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 #ifdef EC_NISTP224_64_GCC_128
68 #include <openssl/err.h>
71 #if defined(__GNUC__) && (__GNUC__ > 3 || (__GNUC__ == 3 && __GNUC_MINOR__ >= 1))
72 /* even with gcc, the typedef won't work for 32-bit platforms */
73 typedef __uint128_t uint128_t; /* nonstandard; implemented by gcc on 64-bit platforms */
75 #error "Need GCC 3.1 or later to define type uint128_t"
81 /******************************************************************************/
82 /* INTERNAL REPRESENTATION OF FIELD ELEMENTS
84 * Field elements are represented as a_0 + 2^56*a_1 + 2^112*a_2 + 2^168*a_3
85 * where each slice a_i is a 64-bit word, i.e., a field element is an fslice
86 * array a with 4 elements, where a[i] = a_i.
87 * Outputs from multiplications are represented as unreduced polynomials
88 * 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
89 * where each b_i is a 128-bit word. We ensure that inputs to each field
90 * multiplication satisfy a_i < 2^60, so outputs satisfy b_i < 4*2^60*2^60,
91 * and fit into a 128-bit word without overflow. The coefficients are then
92 * again partially reduced to a_i < 2^57. We only reduce to the unique minimal
93 * representation at the end of the computation.
97 typedef uint64_t fslice;
99 /* Field element represented as a byte arrary.
100 * 28*8 = 224 bits is also the group order size for the elliptic curve. */
101 typedef u8 felem_bytearray[28];
103 static const felem_bytearray nistp224_curve_params[5] = {
104 {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, /* p */
105 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0x00,0x00,0x00,0x00,
106 0x00,0x00,0x00,0x00,0x00,0x00,0x00,0x01},
107 {0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF, /* a */
108 0xFF,0xFF,0xFF,0xFF,0xFF,0xFE,0xFF,0xFF,0xFF,0xFF,
109 0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFF,0xFE},
110 {0xB4,0x05,0x0A,0x85,0x0C,0x04,0xB3,0xAB,0xF5,0x41, /* b */
111 0x32,0x56,0x50,0x44,0xB0,0xB7,0xD7,0xBF,0xD8,0xBA,
112 0x27,0x0B,0x39,0x43,0x23,0x55,0xFF,0xB4},
113 {0xB7,0x0E,0x0C,0xBD,0x6B,0xB4,0xBF,0x7F,0x32,0x13, /* x */
114 0x90,0xB9,0x4A,0x03,0xC1,0xD3,0x56,0xC2,0x11,0x22,
115 0x34,0x32,0x80,0xD6,0x11,0x5C,0x1D,0x21},
116 {0xbd,0x37,0x63,0x88,0xb5,0xf7,0x23,0xfb,0x4c,0x22, /* y */
117 0xdf,0xe6,0xcd,0x43,0x75,0xa0,0x5a,0x07,0x47,0x64,
118 0x44,0xd5,0x81,0x99,0x85,0x00,0x7e,0x34}
121 /* Precomputed multiples of the standard generator
122 * b_0*G + b_1*2^56*G + b_2*2^112*G + b_3*2^168*G for
123 * (b_3, b_2, b_1, b_0) in [0,15], i.e., gmul[0] = point_at_infinity,
124 * gmul[1] = G, gmul[2] = 2^56*G, gmul[3] = 2^56*G + G, etc.
125 * Points are given in Jacobian projective coordinates: words 0-3 represent the
126 * X-coordinate (slice a_0 is word 0, etc.), words 4-7 represent the
127 * Y-coordinate and words 8-11 represent the Z-coordinate. */
128 static const fslice gmul[16][3][4] = {
129 {{0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000},
130 {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000},
131 {0x00000000000000, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
132 {{0x3280d6115c1d21, 0xc1d356c2112234, 0x7f321390b94a03, 0xb70e0cbd6bb4bf},
133 {0xd5819985007e34, 0x75a05a07476444, 0xfb4c22dfe6cd43, 0xbd376388b5f723},
134 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
135 {{0xfd9675666ebbe9, 0xbca7664d40ce5e, 0x2242df8d8a2a43, 0x1f49bbb0f99bc5},
136 {0x29e0b892dc9c43, 0xece8608436e662, 0xdc858f185310d0, 0x9812dd4eb8d321},
137 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
138 {{0x6d3e678d5d8eb8, 0x559eed1cb362f1, 0x16e9a3bbce8a3f, 0xeedcccd8c2a748},
139 {0xf19f90ed50266d, 0xabf2b4bf65f9df, 0x313865468fafec, 0x5cb379ba910a17},
140 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
141 {{0x0641966cab26e3, 0x91fb2991fab0a0, 0xefec27a4e13a0b, 0x0499aa8a5f8ebe},
142 {0x7510407766af5d, 0x84d929610d5450, 0x81d77aae82f706, 0x6916f6d4338c5b},
143 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
144 {{0xea95ac3b1f15c6, 0x086000905e82d4, 0xdd323ae4d1c8b1, 0x932b56be7685a3},
145 {0x9ef93dea25dbbf, 0x41665960f390f0, 0xfdec76dbe2a8a7, 0x523e80f019062a},
146 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
147 {{0x822fdd26732c73, 0xa01c83531b5d0f, 0x363f37347c1ba4, 0xc391b45c84725c},
148 {0xbbd5e1b2d6ad24, 0xddfbcde19dfaec, 0xc393da7e222a7f, 0x1efb7890ede244},
149 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
150 {{0x4c9e90ca217da1, 0xd11beca79159bb, 0xff8d33c2c98b7c, 0x2610b39409f849},
151 {0x44d1352ac64da0, 0xcdbb7b2c46b4fb, 0x966c079b753c89, 0xfe67e4e820b112},
152 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
153 {{0xe28cae2df5312d, 0xc71b61d16f5c6e, 0x79b7619a3e7c4c, 0x05c73240899b47},
154 {0x9f7f6382c73e3a, 0x18615165c56bda, 0x641fab2116fd56, 0x72855882b08394},
155 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
