1 /* crypto/ec/ecp_nistp224.c */
3 * Written by Emilia Kasper (Google) for the OpenSSL project.
5 /* ====================================================================
6 * Copyright (c) 2000-2010 The OpenSSL Project. All rights reserved.
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51 * ====================================================================
53 * This product includes cryptographic software written by Eric Young
54 * (eay@cryptsoft.com). This product includes software written by Tim
55 * Hudson (tjh@cryptsoft.com).
60 * A 64-bit implementation of the NIST P-224 elliptic curve point multiplication
62 * Inspired by Daniel J. Bernstein's public domain nistp224 implementation
63 * and Adam Langley's public domain 64-bit C implementation of curve25519
65 #include <openssl/opensslconf.h>
66 #ifndef OPENSSL_NO_EC_NISTP224_64_GCC_128
69 #include <openssl/err.h>
72 #if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
73 /* even with gcc, the typedef won't work for 32-bit platforms */
74 typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit platforms */
76 #error "Need GCC 3.1 or later to define type uint128_t"
82 /******************************************************************************/
83 /* INTERNAL REPRESENTATION OF FIELD ELEMENTS
85 * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
86 * where each slice a_i is a 64-bit word, i.e., a field element is an fslice
87 * array a with 4 elements, where a[i] = a_i.
88 * Outputs from multiplications are represented as unreduced polynomials
89 * b_0 + 2^56*b_1 + 2^112*b_2 + 2^168*b_3 + 2^224*b_4 + 2^280*b_5 + 2^336*b_6
90 * where each b_i is a 128-bit word. We ensure that inputs to each field
91 * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60,
92 * and fit into a 128-bit word without overflow. The coefficients are then
93 * again partially reduced to a_i < 2^57. We only reduce to the unique minimal
94 * representation at the end of the computation.
98 typedef uint64_t fslice;
100 /* Field element represented as a byte arrary.
101 * 28*8 = 224 bits is also the group order size for the elliptic curve. */
102 typedef u8 felem_bytearray[28];
104 static const felem_bytearray nistp224_curve_params[5] = {
105 {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, /* p */
106 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0x00,0x00,0x00,0x00,
107 0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x01},
108 {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, /* a */
109 0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFF,0xFF,
110 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE},
111 {0xB4,0x05,0x0A,0x85,0x0C,0x04,0xB3,0xAB,0xF5,0x41, /* b */
112 0x32,0x56,0x50,0x44,0xB0,0xB7,0xD7,0xBF,0xD8,0xBA,
113 0x27,0x0B,0x39,0x43,0x23,0x55,0xFF,0xB4},
114 {0xB7,0x0E,0x0C,0xBD,0x6B,0xB4,0xBF,0x7F,0x32,0x13, /* x */
115 0x90,0xB9,0x4A,0x03,0xC1,0xD3,0x56,0xC2,0x11,0x22,
116 0x34,0x32,0x80,0xD6,0x11,0x5C,0x1D,0x21},
117 {0xbd,0x37,0x63,0x88,0xb5,0xf7,0x23,0xfb,0x4c,0x22, /* y */
118 0xdf,0xe6,0xcd,0x43,0x75,0xa0,0x5a,0x07,0x47,0x64,
119 0x44,0xd5,0x81,0x99,0x85,0x00,0x7e,0x34}
122 /* Precomputed multiples of the standard generator
123 * b_0*G + b_1*2^56*G + b_2*2^112*G + b_3*2^168*G for
124 * (b_3, b_2, b_1, b_0) in [0,15], i.e., gmul[0] = point_at_infinity,
125 * gmul[1] = G, gmul[2] = 2^56*G, gmul[3] = 2^56*G + G, etc.
126 * Points are given in Jacobian projective coordinates: words 0-3 represent the
127 * X-coordinate (slice a_0 is word 0, etc.), words 4-7 represent the
128 * Y-coordinate and words 8-11 represent the Z-coordinate. */
129 static const fslice gmul[16][3][4] = {
130 {{0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000},
131 {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000},
132 {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
133 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
134 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
135 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
136 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
137 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
138 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
139 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
140 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
141 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
142 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
143 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
144 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
145 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
146 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
147 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
148 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
149 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
150 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
151 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
152 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
153 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
154 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
155 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
156 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
157 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
158 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
159 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
160 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
161 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
162 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
163 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
