1 /*
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.
5 *
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
10 *
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.
15 *
16 * Reference:
17 * S.Gueron and V.Krasnov, "Fast Prime Field Elliptic Curve Cryptography with
18 * 256 Bit Primes"
19 */
20
21 #include <string.h>
22
23 #include "internal/cryptlib.h"
24 #include "crypto/bn.h"
25 #include "ec_local.h"
26 #include "internal/refcount.h"
27
28 #if BN_BITS2 != 64
29 # define TOBN(hi,lo) lo,hi
30 #else
31 # define TOBN(hi,lo) ((BN_ULONG)hi<<32|lo)
32 #endif
33
34 #if defined(__GNUC__)
35 # define ALIGN32 __attribute((aligned(32)))
36 #elif defined(_MSC_VER)
37 # define ALIGN32 __declspec(align(32))
38 #else
39 # define ALIGN32
40 #endif
41
42 #define ALIGNPTR(p,N) ((unsigned char *)p+N-(size_t)p%N)
43 #define P256_LIMBS (256/BN_BITS2)
44
45 typedef unsigned short u16;
46
47 typedef struct {
48 BN_ULONG X[P256_LIMBS];
49 BN_ULONG Y[P256_LIMBS];
50 BN_ULONG Z[P256_LIMBS];
51 } P256_POINT;
52
53 typedef struct {
54 BN_ULONG X[P256_LIMBS];
55 BN_ULONG Y[P256_LIMBS];
56 } P256_POINT_AFFINE;
57
58 typedef P256_POINT_AFFINE PRECOMP256_ROW[64];
59
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 */
64 /*
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.
68 */
69 PRECOMP256_ROW *precomp;
70 void *precomp_storage;
71 CRYPTO_REF_COUNT references;
72 CRYPTO_RWLOCK *lock;
73 };
74
75 /* Functions implemented in assembly */
76 /*
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...
91 */
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]);
102
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);
134
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)
139 };
140
141 static NISTZ256_PRE_COMP *ecp_nistz256_pre_comp_new(const EC_GROUP *group);
142
143 /* Precomputed tables for the default generator */
144 extern const PRECOMP256_ROW ecp_nistz256_precomputed[37];
145
146 /* Recode window to a signed digit, see ecp_nistputil.c for details */
_booth_recode_w5(unsigned int in)147 static unsigned int _booth_recode_w5(unsigned int in)
148 {
149 unsigned int s, d;
150
151 s = ~((in >> 5) - 1);
152 d = (1 << 6) - in - 1;
153 d = (d & s) | (in & ~s);
154 d = (d >> 1) + (d & 1);
155
156 return (d << 1) + (s & 1);
157 }
158
_booth_recode_w7(unsigned int in)159 static unsigned int _booth_recode_w7(unsigned int in)
160 {
161 unsigned int s, d;
162
163 s = ~((in >> 7) - 1);
164 d = (1 << 8) - in - 1;
165 d = (d & s) | (in & ~s);
166 d = (d >> 1) + (d & 1);
167
168 return (d << 1) + (s & 1);
169 }
170
copy_conditional(BN_ULONG dst[P256_LIMBS],const BN_ULONG src[P256_LIMBS],BN_ULONG move)171 static void copy_conditional(BN_ULONG dst[P256_LIMBS],
172 const BN_ULONG src[P256_LIMBS], BN_ULONG move)
173 {
174 BN_ULONG mask1 = 0-move;
175 BN_ULONG mask2 = ~mask1;
176
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);
186 }
187 }
188
is_zero(BN_ULONG in)189 static BN_ULONG is_zero(BN_ULONG in)
190 {
191 in |= (0 - in);
192 in = ~in;
193 in >>= BN_BITS2 - 1;
194 return in;
195 }
196
is_equal(const BN_ULONG a[P256_LIMBS],const BN_ULONG b[P256_LIMBS])197 static BN_ULONG is_equal(const BN_ULONG a[P256_LIMBS],
198 const BN_ULONG b[P256_LIMBS])
199 {
200 BN_ULONG res;
201
202 res = a[0] ^ b[0];
203 res |= a[1] ^ b[1];
204 res |= a[2] ^ b[2];
205 res |= a[3] ^ b[3];
206 if (P256_LIMBS == 8) {
207 res |= a[4] ^ b[4];
208 res |= a[5] ^ b[5];
209 res |= a[6] ^ b[6];
210 res |= a[7] ^ b[7];
211 }
212
213 return is_zero(res);
214 }
215
is_one(const BIGNUM * z)216 static BN_ULONG is_one(const BIGNUM *z)
217 {
218 BN_ULONG res = 0;
219 BN_ULONG *a = bn_get_words(z);
220
221 if (bn_get_top(z) == (P256_LIMBS - P256_LIMBS / 8)) {
222 res = a[0] ^ ONE[0];
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];
230 /*
231 * no check for a[7] (being zero) on 32-bit platforms,
232 * because value of "one" takes only 7 limbs.
233 */
234 }
235 res = is_zero(res);
236 }
237
238 return res;
239 }
240
241 /*
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.)