156 {{0x0469182f161c09, 0x74a98ca8d00fb5, 0xb89da93489a3e0, 0x41c98768fb0c1d},
157 {0xe5ea05fb32da81, 0x3dce9ffbca6855, 0x1cfe2d3fbf59e6, 0x0e5e03408738a7},
158 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
159 {{0xdab22b2333e87f, 0x4430137a5dd2f6, 0xe03ab9f738beb8, 0xcb0c5d0dc34f24},
160 {0x764a7df0c8fda5, 0x185ba5c3fa2044, 0x9281d688bcbe50, 0xc40331df893881},
161 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
162 {{0xb89530796f0f60, 0xade92bd26909a3, 0x1a0c83fb4884da, 0x1765bf22a5a984},
163 {0x772a9ee75db09e, 0x23bc6c67cec16f, 0x4c1edba8b14e2f, 0xe2a215d9611369},
164 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
165 {{0x571e509fb5efb3, 0xade88696410552, 0xc8ae85fada74fe, 0x6c7e4be83bbde3},
166 {0xff9f51160f4652, 0xb47ce2495a6539, 0xa2946c53b582f4, 0x286d2db3ee9a60},
167 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
168 {{0x40bbd5081a44af, 0x0995183b13926c, 0xbcefba6f47f6d0, 0x215619e9cc0057},
169 {0x8bc94d3b0df45e, 0xf11c54a3694f6f, 0x8631b93cdfe8b5, 0xe7e3f4b0982db9},
170 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
171 {{0xb17048ab3e1c7b, 0xac38f36ff8a1d8, 0x1c29819435d2c6, 0xc813132f4c07e9},
172 {0x2891425503b11f, 0x08781030579fea, 0xf5426ba5cc9674, 0x1e28ebf18562bc},
173 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}},
174 {{0x9f31997cc864eb, 0x06cd91d28b5e4c, 0xff17036691a973, 0xf1aef351497c58},
175 {0xdd1f2d600564ff, 0xdead073b1402db, 0x74a684435bd693, 0xeea7471f962558},
176 {0x00000000000001, 0x00000000000000, 0x00000000000000, 0x00000000000000}}
179 /* Precomputation for the group generator. */
181 fslice g_pre_comp[16][3][4];
185 const EC_METHOD *EC_GFp_nistp224_method(void)
187 static const EC_METHOD ret = {
188 NID_X9_62_prime_field,
189 ec_GFp_nistp224_group_init,
190 ec_GFp_simple_group_finish,
191 ec_GFp_simple_group_clear_finish,
192 ec_GFp_nist_group_copy,
193 ec_GFp_nistp224_group_set_curve,
194 ec_GFp_simple_group_get_curve,
195 ec_GFp_simple_group_get_degree,
196 ec_GFp_simple_group_check_discriminant,
197 ec_GFp_simple_point_init,
198 ec_GFp_simple_point_finish,
199 ec_GFp_simple_point_clear_finish,
200 ec_GFp_simple_point_copy,
201 ec_GFp_simple_point_set_to_infinity,
202 ec_GFp_simple_set_Jprojective_coordinates_GFp,
203 ec_GFp_simple_get_Jprojective_coordinates_GFp,
204 ec_GFp_simple_point_set_affine_coordinates,
205 ec_GFp_nistp224_point_get_affine_coordinates,
206 ec_GFp_simple_set_compressed_coordinates,
207 ec_GFp_simple_point2oct,
208 ec_GFp_simple_oct2point,
211 ec_GFp_simple_invert,
212 ec_GFp_simple_is_at_infinity,
213 ec_GFp_simple_is_on_curve,
215 ec_GFp_simple_make_affine,
216 ec_GFp_simple_points_make_affine,
217 ec_GFp_nistp224_points_mul,
218 ec_GFp_nistp224_precompute_mult,
219 ec_GFp_nistp224_have_precompute_mult,
220 ec_GFp_nist_field_mul,
221 ec_GFp_nist_field_sqr,
223 0 /* field_encode */,
224 0 /* field_decode */,
225 0 /* field_set_to_one */ };
230 /* Helper functions to convert field elements to/from internal representation */
231 static void bin28_to_felem(fslice out[4], const u8 in[28])
233 out[0] = *((const uint64_t *)(in)) & 0x00ffffffffffffff;
234 out[1] = (*((const uint64_t *)(in+7))) & 0x00ffffffffffffff;
235 out[2] = (*((const uint64_t *)(in+14))) & 0x00ffffffffffffff;
236 out[3] = (*((const uint64_t *)(in+21))) & 0x00ffffffffffffff;
239 static void felem_to_bin28(u8 out[28], const fslice in[4])
242 for (i = 0; i < 7; ++i)
244 out[i] = in[0]>>(8*i);
245 out[i+7] = in[1]>>(8*i);
246 out[i+14] = in[2]>>(8*i);
247 out[i+21] = in[3]>>(8*i);
251 /* To preserve endianness when using BN_bn2bin and BN_bin2bn */
252 static void flip_endian(u8 *out, const u8 *in, unsigned len)
255 for (i = 0; i < len; ++i)
256 out[i] = in[len-1-i];
259 /* From OpenSSL BIGNUM to internal representation */
260 static int BN_to_felem(fslice out[4], const BIGNUM *bn)
262 felem_bytearray b_in;
263 felem_bytearray b_out;
266 /* BN_bn2bin eats leading zeroes */
267 memset(b_out, 0, sizeof b_out);
268 num_bytes = BN_num_bytes(bn);
269 if (num_bytes > sizeof b_out)
271 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
274 if (BN_is_negative(bn))
276 ECerr(EC_F_BN_TO_FELEM, EC_R_BIGNUM_OUT_OF_RANGE);
279 num_bytes = BN_bn2bin(bn, b_in);
280 flip_endian(b_out, b_in, num_bytes);
281 bin28_to_felem(out, b_out);
285 /* From internal representation to OpenSSL BIGNUM */
286 static BIGNUM *felem_to_BN(BIGNUM *out, const fslice in[4])
288 felem_bytearray b_in, b_out;
289 felem_to_bin28(b_in, in);
290 flip_endian(b_out, b_in, sizeof b_out);
291 return BN_bin2bn(b_out, sizeof b_out, out);
294 /******************************************************************************/
297 * Field operations, using the internal representation of field elements.
298 * NB! These operations are specific to our point multiplication and cannot be
299 * expected to be correct in general - e.g., multiplication with a large scalar
300 * will cause an overflow.