164 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
165 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
166 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
167 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
168 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
169 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
170 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
171 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
172 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
173 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
174 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
175 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
176 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
177 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}
180 /* Precomputation for the group generator. */
182 fslice g_pre_comp[16][3][4];
186 const EC_METHOD *EC_GFp_nistp224_method(void)
188 static const EC_METHOD ret = {
189 NID_X9_62_prime_field,
190 ec_GFp_nistp224_group_init,
191 ec_GFp_simple_group_finish,
192 ec_GFp_simple_group_clear_finish,
193 ec_GFp_nist_group_copy,
194 ec_GFp_nistp224_group_set_curve,
195 ec_GFp_simple_group_get_curve,
196 ec_GFp_simple_group_get_degree,
197 ec_GFp_simple_group_check_discriminant,
198 ec_GFp_simple_point_init,
199 ec_GFp_simple_point_finish,
200 ec_GFp_simple_point_clear_finish,
201 ec_GFp_simple_point_copy,
202 ec_GFp_simple_point_set_to_infinity,
203 ec_GFp_simple_set_Jprojective_coordinates_GFp,
204 ec_GFp_simple_get_Jprojective_coordinates_GFp,
205 ec_GFp_simple_point_set_affine_coordinates,
206 ec_GFp_nistp224_point_get_affine_coordinates,
207 ec_GFp_simple_set_compressed_coordinates,
208 ec_GFp_simple_point2oct,
209 ec_GFp_simple_oct2point,
212 ec_GFp_simple_invert,
213 ec_GFp_simple_is_at_infinity,
214 ec_GFp_simple_is_on_curve,
216 ec_GFp_simple_make_affine,
217 ec_GFp_simple_points_make_affine,
218 ec_GFp_nistp224_points_mul,
219 ec_GFp_nistp224_precompute_mult,
220 ec_GFp_nistp224_have_precompute_mult,
221 ec_GFp_nist_field_mul,
222 ec_GFp_nist_field_sqr,
224 0 /* field_encode */,
225 0 /* field_decode */,
226 0 /* field_set_to_one */ };
231 /* Helper functions to convert field elements to/from internal representation */
232 static void bin28_to_felem(fslice out[4], const u8 in[28])
234 out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
235 out[1] = (*((const uint64_t *)(in+7))) & 0x00ffffffffffffff;
236 out[2] = (*((const uint64_t *)(in+14))) & 0x00ffffffffffffff;
237 out[3] = (*((const uint64_t *)(in+21))) & 0x00ffffffffffffff;
240 static void felem_to_bin28(u8 out[28], const fslice in[4])
243 for (i = 0; i < 7; ++i)
245 out[i] = in[0]>>(8*i);
246 out[i+7] = in[1]>>(8*i);
247 out[i+14] = in[2]>>(8*i);
248 out[i+21] = in[3]>>(8*i);
252 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
253 static void flip_endian(u8 *out, const u8 *in, unsigned len)
256 for (i = 0; i < len; ++i)
257 out[i] = in[len-1-i];
260 /* From OpenSSL BIGNUM to internal representation */
261 static int BN_to_felem(fslice out[4], const BIGNUM *bn)
263 felem_bytearray b_in;
264 felem_bytearray b_out;
267 /* BN_bn2bin eats leading zeroes */
268 memset(b_out, 0, sizeof b_out);
269 num_bytes = BN_num_bytes(bn);
270 if (num_bytes > sizeof b_out)
272 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
275 if (BN_is_negative(bn))
277 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
280 num_bytes = BN_bn2bin(bn, b_in);
281 flip_endian(b_out, b_in, num_bytes);
282 bin28_to_felem(out, b_out);
286 /* From internal representation to OpenSSL BIGNUM */
287 static BIGNUM *felem_to_BN(BIGNUM *out, const fslice in[4])
289 felem_bytearray b_in, b_out;
290 felem_to_bin28(b_in, in);
291 flip_endian(b_out, b_in, sizeof b_out);
292 return BN_bin2bn(b_out, sizeof b_out, out);
295 /******************************************************************************/
298 * Field operations, using the internal representation of field elements.
299 * NB! These operations are specific to our point multiplication and cannot be
300 * expected to be correct in general - e.g., multiplication with a large scalar
301 * will cause an overflow.
305 /* Sum two field elements: out += in */
306 static void felem_sum64(fslice out[4], const fslice in[4])
314 /* Subtract field elements: out -= in */
315 /* Assumes in[i] < 2^57 */
316 static void felem_diff64(fslice out[4], const fslice in[4])
318 static const uint64_t two58p2 = (((uint64_t) 1) << 58) + (((uint64_t) 1) << 2);
319 static const uint64_t two58m2 = (((uint64_t) 1) << 58) - (((uint64_t) 1) << 2);
320 static const uint64_t two58m42m2 = (((uint64_t) 1) << 58) -
321 (((uint64_t) 1) << 42) - (((uint64_t) 1) << 2);
323 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
325 out[1] += two58m42m2;
335 /* Subtract in unreduced 128-bit mode: out128 -= in128 */
336 /* Assumes in[i] < 2^119 */
337 static void felem_diff128(uint128_t out[7], const uint128_t in[4])
339 static const uint128_t two120 = ((uint128_t) 1) << 120;
340 static const uint128_t two120m64 = (((uint128_t) 1) << 120) -
341 (((uint128_t) 1) << 64);
342 static const uint128_t two120m104m64 = (((uint128_t) 1) << 120) -
343 (((uint128_t) 1) << 104) - (((uint128_t) 1) << 64);
345 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
350 out[4] += two120m104m64;
363 /* Subtract in mixed mode: out128 -= in64 */
365 static void felem_diff_128_64(uint128_t out[7], const fslice in[4])
367 static const uint128_t two64p8 = (((uint128_t) 1) << 64) +
368 (((uint128_t) 1) << 8);
369 static const uint128_t two64m8 = (((uint128_t) 1) << 64) -