250 */
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,
256 const P256_POINT *a,
257 const P256_POINT_AFFINE *b);
258 #else
259 /* Point double: r = 2*a */
ecp_nistz256_point_double(P256_POINT * r,const P256_POINT * a)260 static void ecp_nistz256_point_double(P256_POINT *r, const P256_POINT *a)
261 {
262 BN_ULONG S[P256_LIMBS];
263 BN_ULONG M[P256_LIMBS];
264 BN_ULONG Zsqr[P256_LIMBS];
265 BN_ULONG tmp0[P256_LIMBS];
266
267 const BN_ULONG *in_x = a->X;
268 const BN_ULONG *in_y = a->Y;
269 const BN_ULONG *in_z = a->Z;
270
271 BN_ULONG *res_x = r->X;
272 BN_ULONG *res_y = r->Y;
273 BN_ULONG *res_z = r->Z;
274
275 ecp_nistz256_mul_by_2(S, in_y);
276
277 ecp_nistz256_sqr_mont(Zsqr, in_z);
278
279 ecp_nistz256_sqr_mont(S, S);
280
281 ecp_nistz256_mul_mont(res_z, in_z, in_y);
282 ecp_nistz256_mul_by_2(res_z, res_z);
283
284 ecp_nistz256_add(M, in_x, Zsqr);
285 ecp_nistz256_sub(Zsqr, in_x, Zsqr);
286
287 ecp_nistz256_sqr_mont(res_y, S);
288 ecp_nistz256_div_by_2(res_y, res_y);
289
290 ecp_nistz256_mul_mont(M, M, Zsqr);
291 ecp_nistz256_mul_by_3(M, M);
292
293 ecp_nistz256_mul_mont(S, S, in_x);
294 ecp_nistz256_mul_by_2(tmp0, S);
295
296 ecp_nistz256_sqr_mont(res_x, M);
297
298 ecp_nistz256_sub(res_x, res_x, tmp0);
299 ecp_nistz256_sub(S, S, res_x);
300
301 ecp_nistz256_mul_mont(S, S, M);
302 ecp_nistz256_sub(res_y, S, res_y);
303 }
304
305 /* Point addition: r = a+b */
ecp_nistz256_point_add(P256_POINT * r,const P256_POINT * a,const P256_POINT * b)306 static void ecp_nistz256_point_add(P256_POINT *r,
307 const P256_POINT *a, const P256_POINT *b)
308 {
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];
317
318 BN_ULONG res_x[P256_LIMBS];
319 BN_ULONG res_y[P256_LIMBS];
320 BN_ULONG res_z[P256_LIMBS];
321
322 BN_ULONG in1infty, in2infty;
323
324 const BN_ULONG *in1_x = a->X;
325 const BN_ULONG *in1_y = a->Y;
326 const BN_ULONG *in1_z = a->Z;
327
328 const BN_ULONG *in2_x = b->X;
329 const BN_ULONG *in2_y = b->Y;
330 const BN_ULONG *in2_z = b->Z;
331
332 /*
333 * Infinity in encoded as (,,0)
334 */
335 in1infty = (in1_z[0] | in1_z[1] | in1_z[2] | in1_z[3]);
336 if (P256_LIMBS == 8)
337 in1infty |= (in1_z[4] | in1_z[5] | in1_z[6] | in1_z[7]);
338
339 in2infty = (in2_z[0] | in2_z[1] | in2_z[2] | in2_z[3]);
340 if (P256_LIMBS == 8)
341 in2infty |= (in2_z[4] | in2_z[5] | in2_z[6] | in2_z[7]);
342
343 in1infty = is_zero(in1infty);
344 in2infty = is_zero(in2infty);
345
346 ecp_nistz256_sqr_mont(Z2sqr, in2_z); /* Z2^2 */
347 ecp_nistz256_sqr_mont(Z1sqr, in1_z); /* Z1^2 */
348
349 ecp_nistz256_mul_mont(S1, Z2sqr, in2_z); /* S1 = Z2^3 */
350 ecp_nistz256_mul_mont(S2, Z1sqr, in1_z); /* S2 = Z1^3 */
351
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 */
355
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 */
359
360 /*
361 * The formulae are incorrect if the points are equal so we check for
362 * this and do doubling if this happens.
363 *
364 * Points here are in Jacobian projective coordinates (Xi, Yi, Zi)
365 * that are bound to the affine coordinates (xi, yi) by the following
366 * equations:
367 * - xi = Xi / (Zi)^2
368 * - y1 = Yi / (Zi)^3
369 *
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
375 *
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.
380 *
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.
384 *
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.
390 *
391 * We use bitwise operations to avoid potential side-channels introduced by
392 * the short-circuiting behaviour of boolean operators.
393 */
394 if (is_equal(U1, U2) & ~in1infty & ~in2infty & is_equal(S1, S2)) {
395 /*
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
398 * ECDSA signing.
399 */
400 ecp_nistz256_point_double(r, a);
401 return;
402 }
403
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 */
409
410 ecp_nistz256_mul_mont(U2, U1, Hsqr); /* U1*H^2 */
411 ecp_nistz256_mul_by_2(Hsqr, U2); /* 2*U1*H^2 */
412
413 ecp_nistz256_sub(res_x, Rsqr, Hsqr);
414 ecp_nistz256_sub(res_x, res_x, Hcub);
415
416 ecp_nistz256_sub(res_y, U2, res_x);
417
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);
421
422 copy_conditional(res_x, in2_x, in1infty);
423 copy_conditional(res_y, in2_y, in1infty);
424 copy_conditional(res_z, in2_z, in1infty);
425
426 copy_conditional(res_x, in1_x, in2infty);
427 copy_conditional(res_y, in1_y, in2infty);
428 copy_conditional(res_z, in1_z, in2infty);
429
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));
433 }
434
435 /* Point addition when b is known to be affine: r = a+b */
ecp_nistz256_point_add_affine(P256_POINT * r,const P256_POINT * a,const P256_POINT_AFFINE * b)436 static void ecp_nistz256_point_add_affine(P256_POINT *r,
437 const P256_POINT *a,
438 const P256_POINT_AFFINE *b)
439 {
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];
446
447 BN_ULONG res_x[P256_LIMBS];
448 BN_ULONG res_y[P256_LIMBS];
449 BN_ULONG res_z[P256_LIMBS];
450
451 BN_ULONG in1infty, in2infty;
452
453 const BN_ULONG *in1_x = a->X;
454 const BN_ULONG *in1_y = a->Y;
455 const BN_ULONG *in1_z = a->Z;
456
457 const BN_ULONG *in2_x = b->X;
458 const BN_ULONG *in2_y = b->Y;
459
460 /*
461 * Infinity in encoded as (,,0)
462 */
463 in1infty = (in1_z[0] | in1_z[1] | in1_z[2] | in1_z[3]);
464 if (P256_LIMBS == 8)
465 in1infty |= (in1_z[4] | in1_z[5] | in1_z[6] | in1_z[7]);