304 /* Sum two field elements: out += in */
305 static void felem_sum64(fslice out[4], const fslice in[4])
313 /* Subtract field elements: out -= in */
314 /* Assumes in[i] < 2^57 */
315 static void felem_diff64(fslice out[4], const fslice in[4])
317 static const uint64_t two58p2 = (((uint64_t) 1) << 58) + (((uint64_t) 1) << 2);
318 static const uint64_t two58m2 = (((uint64_t) 1) << 58) - (((uint64_t) 1) << 2);
319 static const uint64_t two58m42m2 = (((uint64_t) 1) << 58) -
320 (((uint64_t) 1) << 42) - (((uint64_t) 1) << 2);
322 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
324 out[1] += two58m42m2;
334 /* Subtract in unreduced 128-bit mode: out128 -= in128 */
335 /* Assumes in[i] < 2^119 */
336 static void felem_diff128(uint128_t out[7], const uint128_t in[4])
338 static const uint128_t two120 = ((uint128_t) 1) << 120;
339 static const uint128_t two120m64 = (((uint128_t) 1) << 120) -
340 (((uint128_t) 1) << 64);
341 static const uint128_t two120m104m64 = (((uint128_t) 1) << 120) -
342 (((uint128_t) 1) << 104) - (((uint128_t) 1) << 64);
344 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
349 out[4] += two120m104m64;
362 /* Subtract in mixed mode: out128 -= in64 */
364 static void felem_diff_128_64(uint128_t out[7], const fslice in[4])
366 static const uint128_t two64p8 = (((uint128_t) 1) << 64) +
367 (((uint128_t) 1) << 8);
368 static const uint128_t two64m8 = (((uint128_t) 1) << 64) -
369 (((uint128_t) 1) << 8);
370 static const uint128_t two64m48m8 = (((uint128_t) 1) << 64) -
371 (((uint128_t) 1) << 48) - (((uint128_t) 1) << 8);
373 /* Add 0 mod 2^224-2^96+1 to ensure out > in */
375 out[1] += two64m48m8;
385 /* Multiply a field element by a scalar: out64 = out64 * scalar
386 * The scalars we actually use are small, so results fit without overflow */
387 static void felem_scalar64(fslice out[4], const fslice scalar)
395 /* Multiply an unreduced field element by a scalar: out128 = out128 * scalar
396 * The scalars we actually use are small, so results fit without overflow */
397 static void felem_scalar128(uint128_t out[7], const uint128_t scalar)
408 /* Square a field element: out = in^2 */
409 static void felem_square(uint128_t out[7], const fslice in[4])
411 out[0] = ((uint128_t) in[0]) * in[0];
412 out[1] = ((uint128_t) in[0]) * in[1] * 2;
413 out[2] = ((uint128_t) in[0]) * in[2] * 2 + ((uint128_t) in[1]) * in[1];
414 out[3] = ((uint128_t) in[0]) * in[3] * 2 +
415 ((uint128_t) in[1]) * in[2] * 2;
416 out[4] = ((uint128_t) in[1]) * in[3] * 2 + ((uint128_t) in[2]) * in[2];
417 out[5] = ((uint128_t) in[2]) * in[3] * 2;
418 out[6] = ((uint128_t) in[3]) * in[3];
421 /* Multiply two field elements: out = in1 * in2 */
422 static void felem_mul(uint128_t out[7], const fslice in1[4], const fslice in2[4])
424 out[0] = ((uint128_t) in1[0]) * in2[0];
425 out[1] = ((uint128_t) in1[0]) * in2[1] + ((uint128_t) in1[1]) * in2[0];
426 out[2] = ((uint128_t) in1[0]) * in2[2] + ((uint128_t) in1[1]) * in2[1] +
427 ((uint128_t) in1[2]) * in2[0];
428 out[3] = ((uint128_t) in1[0]) * in2[3] + ((uint128_t) in1[1]) * in2[2] +
429 ((uint128_t) in1[2]) * in2[1] + ((uint128_t) in1[3]) * in2[0];
430 out[4] = ((uint128_t) in1[1]) * in2[3] + ((uint128_t) in1[2]) * in2[2] +
431 ((uint128_t) in1[3]) * in2[1];
432 out[5] = ((uint128_t) in1[2]) * in2[3] + ((uint128_t) in1[3]) * in2[2];
433 out[6] = ((uint128_t) in1[3]) * in2[3];
436 /* Reduce 128-bit coefficients to 64-bit coefficients. Requires in[i] < 2^126,
437 * ensures out[0] < 2^56, out[1] < 2^56, out[2] < 2^56, out[3] < 2^57 */
438 static void felem_reduce(fslice out[4], const uint128_t in[7])
440 static const uint128_t two127p15 = (((uint128_t) 1) << 127) +
441 (((uint128_t) 1) << 15);
442 static const uint128_t two127m71 = (((uint128_t) 1) << 127) -
443 (((uint128_t) 1) << 71);
444 static const uint128_t two127m71m55 = (((uint128_t) 1) << 127) -
445 (((uint128_t) 1) << 71) - (((uint128_t) 1) << 55);
448 /* Add 0 mod 2^224-2^96+1 to ensure all differences are positive */
449 output[0] = in[0] + two127p15;
450 output[1] = in[1] + two127m71m55;
451 output[2] = in[2] + two127m71;
455 /* Eliminate in[4], in[5], in[6] */
456 output[4] += in[6] >> 16;
457 output[3] += (in[6]&0xffff) << 40;
460 output[3] += in[5] >> 16;
461 output[2] += (in[5]&0xffff) << 40;
464 output[2] += output[4] >> 16;
465 output[1] += (output[4]&0xffff) << 40;
466 output[0] -= output[4];
469 /* Carry 2 -> 3 -> 4 */
470 output[3] += output[2] >> 56;
471 output[2] &= 0x00ffffffffffffff;
473 output[4] += output[3] >> 56;
474 output[3] &= 0x00ffffffffffffff;
476 /* Now output[2] < 2^56, output[3] < 2^56 */
478 /* Eliminate output[4] */
479 output[2] += output[4] >> 16;
480 output[1] += (output[4]&0xffff) << 40;
481 output[0] -= output[4];
483 /* Carry 0 -> 1 -> 2 -> 3 */
484 output[1] += output[0] >> 56;
485 out[0] = output[0] & 0x00ffffffffffffff;
487 output[2] += output[1] >> 56;
488 out[1] = output[1] & 0x00ffffffffffffff;
489 output[3] += output[2] >> 56;
490 out[2] = output[2] & 0x00ffffffffffffff;
492 /* out[0] < 2^56, out[1] < 2^56, out[2] < 2^56,
493 * out[3] < 2^57 (due to final carry) */
497 /* Reduce to unique minimal representation */
498 static void felem_contract(fslice out[4], const fslice in[4])
500 static const int64_t two56 = ((uint64_t) 1) << 56;
501 /* 0 <= in < 2^225 */
502 /* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */
504 tmp[0] = (int64_t) in[0] - (in[3] >> 56);
505 tmp[1] = (int64_t) in[1] + ((in[3] >> 16) & 0x0000010000000000);
506 tmp[2] = (int64_t) in[2];
507 tmp[3] = (int64_t) in[3] & 0x00ffffffffffffff;
509 /* eliminate negative coefficients */
525 tmp[1] -= (1 & a) << 40;
527 /* carry 1 -> 2 -> 3 */
528 tmp[2] += tmp[1] >> 56;
529 tmp[1] &= 0x00ffffffffffffff;
531 tmp[3] += tmp[2] >> 56;
532 tmp[2] &= 0x00ffffffffffffff;
534 /* 0 <= in < 2^224 + 2^96 - 1 */
535 /* if in > 2^224 , reduce in = in - 2^224 + 2^96 - 1 */
536 tmp[0] -= (tmp[3] >> 56);
537 tmp[1] += ((tmp[3] >> 16) & 0x0000010000000000);
538 tmp[3] &= 0x00ffffffffffffff;
540 /* eliminate negative coefficients */
556 tmp[1] -= (1 & a) << 40;
558 /* carry 1 -> 2 -> 3 */
559 tmp[2] += tmp[1] >> 56;
560 tmp[1] &= 0x00ffffffffffffff;
562 tmp[3] += tmp[2] >> 56;
563 tmp[2] &= 0x00ffffffffffffff;
565 /* Now 0 <= in < 2^224 */
567 /* if in > 2^224 - 2^96, reduce */
568 /* a = 0 iff in > 2^224 - 2^96, i.e.,
569 * the high 128 bits are all 1 and the lower part is non-zero */
570 a = (tmp[3] + 1) | (tmp[2] + 1) |
571 ((tmp[1] | 0x000000ffffffffff) + 1) |
572 ((((tmp[1] & 0xffff) - 1) >> 63) & ((tmp[0] - 1) >> 63));
573 /* turn a into an all-one mask (if a = 0) or an all-zero mask */
574 a = ((a & 0x00ffffffffffffff) - 1) >> 63;
575 /* subtract 2^224 - 2^96 + 1 if a is all-one*/
576 tmp[3] &= a ^ 0xffffffffffffffff;
577 tmp[2] &= a ^ 0xffffffffffffffff;
578 tmp[1] &= (a ^ 0xffffffffffffffff) | 0x000000ffffffffff;
580 /* eliminate negative coefficients: if tmp[0] is negative, tmp[1] must be
581 * non-zero, so we only need one step */
592 /* Zero-check: returns 1 if input is 0, and 0 otherwise.