370 (((uint128_t) 1) << 8);
371 static const uint128_t two64m48m8 = (((uint128_t) 1) << 64) -
372 (((uint128_t) 1) << 48) - (((uint128_t) 1) << 8);
374 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
376 out[1] += two64m48m8;
386 /* Multiply a field element by a scalar: out64 = out64 * scalar
387 * The scalars we actually use are small, so results fit without overflow */
388 static void felem_scalar64(fslice out[4], const fslice scalar)
396 /* Multiply an unreduced field element by a scalar: out128 = out128 * scalar
397 * The scalars we actually use are small, so results fit without overflow */
398 static void felem_scalar128(uint128_t out[7], const uint128_t scalar)
409 /* Square a field element: out = in^2 */
410 static void felem_square(uint128_t out[7], const fslice in[4])
412 out[0] = ((uint128_t) in[0]) * in[0];
413 out[1] = ((uint128_t) in[0]) * in[1] * 2;
414 out[2] = ((uint128_t) in[0]) * in[2] * 2 + ((uint128_t) in[1]) * in[1];
415 out[3] = ((uint128_t) in[0]) * in[3] * 2 +
416 ((uint128_t) in[1]) * in[2] * 2;
417 out[4] = ((uint128_t) in[1]) * in[3] * 2 + ((uint128_t) in[2]) * in[2];
418 out[5] = ((uint128_t) in[2]) * in[3] * 2;
419 out[6] = ((uint128_t) in[3]) * in[3];
422 /* Multiply two field elements: out = in1 * in2 */
423 static void felem_mul(uint128_t out[7], const fslice in1[4], const fslice in2[4])
425 out[0] = ((uint128_t) in1[0]) * in2[0];
426 out[1] = ((uint128_t) in1[0]) * in2[1] + ((uint128_t) in1[1]) * in2[0];
427 out[2] = ((uint128_t) in1[0]) * in2[2] + ((uint128_t) in1[1]) * in2[1] +
428 ((uint128_t) in1[2]) * in2[0];
429 out[3] = ((uint128_t) in1[0]) * in2[3] + ((uint128_t) in1[1]) * in2[2] +
430 ((uint128_t) in1[2]) * in2[1] + ((uint128_t) in1[3]) * in2[0];
431 out[4] = ((uint128_t) in1[1]) * in2[3] + ((uint128_t) in1[2]) * in2[2] +
432 ((uint128_t) in1[3]) * in2[1];
433 out[5] = ((uint128_t) in1[2]) * in2[3] + ((uint128_t) in1[3]) * in2[2];
434 out[6] = ((uint128_t) in1[3]) * in2[3];
437 /* Reduce 128-bit coefficients to 64-bit coefficients. Requires in[i] < 2^126,
438 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] < 2^57 */
439 static void felem_reduce(fslice out[4], const uint128_t in[7])
441 static const uint128_t two127p15 = (((uint128_t) 1) << 127) +
442 (((uint128_t) 1) << 15);
443 static const uint128_t two127m71 = (((uint128_t) 1) << 127) -
444 (((uint128_t) 1) << 71);
445 static const uint128_t two127m71m55 = (((uint128_t) 1) << 127) -
446 (((uint128_t) 1) << 71) - (((uint128_t) 1) << 55);
449 /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
450 output[0] = in[0] + two127p15;
451 output[1] = in[1] + two127m71m55;
452 output[2] = in[2] + two127m71;
456 /* Eliminate in[4], in[5], in[6] */
457 output[4] += in[6] >> 16;
458 output[3] += (in[6]&0xffff) << 40;
461 output[3] += in[5] >> 16;
462 output[2] += (in[5]&0xffff) << 40;
465 output[2] += output[4] >> 16;
466 output[1] += (output[4]&0xffff) << 40;
467 output[0] -= output[4];
470 /* Carry 2 -> 3 -> 4 */
471 output[3] += output[2] >> 56;
472 output[2] &= 0x00ffffffffffffff;
474 output[4] += output[3] >> 56;
475 output[3] &= 0x00ffffffffffffff;
477 /* Now output[2] < 2^56, output[3] < 2^56 */
479 /* Eliminate output[4] */
480 output[2] += output[4] >> 16;
481 output[1] += (output[4]&0xffff) << 40;
482 output[0] -= output[4];
484 /* Carry 0 -> 1 -> 2 -> 3 */
485 output[1] += output[0] >> 56;
486 out[0] = output[0] & 0x00ffffffffffffff;
488 output[2] += output[1] >> 56;
489 out[1] = output[1] & 0x00ffffffffffffff;
490 output[3] += output[2] >> 56;
491 out[2] = output[2] & 0x00ffffffffffffff;
493 /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
494 * out[3] < 2^57 (due to final carry) */
498 /* Reduce to unique minimal representation */
499 static void felem_contract(fslice out[4], const fslice in[4])
501 static const int64_t two56 = ((uint64_t) 1) << 56;
502 /* 0 <= in < 2^225 */
503 /* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */
505 tmp[0] = (int64_t) in[0] - (in[3] >> 56);
506 tmp[1] = (int64_t) in[1] + ((in[3] >> 16) & 0x0000010000000000);
507 tmp[2] = (int64_t) in[2];
508 tmp[3] = (int64_t) in[3] & 0x00ffffffffffffff;
510 /* eliminate negative coefficients */
526 tmp[1] -= (1 & a) << 40;
528 /* carry 1 -> 2 -> 3 */
529 tmp[2] += tmp[1] >> 56;
530 tmp[1] &= 0x00ffffffffffffff;
532 tmp[3] += tmp[2] >> 56;
533 tmp[2] &= 0x00ffffffffffffff;
535 /* 0 <= in < 2^224 + 2^96 - 1 */
536 /* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */
537 tmp[0] -= (tmp[3] >> 56);
538 tmp[1] += ((tmp[3] >> 16) & 0x0000010000000000);
539 tmp[3] &= 0x00ffffffffffffff;
541 /* eliminate negative coefficients */
557 tmp[1] -= (1 & a) << 40;
559 /* carry 1 -> 2 -> 3 */
560 tmp[2] += tmp[1] >> 56;
561 tmp[1] &= 0x00ffffffffffffff;
563 tmp[3] += tmp[2] >> 56;
564 tmp[2] &= 0x00ffffffffffffff;
566 /* Now 0 <= in < 2^224 */
568 /* if in > 2^224 - 2^96, reduce */
569 /* a = 0 iff in > 2^224 - 2^96, i.e.,
570 * the high 128 bits are all 1 and the lower part is non-zero */
571 a = (tmp[3] + 1) | (tmp[2] + 1) |
572 ((tmp[1] | 0x000000ffffffffff) + 1) |
573 ((((tmp[1] & 0xffff) - 1) >> 63) & ((tmp[0] - 1) >> 63));
574 /* turn a into an all-one mask (if a = 0) or an all-zero mask */
575 a = ((a & 0x00ffffffffffffff) - 1) >> 63;
576 /* subtract 2^224 - 2^96 + 1 if a is all-one*/
577 tmp[3] &= a ^ 0xffffffffffffffff;
578 tmp[2] &= a ^ 0xffffffffffffffff;
579 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
581 /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must be
582 * non-zero, so we only need one step */
593 /* Zero-check: returns 1 if input is 0, and 0 otherwise.