466
467 /*
468 * In affine representation we encode infinity as (0,0), which is
469 * not on the curve, so it is OK
470 */
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]);
473 if (P256_LIMBS == 8)
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]);
476
477 in1infty = is_zero(in1infty);
478 in2infty = is_zero(in2infty);
479
480 ecp_nistz256_sqr_mont(Z1sqr, in1_z); /* Z1^2 */
481
482 ecp_nistz256_mul_mont(U2, in2_x, Z1sqr); /* U2 = X2*Z1^2 */
483 ecp_nistz256_sub(H, U2, in1_x); /* H = U2 - U1 */
484
485 ecp_nistz256_mul_mont(S2, Z1sqr, in1_z); /* S2 = Z1^3 */
486
487 ecp_nistz256_mul_mont(res_z, H, in1_z); /* Z3 = H*Z1*Z2 */
488
489 ecp_nistz256_mul_mont(S2, S2, in2_y); /* S2 = Y2*Z1^3 */
490 ecp_nistz256_sub(R, S2, in1_y); /* R = S2 - S1 */
491
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 */
495
496 ecp_nistz256_mul_mont(U2, in1_x, Hsqr); /* U1*H^2 */
497 ecp_nistz256_mul_by_2(Hsqr, U2); /* 2*U1*H^2 */
498
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);
502
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);
506
507 copy_conditional(res_x, in2_x, in1infty);
508 copy_conditional(res_x, in1_x, in2infty);
509
510 copy_conditional(res_y, in2_y, in1infty);
511 copy_conditional(res_y, in1_y, in2infty);
512
513 copy_conditional(res_z, ONE, in1infty);
514 copy_conditional(res_z, in1_z, in2infty);
515
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));
519 }
520 #endif
521
522 /* r = in^-1 mod p */
ecp_nistz256_mod_inverse(BN_ULONG r[P256_LIMBS],const BN_ULONG in[P256_LIMBS])523 static void ecp_nistz256_mod_inverse(BN_ULONG r[P256_LIMBS],
524 const BN_ULONG in[P256_LIMBS])
525 {
526 /*
527 * The poly is ffffffff 00000001 00000000 00000000 00000000 ffffffff
528 * ffffffff ffffffff We use FLT and used poly-2 as exponent
529 */
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];
536 int i;
537
538 ecp_nistz256_sqr_mont(res, in);
539 ecp_nistz256_mul_mont(p2, res, in); /* 3*p */
540
541 ecp_nistz256_sqr_mont(res, p2);
542 ecp_nistz256_sqr_mont(res, res);
543 ecp_nistz256_mul_mont(p4, res, p2); /* f*p */
544
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 */
550
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 */
555
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 */
560
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);
565
566 for (i = 0; i < 32 * 4; i++)
567 ecp_nistz256_sqr_mont(res, res);
568 ecp_nistz256_mul_mont(res, res, p32);
569
570 for (i = 0; i < 32; i++)
571 ecp_nistz256_sqr_mont(res, res);
572 ecp_nistz256_mul_mont(res, res, p32);
573
574 for (i = 0; i < 16; i++)
575 ecp_nistz256_sqr_mont(res, res);
576 ecp_nistz256_mul_mont(res, res, p16);
577
578 for (i = 0; i < 8; i++)
579 ecp_nistz256_sqr_mont(res, res);
580 ecp_nistz256_mul_mont(res, res, p8);
581
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);
587
588 ecp_nistz256_sqr_mont(res, res);
589 ecp_nistz256_sqr_mont(res, res);
590 ecp_nistz256_mul_mont(res, res, p2);
591
592 ecp_nistz256_sqr_mont(res, res);
593 ecp_nistz256_sqr_mont(res, res);
594 ecp_nistz256_mul_mont(res, res, in);
595
596 memcpy(r, res, sizeof(res));
597 }
598
599 /*
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.
602 */
ecp_nistz256_bignum_to_field_elem(BN_ULONG out[P256_LIMBS],const BIGNUM * in)603 __owur static int ecp_nistz256_bignum_to_field_elem(BN_ULONG out[P256_LIMBS],
604 const BIGNUM *in)
605 {
606 return bn_copy_words(out, in, P256_LIMBS);
607 }
608
609 /* r = sum(scalar[i]*point[i]) */
ecp_nistz256_windowed_mul(const EC_GROUP * group,P256_POINT * r,const BIGNUM ** scalar,const EC_POINT ** point,size_t num,BN_CTX * ctx)610 __owur static int ecp_nistz256_windowed_mul(const EC_GROUP *group,
611 P256_POINT *r,
612 const BIGNUM **scalar,
613 const EC_POINT **point,
614 size_t num, BN_CTX *ctx)
615 {
616 size_t i;
617 int j, ret = 0;
618 unsigned int idx;
619 unsigned char (*p_str)[33] = NULL;
620 const unsigned int window_size = 5;
621 const unsigned int mask = (1 << (window_size + 1)) - 1;
622 unsigned int wvalue;
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;
627
628 if ((num * 16 + 6) > OPENSSL_MALLOC_MAX_NELEMS(P256_POINT)
629 || (table_storage =
630 OPENSSL_malloc((num * 16 + 5) * sizeof(P256_POINT) + 64)) == NULL
631 || (p_str =
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);
635 goto err;
636 }
637
638 table = (void *)ALIGNPTR(table_storage, 64);
639 temp = (P256_POINT *)(table + num);
640
641 for (i = 0; i < num; i++) {
642 P256_POINT *row = table[i];
643
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])) {
646 BIGNUM *mod;
647
648 if ((mod = BN_CTX_get(ctx)) == NULL)
649 goto err;
650 if (!BN_nnmod(mod, scalar[i], group->order, ctx)) {
651 ECerr(EC_F_ECP_NISTZ256_WINDOWED_MUL, ERR_R_BN_LIB);
652 goto err;
653 }
654 scalars[i] = mod;
655 } else
656 scalars[i] = scalar[i];
657
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];
660
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);
665 if (BN_BYTES == 8) {
666 d >>= 8;
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);
671 }
672 }
673 for (; j < 33; j++)
674 p_str[i][j] = 0;
675
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);
681 goto err;
682 }
683
684 /*
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
687 * of -1 in table.