593 * We know that field elements are reduced to in < 2^225,
594 * so we only need to check three cases: 0, 2^224 - 2^96 + 1,
595 * and 2^225 - 2^97 + 2 */
596 static fslice felem_is_zero(const fslice in[4])
598 fslice zero, two224m96p1, two225m97p2;
600 zero = in[0] | in[1] | in[2] | in[3];
601 zero = (((int64_t)(zero) - 1) >> 63) & 1;
602 two224m96p1 = (in[0] ^ 1) | (in[1] ^ 0x00ffff0000000000)
603 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x00ffffffffffffff);
604 two224m96p1 = (((int64_t)(two224m96p1) - 1) >> 63) & 1;
605 two225m97p2 = (in[0] ^ 2) | (in[1] ^ 0x00fffe0000000000)
606 | (in[2] ^ 0x00ffffffffffffff) | (in[3] ^ 0x01ffffffffffffff);
607 two225m97p2 = (((int64_t)(two225m97p2) - 1) >> 63) & 1;
608 return (zero | two224m96p1 | two225m97p2);
611 /* Invert a field element */
612 /* Computation chain copied from djb's code */
613 static void felem_inv(fslice out[4], const fslice in[4])
615 fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4];
619 felem_square(tmp, in); felem_reduce(ftmp, tmp); /* 2 */
620 felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp); /* 2^2 - 1 */
621 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp); /* 2^3 - 2 */
622 felem_mul(tmp, in, ftmp); felem_reduce(ftmp, tmp); /* 2^3 - 1 */
623 felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp); /* 2^4 - 2 */
624 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp); /* 2^5 - 4 */
625 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp); /* 2^6 - 8 */
626 felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp); /* 2^6 - 1 */
627 felem_square(tmp, ftmp); felem_reduce(ftmp2, tmp); /* 2^7 - 2 */
628 for (i = 0; i < 5; ++i) /* 2^12 - 2^6 */
630 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
632 felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp2, tmp); /* 2^12 - 1 */
633 felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp); /* 2^13 - 2 */
634 for (i = 0; i < 11; ++i) /* 2^24 - 2^12 */
636 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
638 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp2, tmp); /* 2^24 - 1 */
639 felem_square(tmp, ftmp2); felem_reduce(ftmp3, tmp); /* 2^25 - 2 */
640 for (i = 0; i < 23; ++i) /* 2^48 - 2^24 */
642 felem_square(tmp, ftmp3); felem_reduce(ftmp3, tmp);
644 felem_mul(tmp, ftmp3, ftmp2); felem_reduce(ftmp3, tmp); /* 2^48 - 1 */
645 felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp); /* 2^49 - 2 */
646 for (i = 0; i < 47; ++i) /* 2^96 - 2^48 */
648 felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
650 felem_mul(tmp, ftmp3, ftmp4); felem_reduce(ftmp3, tmp); /* 2^96 - 1 */
651 felem_square(tmp, ftmp3); felem_reduce(ftmp4, tmp); /* 2^97 - 2 */
652 for (i = 0; i < 23; ++i) /* 2^120 - 2^24 */
654 felem_square(tmp, ftmp4); felem_reduce(ftmp4, tmp);
656 felem_mul(tmp, ftmp2, ftmp4); felem_reduce(ftmp2, tmp); /* 2^120 - 1 */
657 for (i = 0; i < 6; ++i) /* 2^126 - 2^6 */
659 felem_square(tmp, ftmp2); felem_reduce(ftmp2, tmp);
661 felem_mul(tmp, ftmp2, ftmp); felem_reduce(ftmp, tmp); /* 2^126 - 1 */
662 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp); /* 2^127 - 2 */
663 felem_mul(tmp, ftmp, in); felem_reduce(ftmp, tmp); /* 2^127 - 1 */
664 for (i = 0; i < 97; ++i) /* 2^224 - 2^97 */
666 felem_square(tmp, ftmp); felem_reduce(ftmp, tmp);
668 felem_mul(tmp, ftmp, ftmp3); felem_reduce(out, tmp); /* 2^224 - 2^96 - 1 */
671 /* Copy in constant time:
672 * if icopy == 1, copy in to out,
673 * if icopy == 0, copy out to itself. */
675 copy_conditional(fslice *out, const fslice *in, unsigned len, fslice icopy)
678 /* icopy is a (64-bit) 0 or 1, so copy is either all-zero or all-one */
679 const fslice copy = -icopy;
680 for (i = 0; i < len; ++i)
682 const fslice tmp = copy & (in[i] ^ out[i]);
687 /* Copy in constant time:
688 * if isel == 1, copy in2 to out,
689 * if isel == 0, copy in1 to out. */
690 static void select_conditional(fslice *out, const fslice *in1, const fslice *in2,
691 unsigned len, fslice isel)
694 /* isel is a (64-bit) 0 or 1, so sel is either all-zero or all-one */
695 const fslice sel = -isel;
696 for (i = 0; i < len; ++i)
698 const fslice tmp = sel & (in1[i] ^ in2[i]);
699 out[i] = in1[i] ^ tmp;
703 /******************************************************************************/
704 /* ELLIPTIC CURVE POINT OPERATIONS
706 * Points are represented in Jacobian projective coordinates:
707 * (X, Y, Z) corresponds to the affine point (X/Z^2, Y/Z^3),
708 * or to the point at infinity if Z == 0.