594 * We know that field elements are reduced to in < 2^225,
595 * so we only need to check three cases: 0, 2^224 - 2^96 + 1,
596 * and 2^225 - 2^97 + 2 */
597 static fslice felem_is_zero(const fslice in[4])
599 fslice zero, two224m96p1, two225m97p2;
601 zero = in[0] | in[1] | in[2] | in[3];
602 zero = (((int64_t)(zero) - 1) >> 63) & 1;
603 two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
604 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
605 two224m96p1 = (((int64_t)(two224m96p1) - 1) >> 63) & 1;
606 two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
607 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
608 two225m97p2 = (((int64_t)(two225m97p2) - 1) >> 63) & 1;
609 return (zero | two224m96p1 | two225m97p2);
612 /* Invert a field element */
613 /* Computation chain copied from djb's code */
614 static void felem_inv(fslice out[4], const fslice in[4])
616 fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4];
620 felem_square(tmp, in); felem_reduce(ftmp, tmp); /* 2 */
621 felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp); /* 2^2 - 1 */
622 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp); /* 2^3 - 2 */
623 felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp); /* 2^3 - 1 */
624 felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
625 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
626 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
627 felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp); /* 2^6 - 1 */
628 felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
629 for (i = 0; i < 5; ++i) /* 2^12 - 2^6 */
631 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
633 felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
634 felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
635 for (i = 0; i < 11; ++i) /* 2^24 - 2^12 */
637 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
639 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
640 felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
641 for (i = 0; i < 23; ++i) /* 2^48 - 2^24 */
643 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
645 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
646 felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
647 for (i = 0; i < 47; ++i) /* 2^96 - 2^48 */
649 felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
651 felem_mul(tmp, ftmp3, ftmp4); felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
652 felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
653 for (i = 0; i < 23; ++i) /* 2^120 - 2^24 */
655 felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
657 felem_mul(tmp, ftmp2, ftmp4); felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
658 for (i = 0; i < 6; ++i) /* 2^126 - 2^6 */
660 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
662 felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp); /* 2^126 - 1 */
663 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp); /* 2^127 - 2 */
664 felem_mul(tmp, ftmp, in); felem_reduce(ftmp, tmp); /* 2^127 - 1 */
665 for (i = 0; i < 97; ++i) /* 2^224 - 2^97 */
667 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);
669 felem_mul(tmp, ftmp, ftmp3); felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
672 /* Copy in constant time:
673 * if icopy == 1, copy in to out,
674 * if icopy == 0, copy out to itself. */
676 copy_conditional(fslice *out, const fslice *in, unsigned len, fslice icopy)
679 /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */
680 const fslice copy = -icopy;
681 for (i = 0; i < len; ++i)
683 const fslice tmp = copy & (in[i] ^ out[i]);
688 /* Copy in constant time:
689 * if isel == 1, copy in2 to out,
690 * if isel == 0, copy in1 to out. */
691 static void select_conditional(fslice *out, const fslice *in1, const fslice *in2,
692 unsigned len, fslice isel)
695 /* isel is a (64-bit) 0 or 1, so sel is either all-zero or all-one */
696 const fslice sel = -isel;
697 for (i = 0; i < len; ++i)
699 const fslice tmp = sel & (in1[i] ^ in2[i]);
700 out[i] = in1[i] ^ tmp;
704 /******************************************************************************/
705 /* ELLIPTIC CURVE POINT OPERATIONS
707 * Points are represented in Jacobian projective coordinates:
708 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
709 * or to the point at infinity if Z == 0.
713 /* Double an elliptic curve point:
714 * (X', Y', Z') = 2 * (X, Y, Z), where
715 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
716 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
717 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
718 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
719 * while x_out == y_in is not (maybe this works, but it's not tested). */
721 point_double(fslice x_out[4], fslice y_out[4], fslice z_out[4],
722 const fslice x_in[4], const fslice y_in[4], const fslice z_in[4])
724 uint128_t tmp[7], tmp2[7];
729 fslice ftmp[4], ftmp2[4];
730 memcpy(ftmp, x_in, 4 * sizeof(fslice));
731 memcpy(ftmp2, x_in, 4 * sizeof(fslice));
734 felem_square(tmp, z_in);
735 felem_reduce(delta, tmp);
738 felem_square(tmp, y_in);
739 felem_reduce(gamma, tmp);
742 felem_mul(tmp, x_in, gamma);
743 felem_reduce(beta, tmp);
745 /* alpha = 3*(x-delta)*(x+delta) */
746 felem_diff64(ftmp, delta);
747 /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
748 felem_sum64(ftmp2, delta);
749 /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
750 felem_scalar64(ftmp2, 3);
751 /* ftmp2[i] < 3 * 2^58 < 2^60 */
752 felem_mul(tmp, ftmp, ftmp2);
753 /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
754 felem_reduce(alpha, tmp);
756 /* x' = alpha^2 - 8*beta */
757 felem_square(tmp, alpha);
758 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
759 memcpy(ftmp, beta, 4 * sizeof(fslice));
760 felem_scalar64(ftmp, 8);
761 /* ftmp[i] < 8 * 2^57 = 2^60 */
762 felem_diff_128_64(tmp, ftmp);
763 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
764 felem_reduce(x_out, tmp);
766 /* z' = (y + z)^2 - gamma - delta */
767 felem_sum64(delta, gamma);
768 /* delta[i] < 2^57 + 2^57 = 2^58 */
769 memcpy(ftmp, y_in, 4 * sizeof(fslice));
770 felem_sum64(ftmp, z_in);
771 /* ftmp[i] < 2^57 + 2^57 = 2^58 */
772 felem_square(tmp, ftmp);
773 /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
774 felem_diff_128_64(tmp, delta);
775 /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
776 felem_reduce(z_out, tmp);
778 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
779 felem_scalar64(beta, 4);
780 /* beta[i] < 4 * 2^57 = 2^59 */
781 felem_diff64(beta, x_out);
782 /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
783 felem_mul(tmp, alpha, beta);
784 /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
785 felem_square(tmp2, gamma);
786 /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
787 felem_scalar128(tmp2, 8);
788 /* tmp2[i] < 8 * 2^116 = 2^119 */
789 felem_diff128(tmp, tmp2);
790 /* tmp[i] < 2^119 + 2^120 < 2^121 */
791 felem_reduce(y_out, tmp);
794 /* Add two elliptic curve points:
795 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
796 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
797 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
798 * Y_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1) * (Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2 - X_3) -
799 * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
800 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2) */
802 /* This function is not entirely constant-time:
803 * it includes a branch for checking whether the two input points are equal,
804 * (while not equal to the point at infinity).
