0; i < 27; i++) {
1439 static const struct { unsigned char p, i; } chain[27] = {
1440 { 32, i_x32 }, { 6, i_101111 }, { 5, i_111 },
1441 { 4, i_11 }, { 5, i_1111 }, { 5, i_10101 },
1442 { 4, i_101 }, { 3, i_101 }, { 3, i_101 },
1443 { 5, i_111 }, { 9, i_101111 }, { 6, i_1111 },
1444 { 2, i_1 }, { 5, i_1 }, { 6, i_1111 },
1445 { 5, i_111 }, { 4, i_111 }, { 5, i_111 },
1446 { 5, i_101 }, { 3, i_11 }, { 10, i_101111 },
1447 { 2, i_11 }, { 5, i_11 }, { 5, i_11 },
1448 { 3, i_1 }, { 7, i_10101 }, { 6, i_1111 }
1451 ecp_nistz256_ord_sqr_mont(out, out, chain[i].p);
1452 ecp_nistz256_ord_mul_mont(out, out, table[chain[i].i]);
1455 ecp_nistz256_ord_mul_mont(out, out, one);
1458 * Can't fail, but check return code to be consistent anyway.
1460 if (!bn_set_words(r, out, P256_LIMBS))
1468 # define ecp_nistz256_inv_mod_ord NULL
1471 const EC_METHOD *EC_GFp_nistz256_method(void)
1473 static const EC_METHOD ret = {
1474 EC_FLAGS_DEFAULT_OCT,
1475 NID_X9_62_prime_field,
1476 ec_GFp_mont_group_init,
1477 ec_GFp_mont_group_finish,
1478 ec_GFp_mont_group_clear_finish,
1479 ec_GFp_mont_group_copy,
1480 ec_GFp_mont_group_set_curve,
1481 ec_GFp_simple_group_get_curve,
1482 ec_GFp_simple_group_get_degree,
1483 ec_group_simple_order_bits,
1484 ec_GFp_simple_group_check_discriminant,
1485 ec_GFp_simple_point_init,
1486 ec_GFp_simple_point_finish,
1487 ec_GFp_simple_point_clear_finish,
1488 ec_GFp_simple_point_copy,
1489 ec_GFp_simple_point_set_to_infinity,
1490 ec_GFp_simple_point_set_affine_coordinates,
1491 ecp_nistz256_get_affine,
1495 ec_GFp_simple_invert,
1496 ec_GFp_simple_is_at_infinity,
1497 ec_GFp_simple_is_on_curve,
1499 ec_GFp_simple_make_affine,
1500 ec_GFp_simple_points_make_affine,
1501 ecp_nistz256_points_mul, /* mul */
1502 ecp_nistz256_mult_precompute, /* precompute_mult */
1503 ecp_nistz256_window_have_precompute_mult, /* have_precompute_mult */
1504 ec_GFp_mont_field_mul,
1505 ec_GFp_mont_field_sqr,
1507 ec_GFp_mont_field_inv,
1508 ec_GFp_mont_field_encode,
1509 ec_GFp_mont_field_decode,
1510 ec_GFp_mont_field_set_to_one,
1511 ec_key_simple_priv2oct,
1512 ec_key_simple_oct2priv,
1513 0, /* set private */
1514 ec_key_simple_generate_key,
1515 ec_key_simple_check_key,
1516 ec_key_simple_generate_public_key,
1519 ecdh_simple_compute_key,
1520 ecdsa_simple_sign_setup,
1521 ecdsa_simple_sign_sig,
1522 ecdsa_simple_verify_sig,
1523 ecp_nistz256_inv_mod_ord, /* can be #define-d NULL */
1524 0, /* blind_coordinates */
1526 0, /* ladder_step */