712 /* Double an elliptic curve point:
713 * (X', Y', Z') = 2 * (X, Y, Z), where
714 * X' = (3 * (X - Z^2) * (X + Z^2))^2 - 8 * X * Y^2
715 * Y' = 3 * (X - Z^2) * (X + Z^2) * (4 * X * Y^2 - X') - 8 * Y^2
716 * Z' = (Y + Z)^2 - Y^2 - Z^2 = 2 * Y * Z
717 * Outputs can equal corresponding inputs, i.e., x_out == x_in is allowed,
718 * while x_out == y_in is not (maybe this works, but it's not tested). */
720 point_double(fslice x_out[4], fslice y_out[4], fslice z_out[4],
721 const fslice x_in[4], const fslice y_in[4], const fslice z_in[4])
723 uint128_t tmp[7], tmp2[7];
728 fslice ftmp[4], ftmp2[4];
729 memcpy(ftmp, x_in, 4 * sizeof(fslice));
730 memcpy(ftmp2, x_in, 4 * sizeof(fslice));
733 felem_square(tmp, z_in);
734 felem_reduce(delta, tmp);
737 felem_square(tmp, y_in);
738 felem_reduce(gamma, tmp);
741 felem_mul(tmp, x_in, gamma);
742 felem_reduce(beta, tmp);
744 /* alpha = 3*(x-delta)*(x+delta) */
745 felem_diff64(ftmp, delta);
746 /* ftmp[i] < 2^57 + 2^58 + 2 < 2^59 */
747 felem_sum64(ftmp2, delta);
748 /* ftmp2[i] < 2^57 + 2^57 = 2^58 */
749 felem_scalar64(ftmp2, 3);
750 /* ftmp2[i] < 3 * 2^58 < 2^60 */
751 felem_mul(tmp, ftmp, ftmp2);
752 /* tmp[i] < 2^60 * 2^59 * 4 = 2^121 */
753 felem_reduce(alpha, tmp);
755 /* x' = alpha^2 - 8*beta */
756 felem_square(tmp, alpha);
757 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
758 memcpy(ftmp, beta, 4 * sizeof(fslice));
759 felem_scalar64(ftmp, 8);
760 /* ftmp[i] < 8 * 2^57 = 2^60 */
761 felem_diff_128_64(tmp, ftmp);
762 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
763 felem_reduce(x_out, tmp);
765 /* z' = (y + z)^2 - gamma - delta */
766 felem_sum64(delta, gamma);
767 /* delta[i] < 2^57 + 2^57 = 2^58 */
768 memcpy(ftmp, y_in, 4 * sizeof(fslice));
769 felem_sum64(ftmp, z_in);
770 /* ftmp[i] < 2^57 + 2^57 = 2^58 */
771 felem_square(tmp, ftmp);
772 /* tmp[i] < 4 * 2^58 * 2^58 = 2^118 */
773 felem_diff_128_64(tmp, delta);
774 /* tmp[i] < 2^118 + 2^64 + 8 < 2^119 */
775 felem_reduce(z_out, tmp);
777 /* y' = alpha*(4*beta - x') - 8*gamma^2 */
778 felem_scalar64(beta, 4);
779 /* beta[i] < 4 * 2^57 = 2^59 */
780 felem_diff64(beta, x_out);
781 /* beta[i] < 2^59 + 2^58 + 2 < 2^60 */
782 felem_mul(tmp, alpha, beta);
783 /* tmp[i] < 4 * 2^57 * 2^60 = 2^119 */
784 felem_square(tmp2, gamma);
785 /* tmp2[i] < 4 * 2^57 * 2^57 = 2^116 */
786 felem_scalar128(tmp2, 8);
787 /* tmp2[i] < 8 * 2^116 = 2^119 */
788 felem_diff128(tmp, tmp2);
789 /* tmp[i] < 2^119 + 2^120 < 2^121 */
790 felem_reduce(y_out, tmp);
793 /* Add two elliptic curve points:
794 * (X_1, Y_1, Z_1) + (X_2, Y_2, Z_2) = (X_3, Y_3, Z_3), where
795 * X_3 = (Z_1^3 * Y_2 - Z_2^3 * Y_1)^2 - (Z_1^2 * X_2 - Z_2^2 * X_1)^3 -
796 * 2 * Z_2^2 * X_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^2
797 * 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) -
798 * Z_2^3 * Y_1 * (Z_1^2 * X_2 - Z_2^2 * X_1)^3
799 * Z_3 = (Z_1^2 * X_2 - Z_2^2 * X_1) * (Z_1 * Z_2) */
801 /* This function is not entirely constant-time:
802 * it includes a branch for checking whether the two input points are equal,
803 * (while not equal to the point at infinity).
804 * This case never happens during single point multiplication,
805 * so there is no timing leak for ECDH or ECDSA signing. */
806 static void point_add(fslice x3[4], fslice y3[4], fslice z3[4],
807 const fslice x1[4], const fslice y1[4], const fslice z1[4],
808 const fslice x2[4], const fslice y2[4], const fslice z2[4])
810 fslice ftmp[4], ftmp2[4], ftmp3[4], ftmp4[4], ftmp5[4];
811 uint128_t tmp[7], tmp2[7];
812 fslice z1_is_zero, z2_is_zero, x_equal, y_equal;
815 felem_square(tmp, z1);
816 felem_reduce(ftmp, tmp);
819 felem_square(tmp, z2);
820 felem_reduce(ftmp2, tmp);
823 felem_mul(tmp, ftmp, z1);
824 felem_reduce(ftmp3, tmp);
827 felem_mul(tmp, ftmp2, z2);
828 felem_reduce(ftmp4, tmp);
830 /* ftmp3 = z1^3*y2 */
831 felem_mul(tmp, ftmp3, y2);
832 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
834 /* ftmp4 = z2^3*y1 */
835 felem_mul(tmp2, ftmp4, y1);
836 felem_reduce(ftmp4, tmp2);
838 /* ftmp3 = z1^3*y2 - z2^3*y1 */
839 felem_diff_128_64(tmp, ftmp4);
840 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
841 felem_reduce(ftmp3, tmp);
844 felem_mul(tmp, ftmp, x2);
845 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
848 felem_mul(tmp2, ftmp2, x1);
849 felem_reduce(ftmp2, tmp2);
851 /* ftmp = z1^2*x2 - z2^2*x1 */
852 felem_diff128(tmp, tmp2);
853 /* tmp[i] < 2^116 + 2^64 + 8 < 2^117 */
854 felem_reduce(ftmp, tmp);
856 /* the formulae are incorrect if the points are equal
857 * so we check for this and do doubling if this happens */
858 x_equal = felem_is_zero(ftmp);
859 y_equal = felem_is_zero(ftmp3);
860 z1_is_zero = felem_is_zero(z1);
861 z2_is_zero = felem_is_zero(z2);
862 /* In affine coordinates, (X_1, Y_1) == (X_2, Y_2) */
863 if (x_equal && y_equal && !z1_is_zero && !z2_is_zero)
865 point_double(x3, y3, z3, x1, y1, z1);
870 felem_mul(tmp, z1, z2);
871 felem_reduce(ftmp5, tmp);
873 /* z3 = (z1^2*x2 - z2^2*x1)*(z1*z2) */
874 felem_mul(tmp, ftmp, ftmp5);
875 felem_reduce(z3, tmp);
877 /* ftmp = (z1^2*x2 - z2^2*x1)^2 */
878 memcpy(ftmp5, ftmp, 4 * sizeof(fslice));
879 felem_square(tmp, ftmp);
880 felem_reduce(ftmp, tmp);
882 /* ftmp5 = (z1^2*x2 - z2^2*x1)^3 */
883 felem_mul(tmp, ftmp, ftmp5);
884 felem_reduce(ftmp5, tmp);
886 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
887 felem_mul(tmp, ftmp2, ftmp);
888 felem_reduce(ftmp2, tmp);
890 /* ftmp4 = z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
891 felem_mul(tmp, ftmp4, ftmp5);
892 /* tmp[i] < 4 * 2^57 * 2^57 = 2^116 */