805 * This case never happens during single point multiplication,
806 * so there is no timing leak for ECDH or ECDSA signing. */
807 static void point_add(fslice x3[4], fslice y3[4], fslice z3[4],
808 const fslice x1[4], const fslice y1[4], const fslice z1[4],
809 const fslice x2[4], const fslice y2[4], const fslice z2[4])
811 fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4], ftmp5[4];
812 uint128_t tmp[7], tmp2[7];
813 fslice z1_is_zero, z2_is_zero, x_equal, y_equal;
816 felem_square(tmp, z1);
817 felem_reduce(ftmp, tmp);
820 felem_square(tmp, z2);
821 felem_reduce(ftmp2, tmp);
824 felem_mul(tmp, ftmp, z1);
825 felem_reduce(ftmp3, tmp);
828 felem_mul(tmp, ftmp2, z2);
829 felem_reduce(ftmp4, tmp);
831 /* ftmp3 = z1^3*y2 */
832 felem_mul(tmp, ftmp3, y2);
833 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
835 /* ftmp4 = z2^3*y1 */
836 felem_mul(tmp2, ftmp4, y1);
837 felem_reduce(ftmp4, tmp2);
839 /* ftmp3 = z1^3*y2 - z2^3*y1 */
840 felem_diff_128_64(tmp, ftmp4);
841 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
842 felem_reduce(ftmp3, tmp);
845 felem_mul(tmp, ftmp, x2);
846 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
849 felem_mul(tmp2, ftmp2, x1);
850 felem_reduce(ftmp2, tmp2);
852 /* ftmp = z1^2*x2 - z2^2*x1 */
853 felem_diff128(tmp, tmp2);
854 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
855 felem_reduce(ftmp, tmp);
857 /* the formulae are incorrect if the points are equal
858 * so we check for this and do doubling if this happens */
859 x_equal = felem_is_zero(ftmp);
860 y_equal = felem_is_zero(ftmp3);
861 z1_is_zero = felem_is_zero(z1);
862 z2_is_zero = felem_is_zero(z2);
863 /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
864 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero)
866 point_double(x3, y3, z3, x1, y1, z1);
871 felem_mul(tmp, z1, z2);
872 felem_reduce(ftmp5, tmp);
874 /* z3 = (z1^2*x2 - z2^2*x1)*(z1*z2) */
875 felem_mul(tmp, ftmp, ftmp5);
876 felem_reduce(z3, tmp);
878 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
879 memcpy(ftmp5, ftmp, 4 * sizeof(fslice));
880 felem_square(tmp, ftmp);
881 felem_reduce(ftmp, tmp);
883 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
884 felem_mul(tmp, ftmp, ftmp5);
885 felem_reduce(ftmp5, tmp);
887 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
888 felem_mul(tmp, ftmp2, ftmp);
889 felem_reduce(ftmp2, tmp);
891 /* ftmp4 = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
892 felem_mul(tmp, ftmp4, ftmp5);
893 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
895 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
896 felem_square(tmp2, ftmp3);
897 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
899 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
900 felem_diff_128_64(tmp2, ftmp5);
901 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
903 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
904 memcpy(ftmp5, ftmp2, 4 * sizeof(fslice));
905 felem_scalar64(ftmp5, 2);
906 /* ftmp5[i] < 2 * 2^57 = 2^58 */
908 /* x3 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
909 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
910 felem_diff_128_64(tmp2, ftmp5);
911 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
912 felem_reduce(x3, tmp2);
914 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3 */
915 felem_diff64(ftmp2, x3);
916 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
918 /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3) */
919 felem_mul(tmp2, ftmp3, ftmp2);
920 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
922 /* y3 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3) -
923 z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
924 felem_diff128(tmp2, tmp);
925 /* tmp2[i] < 2^118 + 2^120 < 2^121 */
926 felem_reduce(y3, tmp2);
928 /* the result (x3, y3, z3) is incorrect if one of the inputs is the
929 * point at infinity, so we need to check for this separately */
931 /* if point 1 is at infinity, copy point 2 to output, and vice versa */
932 copy_conditional(x3, x2, 4, z1_is_zero);
933 copy_conditional(x3, x1, 4, z2_is_zero);
934 copy_conditional(y3, y2, 4, z1_is_zero);
935 copy_conditional(y3, y1, 4, z2_is_zero);
936 copy_conditional(z3, z2, 4, z1_is_zero);
937 copy_conditional(z3, z1, 4, z2_is_zero);
940 /* Select a point from an array of 16 precomputed point multiples,
941 * in constant time: for bits = {b_0, b_1, b_2, b_3}, return the point
942 * pre_comp[8*b_3 + 4*b_2 + 2*b_1 + b_0] */
943 static void select_point(const fslice bits[4], const fslice pre_comp[16][3][4],
947 select_conditional(tmp[0], pre_comp[7][0], pre_comp[15][0], 12, bits[3]);
948 select_conditional(tmp[1], pre_comp[3][0], pre_comp[11][0], 12, bits[3]);
949 select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]);
950 select_conditional(tmp[0], pre_comp[5][0], pre_comp[13][0], 12, bits[3]);
951 select_conditional(tmp[1], pre_comp[1][0], pre_comp[9][0], 12, bits[3]);
952 select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]);
953 select_conditional(tmp[4], tmp[3], tmp[2], 12, bits[1]);
954 select_conditional(tmp[0], pre_comp[6][0], pre_comp[14][0], 12, bits[3]);
955 select_conditional(tmp[1], pre_comp[2][0], pre_comp[10][0], 12, bits[3]);
956 select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]);
957 select_conditional(tmp[0], pre_comp[4][0], pre_comp[12][0], 12, bits[3]);
958 select_conditional(tmp[1], pre_comp[0][0], pre_comp[8][0], 12, bits[3]);
959 select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]);
960 select_conditional(tmp[1], tmp[3], tmp[2], 12, bits[1]);
961 select_conditional(out, tmp[1], tmp[4], 12, bits[0]);
964 /* Interleaved point multiplication using precomputed point multiples:
965 * The small point multiples 0*P, 1*P, ..., 15*P are in pre_comp[],
966 * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple
967 * of the generator, using certain (large) precomputed multiples in g_pre_comp.