688 */
689
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);
721 }
722
723 idx = 255;
724
725 wvalue = p_str[0][(idx - 1) / 8];
726 wvalue = (wvalue >> ((idx - 1) % 8)) & mask;
727
728 /*
729 * We gather to temp[0], because we know it's position relative
730 * to table
731 */
732 ecp_nistz256_gather_w5(&temp[0], table[0], _booth_recode_w5(wvalue) >> 1);
733 memcpy(r, &temp[0], sizeof(temp[0]));
734
735 while (idx >= 5) {
736 for (i = (idx == 255 ? 1 : 0); i < num; i++) {
737 unsigned int off = (idx - 1) / 8;
738
739 wvalue = p_str[i][off] | p_str[i][off + 1] << 8;
740 wvalue = (wvalue >> ((idx - 1) % 8)) & mask;
741
742 wvalue = _booth_recode_w5(wvalue);
743
744 ecp_nistz256_gather_w5(&temp[0], table[i], wvalue >> 1);
745
746 ecp_nistz256_neg(temp[1].Y, temp[0].Y);
747 copy_conditional(temp[0].Y, temp[1].Y, (wvalue & 1));
748
749 ecp_nistz256_point_add(r, r, &temp[0]);
750 }
751
752 idx -= window_size;
753
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);
759 }
760
761 /* Final window */
762 for (i = 0; i < num; i++) {
763 wvalue = p_str[i][0];
764 wvalue = (wvalue << 1) & mask;
765
766 wvalue = _booth_recode_w5(wvalue);
767
768 ecp_nistz256_gather_w5(&temp[0], table[i], wvalue >> 1);
769
770 ecp_nistz256_neg(temp[1].Y, temp[0].Y);
771 copy_conditional(temp[0].Y, temp[1].Y, wvalue & 1);
772
773 ecp_nistz256_point_add(r, r, &temp[0]);
774 }
775
776 ret = 1;
777 err:
778 OPENSSL_free(table_storage);
779 OPENSSL_free(p_str);
780 OPENSSL_free(scalars);
781 return ret;
782 }
783
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)
788 };
789
790 static const BN_ULONG def_yG[P256_LIMBS] = {
791 TOBN(0xddf25357, 0xce95560a), TOBN(0x8b4ab8e4, 0xba19e45c),
792 TOBN(0xd2e88688, 0xdd21f325), TOBN(0x8571ff18, 0x25885d85)
793 };
794
795 /*
796 * ecp_nistz256_is_affine_G returns one if |generator| is the standard, P-256
797 * generator.
798 */
ecp_nistz256_is_affine_G(const EC_POINT * generator)799 static int ecp_nistz256_is_affine_G(const EC_POINT *generator)
800 {
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);
806 }
807
ecp_nistz256_mult_precompute(EC_GROUP * group,BN_CTX * ctx)808 __owur static int ecp_nistz256_mult_precompute(EC_GROUP *group, BN_CTX *ctx)
809 {
810 /*
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.
815 */
816 const BIGNUM *order;
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;
822 size_t w;
823
824 PRECOMP256_ROW *preComputedTable = NULL;
825 unsigned char *precomp_storage = NULL;
826
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);
832 return 0;
833 }
834
835 if (ecp_nistz256_is_affine_G(generator)) {
836 /*
837 * No need to calculate tables for the standard generator because we
838 * have them statically.
839 */
840 return 1;
841 }
842
843 if ((pre_comp = ecp_nistz256_pre_comp_new(group)) == NULL)
844 return 0;
845
846 if (ctx == NULL) {
847 ctx = new_ctx = BN_CTX_new();
848 if (ctx == NULL)
849 goto err;
850 }
851
852 BN_CTX_start(ctx);
853
854 order = EC_GROUP_get0_order(group);
855 if (order == NULL)
856 goto err;
857
858 if (BN_is_zero(order)) {
859 ECerr(EC_F_ECP_NISTZ256_MULT_PRECOMPUTE, EC_R_UNKNOWN_ORDER);
860 goto err;
861 }
862
863 w = 7;
864
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);
868 goto err;
869 }
870
871 preComputedTable = (void *)ALIGNPTR(precomp_storage, 64);
872
873 P = EC_POINT_new(group);
874 T = EC_POINT_new(group);
875 if (P == NULL || T == NULL)
876 goto err;
877
878 /*
879 * The zero entry is implicitly infinity, and we skip it, storing other
880 * values with -1 offset.
881 */
882 if (!EC_POINT_copy(T, generator))
883 goto err;
884
885 for (k = 0; k < 64; k++) {
886 if (!EC_POINT_copy(P, T))
887 goto err;
888 for (j = 0; j < 37; j++) {
889 P256_POINT_AFFINE temp;
890 /*
891 * It would be faster to use EC_POINTs_make_affine and
892 * make multiple points affine at the same time.
893 */
894 if (!EC_POINT_make_affine(group, P, ctx))
895 goto err;
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);
900 goto err;
901 }
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))
905 goto err;
906 }
907 }
908 if (!EC_POINT_add(group, T, T, generator, ctx))
909 goto err;
910 }
911
912 pre_comp->group = group;
913 pre_comp->w = w;
914 pre_comp->precomp = preComputedTable;
915 pre_comp->precomp_storage = precomp_storage;
916 precomp_storage = NULL;
917 SETPRECOMP(group, nistz256, pre_comp);
918 pre_comp = NULL;
919 ret = 1;
920
921 err:
922 BN_CTX_end(ctx);
923 BN_CTX_free(new_ctx);
924
925 EC_nistz256_pre_comp_free(pre_comp);
926 OPENSSL_free(precomp_storage);
927 EC_POINT_free(P);
928 EC_POINT_free(T);
929 return ret;
930 }
931
ecp_nistz256_set_from_affine(EC_POINT * out,const EC_GROUP * group,const P256_POINT_AFFINE * in,BN_CTX * ctx)932 __owur static int ecp_nistz256_set_from_affine(EC_POINT *out, const EC_GROUP *group,
933 const P256_POINT_AFFINE *in,
934 BN_CTX *ctx)
935 {
936 int ret = 0;
937
938 if ((ret = bn_set_words(out->X, in->X, P256_LIMBS))
939 && (ret = bn_set_words(out->Y, in->Y, P256_LIMBS))
940 && (ret = bn_set_words(out->Z, ONE, P256_LIMBS)))
941 out->Z_is_one = 1;
942
943 return ret;
944 }
945
946 /* r = scalar*G + sum(scalars[i]*points[i]) */
ecp_nistz256_points_mul(const EC_GROUP * group,EC_POINT * r,const BIGNUM * scalar,size_t num,const EC_POINT * points[],const BIGNUM * scalars[],BN_CTX * ctx)947 __owur static int ecp_nistz256_points_mul(const EC_GROUP *group,
948 EC_POINT *r,
949 const BIGNUM *scalar,
950 size_t num,
951 const EC_POINT *points[],
952 const BIGNUM *scalars[], BN_CTX *ctx)
953 {
954 int i = 0, ret = 0, no_precomp_for_generator = 0, p_is_infinity = 0;
955 unsigned char p_str[33] = { 0 };
956 const PRECOMP256_ROW *preComputedTable = NULL;
957 const NISTZ256_PRE_COMP *pre_comp = NULL;
958 const EC_POINT *generator = NULL;
959 const BIGNUM **new_scalars = NULL;
960 const EC_POINT **new_points = NULL;
961 unsigned int idx = 0;
962 const unsigned int window_size = 7;
963 const unsigned int mask = (1 << (window_size + 1)) - 1;
964 unsigned int wvalue;
965 ALIGN32 union {
966 P256_POINT p;
967 P256_POINT_AFFINE a;
968 } t, p;
969 BIGNUM *tmp_scalar;
970
971 if ((num + 1) == 0 || (num + 1) > OPENSSL_MALLOC_MAX_NELEMS(void *)) {
972 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL, ERR_R_MALLOC_FAILURE);
973 return 0;
974 }
975
976 BN_CTX_start(ctx);
977
978 if (scalar) {
979 generator = EC_GROUP_get0_generator(group);
980 if (generator == NULL) {
981 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL, EC_R_UNDEFINED_GENERATOR);
982 goto err;
983 }
984
985 /* look if we can use precomputed multiples of generator */
986 pre_comp = group->pre_comp.nistz256;
987
988 if (pre_comp) {
989 /*
990 * If there is a precomputed table for the generator, check that
991 * it was generated with the same generator.