894 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 */
895 felem_square(tmp2, ftmp3);
896 /* tmp2[i] < 4 * 2^57 * 2^57 < 2^116 */
898 /* tmp2 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 */
899 felem_diff_128_64(tmp2, ftmp5);
900 /* tmp2[i] < 2^116 + 2^64 + 8 < 2^117 */
902 /* ftmp5 = 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
903 memcpy(ftmp5, ftmp2, 4 * sizeof(fslice));
904 felem_scalar64(ftmp5, 2);
905 /* ftmp5[i] < 2 * 2^57 = 2^58 */
907 /* x3 = (z1^3*y2 - z2^3*y1)^2 - (z1^2*x2 - z2^2*x1)^3 -
908 2*z2^2*x1*(z1^2*x2 - z2^2*x1)^2 */
909 felem_diff_128_64(tmp2, ftmp5);
910 /* tmp2[i] < 2^117 + 2^64 + 8 < 2^118 */
911 felem_reduce(x3, tmp2);
913 /* ftmp2 = z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3 */
914 felem_diff64(ftmp2, x3);
915 /* ftmp2[i] < 2^57 + 2^58 + 2 < 2^59 */
917 /* tmp2 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3) */
918 felem_mul(tmp2, ftmp3, ftmp2);
919 /* tmp2[i] < 4 * 2^57 * 2^59 = 2^118 */
921 /* y3 = (z1^3*y2 - z2^3*y1)*(z2^2*x1*(z1^2*x2 - z2^2*x1)^2 - x3) -
922 z2^3*y1*(z1^2*x2 - z2^2*x1)^3 */
923 felem_diff128(tmp2, tmp);
924 /* tmp2[i] < 2^118 + 2^120 < 2^121 */
925 felem_reduce(y3, tmp2);
927 /* the result (x3, y3, z3) is incorrect if one of the inputs is the
928 * point at infinity, so we need to check for this separately */
930 /* if point 1 is at infinity, copy point 2 to output, and vice versa */
931 copy_conditional(x3, x2, 4, z1_is_zero);
932 copy_conditional(x3, x1, 4, z2_is_zero);
933 copy_conditional(y3, y2, 4, z1_is_zero);
934 copy_conditional(y3, y1, 4, z2_is_zero);
935 copy_conditional(z3, z2, 4, z1_is_zero);
936 copy_conditional(z3, z1, 4, z2_is_zero);
939 /* Select a point from an array of 16 precomputed point multiples,
940 * in constant time: for bits = {b_0, b_1, b_2, b_3}, return the point
941 * pre_comp[8*b_3 + 4*b_2 + 2*b_1 + b_0] */
942 static void select_point(const fslice bits[4], const fslice pre_comp[16][3][4],
946 select_conditional(tmp[0], pre_comp[7][0], pre_comp[15][0], 12, bits[3]);
947 select_conditional(tmp[1], pre_comp[3][0], pre_comp[11][0], 12, bits[3]);
948 select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]);
949 select_conditional(tmp[0], pre_comp[5][0], pre_comp[13][0], 12, bits[3]);
950 select_conditional(tmp[1], pre_comp[1][0], pre_comp[9][0], 12, bits[3]);
951 select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]);
952 select_conditional(tmp[4], tmp[3], tmp[2], 12, bits[1]);
953 select_conditional(tmp[0], pre_comp[6][0], pre_comp[14][0], 12, bits[3]);
954 select_conditional(tmp[1], pre_comp[2][0], pre_comp[10][0], 12, bits[3]);
955 select_conditional(tmp[2], tmp[1], tmp[0], 12, bits[2]);
956 select_conditional(tmp[0], pre_comp[4][0], pre_comp[12][0], 12, bits[3]);
957 select_conditional(tmp[1], pre_comp[0][0], pre_comp[8][0], 12, bits[3]);
958 select_conditional(tmp[3], tmp[1], tmp[0], 12, bits[2]);
959 select_conditional(tmp[1], tmp[3], tmp[2], 12, bits[1]);
960 select_conditional(out, tmp[1], tmp[4], 12, bits[0]);
963 /* Interleaved point multiplication using precomputed point multiples:
964 * The small point multiples 0*P, 1*P, ..., 15*P are in pre_comp[],
965 * the scalars in scalars[]. If g_scalar is non-NULL, we also add this multiple
966 * of the generator, using certain (large) precomputed multiples in g_pre_comp.
967 * Output point (X, Y, Z) is stored in x_out, y_out, z_out */
968 static void batch_mul(fslice x_out[4], fslice y_out[4], fslice z_out[4],
969 const felem_bytearray scalars[], const unsigned num_points, const u8 *g_scalar,
970 const fslice pre_comp[][16][3][4], const fslice g_pre_comp[16][3][4])
973 unsigned gen_mul = (g_scalar != NULL);
974 fslice nq[12], nqt[12], tmp[12];
978 /* set nq to the point at infinity */
979 memset(nq, 0, 12 * sizeof(fslice));
981 /* Loop over all scalars msb-to-lsb, 4 bits at a time: for each nibble,
982 * double 4 times, then add the precomputed point multiples.
983 * If we are also adding multiples of the generator, then interleave
984 * these additions with the last 56 doublings. */
985 for (i = (num_points ? 28 : 7); i > 0; --i)
987 for (j = 0; j < 8; ++j)
990 point_double(nq, nq+4, nq+8, nq, nq+4, nq+8);
991 /* add multiples of the generator */
992 if ((gen_mul) && (i <= 7))
994 bits[3] = (g_scalar[i+20] >> (7-j)) & 1;
995 bits[2] = (g_scalar[i+13] >> (7-j)) & 1;
996 bits[1] = (g_scalar[i+6] >> (7-j)) & 1;
997 bits[0] = (g_scalar[i-1] >> (7-j)) & 1;
998 /* select the point to add, in constant time */
999 select_point(bits, g_pre_comp, tmp);
1000 memcpy(nqt, nq, 12 * sizeof(fslice));
1001 point_add(nq, nq+4, nq+8, nqt, nqt+4, nqt+8,
1004 /* do an addition after every 4 doublings */
1007 /* loop over all scalars */
1008 for (num = 0; num < num_points; ++num)
1010 byte = scalars[num][i-1];
1011 bits[3] = (byte >> (10-j)) & 1;
1012 bits[2] = (byte >> (9-j)) & 1;
1013 bits[1] = (byte >> (8-j)) & 1;
1014 bits[0] = (byte >> (7-j)) & 1;
1015 /* select the point to add */
1017 pre_comp[num], tmp);
1018 memcpy(nqt, nq, 12 * sizeof(fslice));
1019 point_add(nq, nq+4, nq+8, nqt, nqt+4,
1020 nqt+8, tmp, tmp+4, tmp+8);
1025 memcpy(x_out, nq, 4 * sizeof(fslice));
1026 memcpy(y_out, nq+4, 4 * sizeof(fslice));
1027 memcpy(z_out, nq+8, 4 * sizeof(fslice));
1030 /******************************************************************************/
1031 /* FUNCTIONS TO MANAGE PRECOMPUTATION
1034 static NISTP224_PRE_COMP *nistp224_pre_comp_new()
1036 NISTP224_PRE_COMP *ret = NULL;
1037 ret = (NISTP224_PRE_COMP *)OPENSSL_malloc(sizeof(NISTP224_PRE_COMP));
1040 ECerr(EC_F_NISTP224_PRE_COMP_NEW, ERR_R_MALLOC_FAILURE);