968 * Output point (X, Y, Z) is stored in x_out, y_out, z_out */
969 static void batch_mul(fslice x_out[4], fslice y_out[4], fslice z_out[4],
970 const felem_bytearray scalars[], const unsigned num_points, const u8 *g_scalar,
971 const fslice pre_comp[][16][3][4], const fslice g_pre_comp[16][3][4])
974 unsigned gen_mul = (g_scalar != NULL);
975 fslice nq[12], nqt[12], tmp[12];
979 /* set nq to the point at infinity */
980 memset(nq, 0, 12 * sizeof(fslice));
982 /* Loop over all scalars msb-to-lsb, 4 bits at a time: for each nibble,
983 * double 4 times, then add the precomputed point multiples.
984 * If we are also adding multiples of the generator, then interleave
985 * these additions with the last 56 doublings. */
986 for (i = (num_points ? 28 : 7); i > 0; --i)
988 for (j = 0; j < 8; ++j)
991 point_double(nq, nq+4, nq+8, nq, nq+4, nq+8);
992 /* add multiples of the generator */
993 if ((gen_mul) && (i <= 7))
995 bits[3] = (g_scalar[i+20] >> (7-j)) & 1;
996 bits[2] = (g_scalar[i+13] >> (7-j)) & 1;
997 bits[1] = (g_scalar[i+6] >> (7-j)) & 1;
998 bits[0] = (g_scalar[i-1] >> (7-j)) & 1;
999 /* select the point to add, in constant time */
1000 select_point(bits, g_pre_comp, tmp);
1001 memcpy(nqt, nq, 12 * sizeof(fslice));
1002 point_add(nq, nq+4, nq+8, nqt, nqt+4, nqt+8,
1005 /* do an addition after every 4 doublings */
1008 /* loop over all scalars */
1009 for (num = 0; num < num_points; ++num)
1011 byte = scalars[num][i-1];
1012 bits[3] = (byte >> (10-j)) & 1;
1013 bits[2] = (byte >> (9-j)) & 1;
1014 bits[1] = (byte >> (8-j)) & 1;
1015 bits[0] = (byte >> (7-j)) & 1;
1016 /* select the point to add */
1018 pre_comp[num], tmp);
1019 memcpy(nqt, nq, 12 * sizeof(fslice));
1020 point_add(nq, nq+4, nq+8, nqt, nqt+4,
1021 nqt+8, tmp, tmp+4, tmp+8);
1026 memcpy(x_out, nq, 4 * sizeof(fslice));
1027 memcpy(y_out, nq+4, 4 * sizeof(fslice));
1028 memcpy(z_out, nq+8, 4 * sizeof(fslice));
1031 /******************************************************************************/
1032 /* FUNCTIONS TO MANAGE PRECOMPUTATION
1035 static NISTP224_PRE_COMP *nistp224_pre_comp_new()
1037 NISTP224_PRE_COMP *ret = NULL;
1038 ret = (NISTP224_PRE_COMP *)OPENSSL_malloc(sizeof(NISTP224_PRE_COMP));
1041 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1044 memset(ret->g_pre_comp, 0, sizeof(ret->g_pre_comp));
1045 ret->references = 1;
1049 static void *nistp224_pre_comp_dup(void *src_)
1051 NISTP224_PRE_COMP *src = src_;
1053 /* no need to actually copy, these objects never change! */
1054 CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP);
1059 static void nistp224_pre_comp_free(void *pre_)
1062 NISTP224_PRE_COMP *pre = pre_;
1067 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1074 static void nistp224_pre_comp_clear_free(void *pre_)
1077 NISTP224_PRE_COMP *pre = pre_;
1082 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1086 OPENSSL_cleanse(pre, sizeof *pre);
1090 /******************************************************************************/
1091 /* OPENSSL EC_METHOD FUNCTIONS
1094 int ec_GFp_nistp224_group_init(EC_GROUP *group)
1097 ret = ec_GFp_simple_group_init(group);
1098 group->a_is_minus3 = 1;
1102 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1103 const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
1106 BN_CTX *new_ctx = NULL;
1107 BIGNUM *curve_p, *curve_a, *curve_b;
1110 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1112 if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1113 ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1114 ((curve_b = BN_CTX_get(ctx)) == NULL)) goto err;
1115 BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1116 BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1117 BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1118 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) ||
1119 (BN_cmp(curve_b, b)))
1121 ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE,
1122 EC_R_WRONG_CURVE_PARAMETERS);
1125 group->field_mod_func = BN_nist_mod_224;
1126 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1129 if (new_ctx != NULL)
1130 BN_CTX_free(new_ctx);
1134 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
1135 * (X', Y') = (X/Z^2, Y/Z^3) */
1136 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1137 const EC_POINT *point, BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
1139 fslice z1[4], z2[4], x_in[4], y_in[4], x_out[4], y_out[4];
1142 if (EC_POINT_is_at_infinity(group, point))
1144 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1145 EC_R_POINT_AT_INFINITY);
1148 if ((!BN_to_felem(x_in, &point->X)) || (!BN_to_felem(y_in, &point->Y)) ||
1149 (!BN_to_felem(z1, &point->Z))) return 0;
1151 felem_square(tmp, z2); felem_reduce(z1, tmp);
1152 felem_mul(tmp, x_in, z1); felem_reduce(x_in, tmp);
1153 felem_contract(x_out, x_in);
1156 if (!felem_to_BN(x, x_out)) {