992 */
993 EC_POINT *pre_comp_generator = EC_POINT_new(group);
994 if (pre_comp_generator == NULL)
995 goto err;
996
997 ecp_nistz256_gather_w7(&p.a, pre_comp->precomp[0], 1);
998 if (!ecp_nistz256_set_from_affine(pre_comp_generator,
999 group, &p.a, ctx)) {
1000 EC_POINT_free(pre_comp_generator);
1001 goto err;
1002 }
1003
1004 if (0 == EC_POINT_cmp(group, generator, pre_comp_generator, ctx))
1005 preComputedTable = (const PRECOMP256_ROW *)pre_comp->precomp;
1006
1007 EC_POINT_free(pre_comp_generator);
1008 }
1009
1010 if (preComputedTable == NULL && ecp_nistz256_is_affine_G(generator)) {
1011 /*
1012 * If there is no precomputed data, but the generator is the
1013 * default, a hardcoded table of precomputed data is used. This
1014 * is because applications, such as Apache, do not use
1015 * EC_KEY_precompute_mult.
1016 */
1017 preComputedTable = ecp_nistz256_precomputed;
1018 }
1019
1020 if (preComputedTable) {
1021 BN_ULONG infty;
1022
1023 if ((BN_num_bits(scalar) > 256)
1024 || BN_is_negative(scalar)) {
1025 if ((tmp_scalar = BN_CTX_get(ctx)) == NULL)
1026 goto err;
1027
1028 if (!BN_nnmod(tmp_scalar, scalar, group->order, ctx)) {
1029 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL, ERR_R_BN_LIB);
1030 goto err;
1031 }
1032 scalar = tmp_scalar;
1033 }
1034
1035 for (i = 0; i < bn_get_top(scalar) * BN_BYTES; i += BN_BYTES) {
1036 BN_ULONG d = bn_get_words(scalar)[i / BN_BYTES];
1037
1038 p_str[i + 0] = (unsigned char)d;
1039 p_str[i + 1] = (unsigned char)(d >> 8);
1040 p_str[i + 2] = (unsigned char)(d >> 16);
1041 p_str[i + 3] = (unsigned char)(d >>= 24);
1042 if (BN_BYTES == 8) {
1043 d >>= 8;
1044 p_str[i + 4] = (unsigned char)d;
1045 p_str[i + 5] = (unsigned char)(d >> 8);
1046 p_str[i + 6] = (unsigned char)(d >> 16);
1047 p_str[i + 7] = (unsigned char)(d >> 24);
1048 }
1049 }
1050
1051 for (; i < 33; i++)
1052 p_str[i] = 0;
1053
1054 /* First window */
1055 wvalue = (p_str[0] << 1) & mask;
1056 idx += window_size;
1057
1058 wvalue = _booth_recode_w7(wvalue);
1059
1060 ecp_nistz256_gather_w7(&p.a, preComputedTable[0],
1061 wvalue >> 1);
1062
1063 ecp_nistz256_neg(p.p.Z, p.p.Y);
1064 copy_conditional(p.p.Y, p.p.Z, wvalue & 1);
1065
1066 /*
1067 * Since affine infinity is encoded as (0,0) and
1068 * Jacobian is (,,0), we need to harmonize them
1069 * by assigning "one" or zero to Z.
1070 */
1071 infty = (p.p.X[0] | p.p.X[1] | p.p.X[2] | p.p.X[3] |
1072 p.p.Y[0] | p.p.Y[1] | p.p.Y[2] | p.p.Y[3]);
1073 if (P256_LIMBS == 8)
1074 infty |= (p.p.X[4] | p.p.X[5] | p.p.X[6] | p.p.X[7] |
1075 p.p.Y[4] | p.p.Y[5] | p.p.Y[6] | p.p.Y[7]);
1076
1077 infty = 0 - is_zero(infty);
1078 infty = ~infty;
1079
1080 p.p.Z[0] = ONE[0] & infty;
1081 p.p.Z[1] = ONE[1] & infty;
1082 p.p.Z[2] = ONE[2] & infty;
1083 p.p.Z[3] = ONE[3] & infty;
1084 if (P256_LIMBS == 8) {
1085 p.p.Z[4] = ONE[4] & infty;
1086 p.p.Z[5] = ONE[5] & infty;
1087 p.p.Z[6] = ONE[6] & infty;
1088 p.p.Z[7] = ONE[7] & infty;
1089 }
1090
1091 for (i = 1; i < 37; i++) {
1092 unsigned int off = (idx - 1) / 8;
1093 wvalue = p_str[off] | p_str[off + 1] << 8;
1094 wvalue = (wvalue >> ((idx - 1) % 8)) & mask;
1095 idx += window_size;
1096
1097 wvalue = _booth_recode_w7(wvalue);
1098
1099 ecp_nistz256_gather_w7(&t.a,
1100 preComputedTable[i], wvalue >> 1);
1101
1102 ecp_nistz256_neg(t.p.Z, t.a.Y);
1103 copy_conditional(t.a.Y, t.p.Z, wvalue & 1);
1104
1105 ecp_nistz256_point_add_affine(&p.p, &p.p, &t.a);
1106 }
1107 } else {
1108 p_is_infinity = 1;
1109 no_precomp_for_generator = 1;
1110 }
1111 } else
1112 p_is_infinity = 1;
1113
1114 if (no_precomp_for_generator) {
1115 /*
1116 * Without a precomputed table for the generator, it has to be
1117 * handled like a normal point.