1043 memset(ret->g_pre_comp, 0, sizeof(ret->g_pre_comp));
1044 ret->references = 1;
1048 static void *nistp224_pre_comp_dup(void *src_)
1050 NISTP224_PRE_COMP *src = src_;
1052 /* no need to actually copy, these objects never change! */
1053 CRYPTO_add(&src->references, 1, CRYPTO_LOCK_EC_PRE_COMP);
1058 static void nistp224_pre_comp_free(void *pre_)
1061 NISTP224_PRE_COMP *pre = pre_;
1066 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1073 static void nistp224_pre_comp_clear_free(void *pre_)
1076 NISTP224_PRE_COMP *pre = pre_;
1081 i = CRYPTO_add(&pre->references, -1, CRYPTO_LOCK_EC_PRE_COMP);
1085 OPENSSL_cleanse(pre, sizeof *pre);
1089 /******************************************************************************/
1090 /* OPENSSL EC_METHOD FUNCTIONS
1093 int ec_GFp_nistp224_group_init(EC_GROUP *group)
1096 ret = ec_GFp_simple_group_init(group);
1097 group->a_is_minus3 = 1;
1101 int ec_GFp_nistp224_group_set_curve(EC_GROUP *group, const BIGNUM *p,
1102 const BIGNUM *a, const BIGNUM *b, BN_CTX *ctx)
1105 BN_CTX *new_ctx = NULL;
1106 BIGNUM *curve_p, *curve_a, *curve_b;
1109 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1111 if (((curve_p = BN_CTX_get(ctx)) == NULL) ||
1112 ((curve_a = BN_CTX_get(ctx)) == NULL) ||
1113 ((curve_b = BN_CTX_get(ctx)) == NULL)) goto err;
1114 BN_bin2bn(nistp224_curve_params[0], sizeof(felem_bytearray), curve_p);
1115 BN_bin2bn(nistp224_curve_params[1], sizeof(felem_bytearray), curve_a);
1116 BN_bin2bn(nistp224_curve_params[2], sizeof(felem_bytearray), curve_b);
1117 if ((BN_cmp(curve_p, p)) || (BN_cmp(curve_a, a)) ||
1118 (BN_cmp(curve_b, b)))
1120 ECerr(EC_F_EC_GFP_NISTP224_GROUP_SET_CURVE,
1121 EC_R_WRONG_CURVE_PARAMETERS);
1124 group->field_mod_func = BN_nist_mod_224;
1125 ret = ec_GFp_simple_group_set_curve(group, p, a, b, ctx);
1128 if (new_ctx != NULL)
1129 BN_CTX_free(new_ctx);
1133 /* Takes the Jacobian coordinates (X, Y, Z) of a point and returns
1134 * (X', Y') = (X/Z^2, Y/Z^3) */
1135 int ec_GFp_nistp224_point_get_affine_coordinates(const EC_GROUP *group,
1136 const EC_POINT *point, BIGNUM *x, BIGNUM *y, BN_CTX *ctx)
1138 fslice z1[4], z2[4], x_in[4], y_in[4], x_out[4], y_out[4];
1141 if (EC_POINT_is_at_infinity(group, point))
1143 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1144 EC_R_POINT_AT_INFINITY);
1147 if ((!BN_to_felem(x_in, &point->X)) || (!BN_to_felem(y_in, &point->Y)) ||
1148 (!BN_to_felem(z1, &point->Z))) return 0;
1150 felem_square(tmp, z2); felem_reduce(z1, tmp);
1151 felem_mul(tmp, x_in, z1); felem_reduce(x_in, tmp);
1152 felem_contract(x_out, x_in);
1155 if (!felem_to_BN(x, x_out)) {
1156 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1161 felem_mul(tmp, z1, z2); felem_reduce(z1, tmp);
1162 felem_mul(tmp, y_in, z1); felem_reduce(y_in, tmp);
1163 felem_contract(y_out, y_in);
1166 if (!felem_to_BN(y, y_out)) {
1167 ECerr(EC_F_EC_GFP_NISTP224_POINT_GET_AFFINE_COORDINATES,
1175 /* Computes scalar*generator + \sum scalars[i]*points[i], ignoring NULL values
1176 * Result is stored in r (r can equal one of the inputs). */
1177 int ec_GFp_nistp224_points_mul(const EC_GROUP *group, EC_POINT *r,
1178 const BIGNUM *scalar, size_t num, const EC_POINT *points[],
1179 const BIGNUM *scalars[], BN_CTX *ctx)
1183 BN_CTX *new_ctx = NULL;
1184 BIGNUM *x, *y, *z, *tmp_scalar;
1185 felem_bytearray g_secret;
1186 felem_bytearray *secrets = NULL;
1187 fslice (*pre_comp)[16][3][4] = NULL;
1188 felem_bytearray tmp;
1190 int have_pre_comp = 0;
1191 size_t num_points = num;
1192 fslice x_in[4], y_in[4], z_in[4], x_out[4], y_out[4], z_out[4];
1193 NISTP224_PRE_COMP *pre = NULL;
1194 fslice (*g_pre_comp)[3][4] = NULL;
1195 EC_POINT *generator = NULL;
1196 const EC_POINT *p = NULL;
1197 const BIGNUM *p_scalar = NULL;
1200 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1202 if (((x = BN_CTX_get(ctx)) == NULL) ||
1203 ((y = BN_CTX_get(ctx)) == NULL) ||
1204 ((z = BN_CTX_get(ctx)) == NULL) ||
1205 ((tmp_scalar = BN_CTX_get(ctx)) == NULL))
1210 pre = EC_EX_DATA_get_data(group->extra_data,
1211 nistp224_pre_comp_dup, nistp224_pre_comp_free,
1212 nistp224_pre_comp_clear_free);
1214 /* we have precomputation, try to use it */
1215 g_pre_comp = pre->g_pre_comp;
1217 /* try to use the standard precomputation */
1218 g_pre_comp = (fslice (*)[3][4]) gmul;
1219 generator = EC_POINT_new(group);
1220 if (generator == NULL)
1222 /* get the generator from precomputation */
1223 if (!felem_to_BN(x, g_pre_comp[1][0]) ||
1224 !felem_to_BN(y, g_pre_comp[1][1]) ||
1225 !felem_to_BN(z, g_pre_comp[1][2]))
1227 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1230 if (!EC_POINT_set_Jprojective_coordinates_GFp(group,
1231 generator, x, y, z, ctx))
1233 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1234 /* precomputation matches generator */
1237 /* we don't have valid precomputation:
1238 * treat the generator as a random point */
1239 num_points = num_points + 1;
1241 secrets = OPENSSL_malloc(num_points * sizeof(felem_bytearray));
1242 pre_comp = OPENSSL_malloc(num_points * 16 * 3 * 4 * sizeof(fslice));
1244 if ((num_points) && ((secrets == NULL) || (pre_comp == NULL)))
1246 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_MALLOC_FAILURE);
1250 /* we treat NULL scalars as 0, and NULL points as points at infinity,
1251 * i.e., they contribute nothing to the linear combination */
1252 memset(secrets, 0, num_points * sizeof(felem_bytearray));
1253 memset(pre_comp, 0, num_points * 16 * 3 * 4 * sizeof(fslice));
1254 for (i = 0; i < num_points; ++i)
1259 p = EC_GROUP_get0_generator(group);
1263 /* the i^th point */
1266 p_scalar = scalars[i];