1157 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1162 felem_mul(tmp, z1, z2); felem_reduce(z1, tmp);
1163 felem_mul(tmp, y_in, z1); felem_reduce(y_in, tmp);
1164 felem_contract(y_out, y_in);
1167 if (!felem_to_BN(y, y_out)) {
1168 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1176 /* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL values
1177 * Result is stored in r (r can equal one of the inputs). */
1178 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1179 const BIGNUM *scalar, size_t num, const EC_POINT *points[],
1180 const BIGNUM *scalars[], BN_CTX *ctx)
1184 BN_CTX *new_ctx = NULL;
1185 BIGNUM *x, *y, *z, *tmp_scalar;
1186 felem_bytearray g_secret;
1187 felem_bytearray *secrets = NULL;
1188 fslice (*pre_comp)[16][3][4] = NULL;
1189 felem_bytearray tmp;
1191 int have_pre_comp = 0;
1192 size_t num_points = num;
1193 fslice x_in[4], y_in[4], z_in[4], x_out[4], y_out[4], z_out[4];
1194 NISTP224_PRE_COMP *pre = NULL;
1195 fslice (*g_pre_comp)[3][4] = NULL;
1196 EC_POINT *generator = NULL;
1197 const EC_POINT *p = NULL;
1198 const BIGNUM *p_scalar = NULL;
1201 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1203 if (((x = BN_CTX_get(ctx)) == NULL) ||
1204 ((y = BN_CTX_get(ctx)) == NULL) ||
1205 ((z = BN_CTX_get(ctx)) == NULL) ||
1206 ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
1211 pre = EC_EX_DATA_get_data(group->extra_data,
1212 nistp224_pre_comp_dup, nistp224_pre_comp_free,
1213 nistp224_pre_comp_clear_free);
1215 /* we have precomputation, try to use it */
1216 g_pre_comp = pre->g_pre_comp;
1218 /* try to use the standard precomputation */
1219 g_pre_comp = (fslice (*)[3][4]) gmul;
1220 generator = EC_POINT_new(group);
1221 if (generator == NULL)
1223 /* get the generator from precomputation */
1224 if (!felem_to_BN(x, g_pre_comp[1][0]) ||
1225 !felem_to_BN(y, g_pre_comp[1][1]) ||
1226 !felem_to_BN(z, g_pre_comp[1][2]))
1228 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1231 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1232 generator, x, y, z, ctx))
1234 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1235 /* precomputation matches generator */
1238 /* we don't have valid precomputation:
1239 * treat the generator as a random point */
1240 num_points = num_points + 1;
1242 secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray));
1243 pre_comp = OPENSSL_malloc(num_points * 16 * 3 * 4 * sizeof(fslice));
1245 if ((num_points) && ((secrets == NULL) || (pre_comp == NULL)))
1247 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1251 /* we treat NULL scalars as 0, and NULL points as points at infinity,
1252 * i.e., they contribute nothing to the linear combination */
1253 memset(secrets, 0, num_points * sizeof(felem_bytearray));
1254 memset(pre_comp, 0, num_points * 16 * 3 * 4 * sizeof(fslice));
1255 for (i = 0; i < num_points; ++i)
1260 p = EC_GROUP_get0_generator(group);
1264 /* the i^th point */
1267 p_scalar = scalars[i];
1269 if ((p_scalar != NULL) && (p != NULL))
1271 num_bytes = BN_num_bytes(p_scalar);
1272 /* reduce scalar to 0 <= scalar < 2^224 */
1273 if ((num_bytes > sizeof(felem_bytearray)) || (BN_is_negative(p_scalar)))
1275 /* this is an unusual input, and we don't guarantee
1276 * constant-timeness */
1277 if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx))
1279 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1282 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1285 BN_bn2bin(p_scalar, tmp);
1286 flip_endian(secrets[i], tmp, num_bytes);
1287 /* precompute multiples */
1288 if ((!BN_to_felem(x_out, &p->X)) ||
1289 (!BN_to_felem(y_out, &p->Y)) ||
1290 (!BN_to_felem(z_out, &p->Z))) goto err;
1291 memcpy(pre_comp[i][1][0], x_out, 4 * sizeof(fslice));
1292 memcpy(pre_comp[i][1][1], y_out, 4 * sizeof(fslice));
1293 memcpy(pre_comp[i][1][2], z_out, 4 * sizeof(fslice));
1294 for (j = 1; j < 8; ++j)
1296 point_double(pre_comp[i][2*j][0],
1297 pre_comp[i][2*j][1],
1298 pre_comp[i][2*j][2],
1302 point_add(pre_comp[i][2*j+1][0],
1303 pre_comp[i][2*j+1][1],
1304 pre_comp[i][2*j+1][2],
1308 pre_comp[i][2*j][0],
1309 pre_comp[i][2*j][1],
1310 pre_comp[i][2*j][2]);
1315 /* the scalar for the generator */
1316 if ((scalar != NULL) && (have_pre_comp))
1318 memset(g_secret, 0, sizeof g_secret);
1319 num_bytes = BN_num_bytes(scalar);
1320 /* reduce scalar to 0 <= scalar < 2^224 */
1321 if ((num_bytes > sizeof(felem_bytearray)) || (BN_is_negative(scalar)))
1323 /* this is an unusual input, and we don't guarantee
1324 * constant-timeness */
1325 if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx))
1327 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1330 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1333 BN_bn2bin(scalar, tmp);
1334 flip_endian(g_secret, tmp, num_bytes);
1335 /* do the multiplication with generator precomputation*/
1336 batch_mul(x_out, y_out, z_out,
1337 (const felem_bytearray (*)) secrets, num_points,