1118 */
1119 new_scalars = OPENSSL_malloc((num + 1) * sizeof(BIGNUM *));
1120 if (new_scalars == NULL) {
1121 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1122 goto err;
1123 }
1124
1125 new_points = OPENSSL_malloc((num + 1) * sizeof(EC_POINT *));
1126 if (new_points == NULL) {
1127 ECerr(EC_F_ECP_NISTZ256_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1128 goto err;
1129 }
1130
1131 memcpy(new_scalars, scalars, num * sizeof(BIGNUM *));
1132 new_scalars[num] = scalar;
1133 memcpy(new_points, points, num * sizeof(EC_POINT *));
1134 new_points[num] = generator;
1135
1136 scalars = new_scalars;
1137 points = new_points;
1138 num++;
1139 }
1140
1141 if (num) {
1142 P256_POINT *out = &t.p;
1143 if (p_is_infinity)
1144 out = &p.p;
1145
1146 if (!ecp_nistz256_windowed_mul(group, out, scalars, points, num, ctx))
1147 goto err;
1148
1149 if (!p_is_infinity)
1150 ecp_nistz256_point_add(&p.p, &p.p, out);
1151 }
1152
1153 /* Not constant-time, but we're only operating on the public output. */
1154 if (!bn_set_words(r->X, p.p.X, P256_LIMBS) ||
1155 !bn_set_words(r->Y, p.p.Y, P256_LIMBS) ||
1156 !bn_set_words(r->Z, p.p.Z, P256_LIMBS)) {
1157 goto err;
1158 }
1159 r->Z_is_one = is_one(r->Z) & 1;
1160
1161 ret = 1;
1162
1163 err:
1164 BN_CTX_end(ctx);
1165 OPENSSL_free(new_points);
1166 OPENSSL_free(new_scalars);
1167 return ret;
1168 }
1169
ecp_nistz256_get_affine(const EC_GROUP * group,const EC_POINT * point,BIGNUM * x,BIGNUM * y,BN_CTX * ctx)1170 __owur static int ecp_nistz256_get_affine(const EC_GROUP *group,
1171 const EC_POINT *point,
1172 BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
1173 {
1174 BN_ULONG z_inv2[P256_LIMBS];
1175 BN_ULONG z_inv3[P256_LIMBS];
1176 BN_ULONG x_aff[P256_LIMBS];
1177 BN_ULONG y_aff[P256_LIMBS];
1178 BN_ULONG point_x[P256_LIMBS], point_y[P256_LIMBS], point_z[P256_LIMBS];
1179 BN_ULONG x_ret[P256_LIMBS], y_ret[P256_LIMBS];
1180
1181 if (EC_POINT_is_at_infinity(group, point)) {
1182 ECerr(EC_F_ECP_NISTZ256_GET_AFFINE, EC_R_POINT_AT_INFINITY);
1183 return 0;
1184 }
1185
1186 if (!ecp_nistz256_bignum_to_field_elem(point_x, point->X) ||
1187 !ecp_nistz256_bignum_to_field_elem(point_y, point->Y) ||
1188 !ecp_nistz256_bignum_to_field_elem(point_z, point->Z)) {
1189 ECerr(EC_F_ECP_NISTZ256_GET_AFFINE, EC_R_COORDINATES_OUT_OF_RANGE);
1190 return 0;
1191 }
1192
1193 ecp_nistz256_mod_inverse(z_inv3, point_z);
1194 ecp_nistz256_sqr_mont(z_inv2, z_inv3);
1195 ecp_nistz256_mul_mont(x_aff, z_inv2, point_x);
1196
1197 if (x != NULL) {
1198 ecp_nistz256_from_mont(x_ret, x_aff);
1199 if (!bn_set_words(x, x_ret, P256_LIMBS))
1200 return 0;
1201 }
1202
1203 if (y != NULL) {
1204 ecp_nistz256_mul_mont(z_inv3, z_inv3, z_inv2);
1205 ecp_nistz256_mul_mont(y_aff, z_inv3, point_y);
1206 ecp_nistz256_from_mont(y_ret, y_aff);
1207 if (!bn_set_words(y, y_ret, P256_LIMBS))
1208 return 0;
1209 }
1210
1211 return 1;
1212 }
1213
ecp_nistz256_pre_comp_new(const EC_GROUP * group)1214 static NISTZ256_PRE_COMP *ecp_nistz256_pre_comp_new(const EC_GROUP *group)
1215 {
1216 NISTZ256_PRE_COMP *ret = NULL;
1217
1218 if (!group)
1219 return NULL;
1220
1221 ret = OPENSSL_zalloc(sizeof(*ret));
1222
1223 if (ret == NULL) {
1224 ECerr(EC_F_ECP_NISTZ256_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1225 return ret;
1226 }
1227
1228 ret->group = group;
1229 ret->w = 6; /* default */
1230 ret->references = 1;
1231
1232 ret->lock = CRYPTO_THREAD_lock_new();
1233 if (ret->lock == NULL) {
1234 ECerr(EC_F_ECP_NISTZ256_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1235 OPENSSL_free(ret);
1236 return NULL;
1237 }
1238 return ret;
1239 }
1240
EC_nistz256_pre_comp_dup(NISTZ256_PRE_COMP * p)1241 NISTZ256_PRE_COMP *EC_nistz256_pre_comp_dup(NISTZ256_PRE_COMP *p)
1242 {
1243 int i;
1244 if (p != NULL)
1245 CRYPTO_UP_REF(&p->references, &i, p->lock);
1246 return p;
1247 }
1248
EC_nistz256_pre_comp_free(NISTZ256_PRE_COMP * pre)1249 void EC_nistz256_pre_comp_free(NISTZ256_PRE_COMP *pre)
1250 {
1251 int i;
1252
1253 if (pre == NULL)
1254 return;
1255
1256 CRYPTO_DOWN_REF(&pre->references, &i, pre->lock);
1257 REF_PRINT_COUNT("EC_nistz256", x);
1258 if (i > 0)
1259 return;
1260 REF_ASSERT_ISNT(i < 0);
1261
1262 OPENSSL_free(pre->precomp_storage);
1263 CRYPTO_THREAD_lock_free(pre->lock);
1264 OPENSSL_free(pre);
1265 }
1266
1267
ecp_nistz256_window_have_precompute_mult(const EC_GROUP * group)1268 static int ecp_nistz256_window_have_precompute_mult(const EC_GROUP *group)
1269 {
1270 /* There is a hard-coded table for the default generator. */
1271 const EC_POINT *generator = EC_GROUP_get0_generator(group);
1272
1273 if (generator != NULL && ecp_nistz256_is_affine_G(generator)) {
1274 /* There is a hard-coded table for the default generator. */
1275 return 1;
1276 }
1277
1278 return HAVEPRECOMP(group, nistz256);
1279 }
1280
1281 #if defined(__x86_64) || defined(__x86_64__) || \
1282 defined(_M_AMD64) || defined(_M_X64) || \
1283 defined(__powerpc64__) || defined(_ARCH_PP64) || \
1284 defined(__aarch64__)
1285 /*
1286 * Montgomery mul modulo Order(P): res = a*b*2^-256 mod Order(P)
1287 */
1288 void ecp_nistz256_ord_mul_mont(BN_ULONG res[P256_LIMBS],
1289 const BN_ULONG a[P256_LIMBS],
1290 const BN_ULONG b[P256_LIMBS]);
1291 void ecp_nistz256_ord_sqr_mont(BN_ULONG res[P256_LIMBS],
1292 const BN_ULONG a[P256_LIMBS],
1293 int rep);
1294
ecp_nistz256_inv_mod_ord(const EC_GROUP * group,BIGNUM * r,const BIGNUM * x,BN_CTX * ctx)1295 static int ecp_nistz256_inv_mod_ord(const EC_GROUP *group, BIGNUM *r,
1296 const BIGNUM *x, BN_CTX *ctx)
1297 {
1298 /* RR = 2^512 mod ord(p256) */
1299 static const BN_ULONG RR[P256_LIMBS] = {
1300 TOBN(0x83244c95,0xbe79eea2), TOBN(0x4699799c,0x49bd6fa6),
1301 TOBN(0x2845b239,0x2b6bec59), TOBN(0x66e12d94,0xf3d95620)
1302 };
1303 /* The constant 1 (unlike ONE that is one in Montgomery representation) */
1304 static const BN_ULONG one[P256_LIMBS] = {
1305 TOBN(0,1), TOBN(0,0), TOBN(0,0), TOBN(0,0)
1306 };
1307 /*
1308 * We don't use entry 0 in the table, so we omit it and address
1309 * with -1 offset.
1310 */
1311 BN_ULONG table[15][P256_LIMBS];
1312 BN_ULONG out[P256_LIMBS], t[P256_LIMBS];
1313 int i, ret = 0;
1314 enum {
1315 i_1 = 0, i_10, i_11, i_101, i_111, i_1010, i_1111,
1316 i_10101, i_101010, i_101111, i_x6, i_x8, i_x16, i_x32
1317 };
1318
1319 /*
1320 * Catch allocation failure early.
1321 */
1322 if (bn_wexpand(r, P256_LIMBS) == NULL) {
1323 ECerr(EC_F_ECP_NISTZ256_INV_MOD_ORD, ERR_R_BN_LIB);
1324 goto err;
1325 }
1326
1327 if ((BN_num_bits(x) > 256) || BN_is_negative(x)) {
1328 BIGNUM *tmp;
1329
1330 if ((tmp = BN_CTX_get(ctx)) == NULL
1331 || !BN_nnmod(tmp, x, group->order, ctx)) {
1332 ECerr(EC_F_ECP_NISTZ256_INV_MOD_ORD, ERR_R_BN_LIB);
1333 goto err;
1334 }
1335 x = tmp;
1336 }
1337
1338 if (!ecp_nistz256_bignum_to_field_elem(t, x)) {
1339 ECerr(EC_F_ECP_NISTZ256_INV_MOD_ORD, EC_R_COORDINATES_OUT_OF_RANGE);
1340 goto err;
1341 }
1342
1343 ecp_nistz256_ord_mul_mont(table[0], t, RR);
1344 #if 0
1345 /*
1346 * Original sparse-then-fixed-window algorithm, retained for reference.
1347 */
1348 for (i = 2; i < 16; i += 2) {
1349 ecp_nistz256_ord_sqr_mont(table[i-1], table[i/2-1], 1);
1350 ecp_nistz256_ord_mul_mont(table[i], table[i-1], table[0]);
1351 }
1352
1353 /*
1354 * The top 128bit of the exponent are highly redudndant, so we
1355 * perform an optimized flow
1356 */
1357 ecp_nistz256_ord_sqr_mont(t, table[15-1], 4); /* f0 */
1358 ecp_nistz256_ord_mul_mont(t, t, table[15-1]); /* ff */
1359
1360 ecp_nistz256_ord_sqr_mont(out, t, 8); /* ff00 */
1361 ecp_nistz256_ord_mul_mont(out, out, t); /* ffff */
1362
1363 ecp_nistz256_ord_sqr_mont(t, out, 16); /* ffff0000 */
1364 ecp_nistz256_ord_mul_mont(t, t, out); /* ffffffff */
1365
1366 ecp_nistz256_ord_sqr_mont(out, t, 64); /* ffffffff0000000000000000 */
1367 ecp_nistz256_ord_mul_mont(out, out, t); /* ffffffff00000000ffffffff */
1368
1369 ecp_nistz256_ord_sqr_mont(out, out, 32); /* ffffffff00000000ffffffff00000000 */
1370 ecp_nistz256_ord_mul_mont(out, out, t); /* ffffffff00000000ffffffffffffffff */
1371
1372 /*
1373 * The bottom 128 bit of the exponent are processed with fixed 4-bit window
1374 */
1375 for(i = 0; i < 32; i++) {
1376 /* expLo - the low 128 bits of the exponent we use (ord(p256) - 2),
1377 * split into nibbles */
1378 static const unsigned char expLo[32] = {
1379 0xb,0xc,0xe,0x6,0xf,0xa,0xa,0xd,0xa,0x7,0x1,0x7,0x9,0xe,0x8,0x4,
1380 0xf,0x3,0xb,0x9,0xc,0xa,0xc,0x2,0xf,0xc,0x6,0x3,0x2,0x5,0x4,0xf
1381 };
1382
1383 ecp_nistz256_ord_sqr_mont(out, out, 4);
1384 /* The exponent is public, no need in constant-time access */
1385 ecp_nistz256_ord_mul_mont(out, out, table[expLo[i]-1]);
1386 }
1387 #else
1388 /*
1389 * https://briansmith.org/ecc-inversion-addition-chains-01#p256_scalar_inversion
1390 *
1391 * Even though this code path spares 12 squarings, 4.5%, and 13
1392 * multiplications, 25%, on grand scale sign operation is not that
1393 * much faster, not more that 2%...