1338 g_secret, (const fslice (*)[16][3][4]) pre_comp,
1339 (const fslice (*)[3][4]) g_pre_comp);
1342 /* do the multiplication without generator precomputation */
1343 batch_mul(x_out, y_out, z_out,
1344 (const felem_bytearray (*)) secrets, num_points,
1345 NULL, (const fslice (*)[16][3][4]) pre_comp, NULL);
1346 /* reduce the output to its unique minimal representation */
1347 felem_contract(x_in, x_out);
1348 felem_contract(y_in, y_out);
1349 felem_contract(z_in, z_out);
1350 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1351 (!felem_to_BN(z, z_in)))
1353 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1356 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1360 if (generator != NULL)
1361 EC_POINT_free(generator);
1362 if (new_ctx != NULL)
1363 BN_CTX_free(new_ctx);
1364 if (secrets != NULL)
1365 OPENSSL_free(secrets);
1366 if (pre_comp != NULL)
1367 OPENSSL_free(pre_comp);
1371 int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1374 NISTP224_PRE_COMP *pre = NULL;
1376 BN_CTX *new_ctx = NULL;
1378 EC_POINT *generator = NULL;
1380 /* throw away old precomputation */
1381 EC_EX_DATA_free_data(&group->extra_data, nistp224_pre_comp_dup,
1382 nistp224_pre_comp_free, nistp224_pre_comp_clear_free);
1384 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1386 if (((x = BN_CTX_get(ctx)) == NULL) ||
1387 ((y = BN_CTX_get(ctx)) == NULL))
1389 /* get the generator */
1390 if (group->generator == NULL) goto err;
1391 generator = EC_POINT_new(group);
1392 if (generator == NULL)
1394 BN_bin2bn(nistp224_curve_params[3], sizeof (felem_bytearray), x);
1395 BN_bin2bn(nistp224_curve_params[4], sizeof (felem_bytearray), y);
1396 if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
1398 if ((pre = nistp224_pre_comp_new()) == NULL)
1400 /* if the generator is the standard one, use built-in precomputation */
1401 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1403 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1407 if ((!BN_to_felem(pre->g_pre_comp[1][0], &group->generator->X)) ||
1408 (!BN_to_felem(pre->g_pre_comp[1][1], &group->generator->Y)) ||
1409 (!BN_to_felem(pre->g_pre_comp[1][2], &group->generator->Z)))
1411 /* compute 2^56*G, 2^112*G, 2^168*G */
1412 for (i = 1; i < 5; ++i)
1414 point_double(pre->g_pre_comp[2*i][0], pre->g_pre_comp[2*i][1],
1415 pre->g_pre_comp[2*i][2], pre->g_pre_comp[i][0],
1416 pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
1417 for (j = 0; j < 55; ++j)
1419 point_double(pre->g_pre_comp[2*i][0],
1420 pre->g_pre_comp[2*i][1],
1421 pre->g_pre_comp[2*i][2],
1422 pre->g_pre_comp[2*i][0],
1423 pre->g_pre_comp[2*i][1],
1424 pre->g_pre_comp[2*i][2]);
1427 /* g_pre_comp[0] is the point at infinity */
1428 memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
1429 /* the remaining multiples */
1430 /* 2^56*G + 2^112*G */
1431 point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
1432 pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
1433 pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
1434 pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
1435 pre->g_pre_comp[2][2]);
1436 /* 2^56*G + 2^168*G */
1437 point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
1438 pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
1439 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
1440 pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
1441 pre->g_pre_comp[2][2]);
1442 /* 2^112*G + 2^168*G */
1443 point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
1444 pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
1445 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
1446 pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
1447 pre->g_pre_comp[4][2]);
1448 /* 2^56*G + 2^112*G + 2^168*G */
1449 point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
1450 pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
1451 pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
1452 pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
1453 pre->g_pre_comp[2][2]);
1454 for (i = 1; i < 8; ++i)
1456 /* odd multiples: add G */
1457 point_add(pre->g_pre_comp[2*i+1][0], pre->g_pre_comp[2*i+1][1],
1458 pre->g_pre_comp[2*i+1][2], pre->g_pre_comp[2*i][0],
1459 pre->g_pre_comp[2*i][1], pre->g_pre_comp[2*i][2],
1460 pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
1461 pre->g_pre_comp[1][2]);
1464 if (!EC_EX_DATA_set_data(&group->extra_data, pre, nistp224_pre_comp_dup,
1465 nistp224_pre_comp_free, nistp224_pre_comp_clear_free))
1471 if (generator != NULL)
1472 EC_POINT_free(generator);
1473 if (new_ctx != NULL)
1474 BN_CTX_free(new_ctx);
1476 nistp224_pre_comp_free(pre);
1480 int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
1482 if (EC_EX_DATA_get_data(group->extra_data, nistp224_pre_comp_dup,
1483 nistp224_pre_comp_free, nistp224_pre_comp_clear_free)
1491 static void *dummy=&dummy;