1394 */
1395
1396 /* pre-calculate powers */
1397 ecp_nistz256_ord_sqr_mont(table[i_10], table[i_1], 1);
1398
1399 ecp_nistz256_ord_mul_mont(table[i_11], table[i_1], table[i_10]);
1400
1401 ecp_nistz256_ord_mul_mont(table[i_101], table[i_11], table[i_10]);
1402
1403 ecp_nistz256_ord_mul_mont(table[i_111], table[i_101], table[i_10]);
1404
1405 ecp_nistz256_ord_sqr_mont(table[i_1010], table[i_101], 1);
1406
1407 ecp_nistz256_ord_mul_mont(table[i_1111], table[i_1010], table[i_101]);
1408
1409 ecp_nistz256_ord_sqr_mont(table[i_10101], table[i_1010], 1);
1410 ecp_nistz256_ord_mul_mont(table[i_10101], table[i_10101], table[i_1]);
1411
1412 ecp_nistz256_ord_sqr_mont(table[i_101010], table[i_10101], 1);
1413
1414 ecp_nistz256_ord_mul_mont(table[i_101111], table[i_101010], table[i_101]);
1415
1416 ecp_nistz256_ord_mul_mont(table[i_x6], table[i_101010], table[i_10101]);
1417
1418 ecp_nistz256_ord_sqr_mont(table[i_x8], table[i_x6], 2);
1419 ecp_nistz256_ord_mul_mont(table[i_x8], table[i_x8], table[i_11]);
1420
1421 ecp_nistz256_ord_sqr_mont(table[i_x16], table[i_x8], 8);
1422 ecp_nistz256_ord_mul_mont(table[i_x16], table[i_x16], table[i_x8]);
1423
1424 ecp_nistz256_ord_sqr_mont(table[i_x32], table[i_x16], 16);
1425 ecp_nistz256_ord_mul_mont(table[i_x32], table[i_x32], table[i_x16]);
1426
1427 /* calculations */
1428 ecp_nistz256_ord_sqr_mont(out, table[i_x32], 64);
1429 ecp_nistz256_ord_mul_mont(out, out, table[i_x32]);
1430
1431 for (i = 0; i < 27; i++) {
1432 static const struct { unsigned char p, i; } chain[27] = {
1433 { 32, i_x32 }, { 6, i_101111 }, { 5, i_111 },
1434 { 4, i_11 }, { 5, i_1111 }, { 5, i_10101 },
1435 { 4, i_101 }, { 3, i_101 }, { 3, i_101 },
1436 { 5, i_111 }, { 9, i_101111 }, { 6, i_1111 },
1437 { 2, i_1 }, { 5, i_1 }, { 6, i_1111 },
1438 { 5, i_111 }, { 4, i_111 }, { 5, i_111 },
1439 { 5, i_101 }, { 3, i_11 }, { 10, i_101111 },
1440 { 2, i_11 }, { 5, i_11 }, { 5, i_11 },
1441 { 3, i_1 }, { 7, i_10101 }, { 6, i_1111 }
1442 };
1443
1444 ecp_nistz256_ord_sqr_mont(out, out, chain[i].p);
1445 ecp_nistz256_ord_mul_mont(out, out, table[chain[i].i]);
1446 }
1447 #endif
1448 ecp_nistz256_ord_mul_mont(out, out, one);
1449
1450 /*
1451 * Can't fail, but check return code to be consistent anyway.
1452 */
1453 if (!bn_set_words(r, out, P256_LIMBS))
1454 goto err;
1455
1456 ret = 1;
1457 err:
1458 return ret;
1459 }
1460 #else
1461 # define ecp_nistz256_inv_mod_ord NULL
1462 #endif
1463
EC_GFp_nistz256_method(void)1464 const EC_METHOD *EC_GFp_nistz256_method(void)
1465 {
1466 static const EC_METHOD ret = {
1467 EC_FLAGS_DEFAULT_OCT,
1468 NID_X9_62_prime_field,
1469 ec_GFp_mont_group_init,
1470 ec_GFp_mont_group_finish,
1471 ec_GFp_mont_group_clear_finish,
1472 ec_GFp_mont_group_copy,
1473 ec_GFp_mont_group_set_curve,
1474 ec_GFp_simple_group_get_curve,
1475 ec_GFp_simple_group_get_degree,
1476 ec_group_simple_order_bits,
1477 ec_GFp_simple_group_check_discriminant,
1478 ec_GFp_simple_point_init,
1479 ec_GFp_simple_point_finish,
1480 ec_GFp_simple_point_clear_finish,
1481 ec_GFp_simple_point_copy,
1482 ec_GFp_simple_point_set_to_infinity,
1483 ec_GFp_simple_set_Jprojective_coordinates_GFp,
1484 ec_GFp_simple_get_Jprojective_coordinates_GFp,
1485 ec_GFp_simple_point_set_affine_coordinates,
1486 ecp_nistz256_get_affine,
1487 0, 0, 0,
1488 ec_GFp_simple_add,
1489 ec_GFp_simple_dbl,
1490 ec_GFp_simple_invert,
1491 ec_GFp_simple_is_at_infinity,
1492 ec_GFp_simple_is_on_curve,
1493 ec_GFp_simple_cmp,
1494 ec_GFp_simple_make_affine,
1495 ec_GFp_simple_points_make_affine,
1496 ecp_nistz256_points_mul, /* mul */
1497 ecp_nistz256_mult_precompute, /* precompute_mult */
1498 ecp_nistz256_window_have_precompute_mult, /* have_precompute_mult */
1499 ec_GFp_mont_field_mul,
1500 ec_GFp_mont_field_sqr,
1501 0, /* field_div */
1502 ec_GFp_mont_field_inv,
1503 ec_GFp_mont_field_encode,
1504 ec_GFp_mont_field_decode,
1505 ec_GFp_mont_field_set_to_one,
1506 ec_key_simple_priv2oct,
1507 ec_key_simple_oct2priv,
1508 0, /* set private */
1509 ec_key_simple_generate_key,
1510 ec_key_simple_check_key,
1511 ec_key_simple_generate_public_key,
1512 0, /* keycopy */
1513 0, /* keyfinish */
1514 ecdh_simple_compute_key,
1515 ecp_nistz256_inv_mod_ord, /* can be #define-d NULL */
1516 0, /* blind_coordinates */
1517 0, /* ladder_pre */
1518 0, /* ladder_step */
1519 0 /* ladder_post */
1520 };
1521
1522 return &ret;
1523 }
1524