1268 if ((p_scalar != NULL) && (p != NULL))
1270 num_bytes = BN_num_bytes(p_scalar);
1271 /* reduce scalar to 0 <= scalar < 2^224 */
1272 if ((num_bytes > sizeof(felem_bytearray)) || (BN_is_negative(p_scalar)))
1274 /* this is an unusual input, and we don't guarantee
1275 * constant-timeness */
1276 if (!BN_nnmod(tmp_scalar, p_scalar, &group->order, ctx))
1278 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1281 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1284 BN_bn2bin(p_scalar, tmp);
1285 flip_endian(secrets[i], tmp, num_bytes);
1286 /* precompute multiples */
1287 if ((!BN_to_felem(x_out, &p->X)) ||
1288 (!BN_to_felem(y_out, &p->Y)) ||
1289 (!BN_to_felem(z_out, &p->Z))) goto err;
1290 memcpy(pre_comp[i][1][0], x_out, 4 * sizeof(fslice));
1291 memcpy(pre_comp[i][1][1], y_out, 4 * sizeof(fslice));
1292 memcpy(pre_comp[i][1][2], z_out, 4 * sizeof(fslice));
1293 for (j = 1; j < 8; ++j)
1295 point_double(pre_comp[i][2*j][0],
1296 pre_comp[i][2*j][1],
1297 pre_comp[i][2*j][2],
1301 point_add(pre_comp[i][2*j+1][0],
1302 pre_comp[i][2*j+1][1],
1303 pre_comp[i][2*j+1][2],
1307 pre_comp[i][2*j][0],
1308 pre_comp[i][2*j][1],
1309 pre_comp[i][2*j][2]);
1314 /* the scalar for the generator */
1315 if ((scalar != NULL) && (have_pre_comp))
1317 memset(g_secret, 0, sizeof g_secret);
1318 num_bytes = BN_num_bytes(scalar);
1319 /* reduce scalar to 0 <= scalar < 2^224 */
1320 if ((num_bytes > sizeof(felem_bytearray)) || (BN_is_negative(scalar)))
1322 /* this is an unusual input, and we don't guarantee
1323 * constant-timeness */
1324 if (!BN_nnmod(tmp_scalar, scalar, &group->order, ctx))
1326 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1329 num_bytes = BN_bn2bin(tmp_scalar, tmp);
1332 BN_bn2bin(scalar, tmp);
1333 flip_endian(g_secret, tmp, num_bytes);
1334 /* do the multiplication with generator precomputation*/
1335 batch_mul(x_out, y_out, z_out,
1336 (const felem_bytearray (*)) secrets, num_points,
1337 g_secret, (const fslice (*)[16][3][4]) pre_comp,
1338 (const fslice (*)[3][4]) g_pre_comp);
1341 /* do the multiplication without generator precomputation */
1342 batch_mul(x_out, y_out, z_out,
1343 (const felem_bytearray (*)) secrets, num_points,
1344 NULL, (const fslice (*)[16][3][4]) pre_comp, NULL);
1345 /* reduce the output to its unique minimal representation */
1346 felem_contract(x_in, x_out);
1347 felem_contract(y_in, y_out);
1348 felem_contract(z_in, z_out);
1349 if ((!felem_to_BN(x, x_in)) || (!felem_to_BN(y, y_in)) ||
1350 (!felem_to_BN(z, z_in)))
1352 ECerr(EC_F_EC_GFP_NISTP224_POINTS_MUL, ERR_R_BN_LIB);
1355 ret = EC_POINT_set_Jprojective_coordinates_GFp(group, r, x, y, z, ctx);
1359 if (generator != NULL)
1360 EC_POINT_free(generator);
1361 if (new_ctx != NULL)
1362 BN_CTX_free(new_ctx);
1363 if (secrets != NULL)
1364 OPENSSL_free(secrets);
1365 if (pre_comp != NULL)
1366 OPENSSL_free(pre_comp);
1370 int ec_GFp_nistp224_precompute_mult(EC_GROUP *group, BN_CTX *ctx)
1373 NISTP224_PRE_COMP *pre = NULL;
1375 BN_CTX *new_ctx = NULL;
1377 EC_POINT *generator = NULL;
1379 /* throw away old precomputation */
1380 EC_EX_DATA_free_data(&group->extra_data, nistp224_pre_comp_dup,
1381 nistp224_pre_comp_free, nistp224_pre_comp_clear_free);
1383 if ((ctx = new_ctx = BN_CTX_new()) == NULL) return 0;
1385 if (((x = BN_CTX_get(ctx)) == NULL) ||
1386 ((y = BN_CTX_get(ctx)) == NULL))
1388 /* get the generator */
1389 if (group->generator == NULL) goto err;
1390 generator = EC_POINT_new(group);
1391 if (generator == NULL)
1393 BN_bin2bn(nistp224_curve_params[3], sizeof (felem_bytearray), x);
1394 BN_bin2bn(nistp224_curve_params[4], sizeof (felem_bytearray), y);
1395 if (!EC_POINT_set_affine_coordinates_GFp(group, generator, x, y, ctx))
1397 if ((pre = nistp224_pre_comp_new()) == NULL)
1399 /* if the generator is the standard one, use built-in precomputation */
1400 if (0 == EC_POINT_cmp(group, generator, group->generator, ctx))
1402 memcpy(pre->g_pre_comp, gmul, sizeof(pre->g_pre_comp));
1406 if ((!BN_to_felem(pre->g_pre_comp[1][0], &group->generator->X)) ||
1407 (!BN_to_felem(pre->g_pre_comp[1][1], &group->generator->Y)) ||
1408 (!BN_to_felem(pre->g_pre_comp[1][2], &group->generator->Z)))
1410 /* compute 2^56*G, 2^112*G, 2^168*G */
1411 for (i = 1; i < 5; ++i)
1413 point_double(pre->g_pre_comp[2*i][0], pre->g_pre_comp[2*i][1],
1414 pre->g_pre_comp[2*i][2], pre->g_pre_comp[i][0],
1415 pre->g_pre_comp[i][1], pre->g_pre_comp[i][2]);
1416 for (j = 0; j < 55; ++j)
1418 point_double(pre->g_pre_comp[2*i][0],
1419 pre->g_pre_comp[2*i][1],
1420 pre->g_pre_comp[2*i][2],
1421 pre->g_pre_comp[2*i][0],
1422 pre->g_pre_comp[2*i][1],
1423 pre->g_pre_comp[2*i][2]);
1426 /* g_pre_comp[0] is the point at infinity */
1427 memset(pre->g_pre_comp[0], 0, sizeof(pre->g_pre_comp[0]));
1428 /* the remaining multiples */
1429 /* 2^56*G + 2^112*G */
1430 point_add(pre->g_pre_comp[6][0], pre->g_pre_comp[6][1],
1431 pre->g_pre_comp[6][2], pre->g_pre_comp[4][0],
1432 pre->g_pre_comp[4][1], pre->g_pre_comp[4][2],
1433 pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
1434 pre->g_pre_comp[2][2]);
1435 /* 2^56*G + 2^168*G */
1436 point_add(pre->g_pre_comp[10][0], pre->g_pre_comp[10][1],
1437 pre->g_pre_comp[10][2], pre->g_pre_comp[8][0],
1438 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
1439 pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
1440 pre->g_pre_comp[2][2]);
1441 /* 2^112*G + 2^168*G */
1442 point_add(pre->g_pre_comp[12][0], pre->g_pre_comp[12][1],
1443 pre->g_pre_comp[12][2], pre->g_pre_comp[8][0],
1444 pre->g_pre_comp[8][1], pre->g_pre_comp[8][2],
1445 pre->g_pre_comp[4][0], pre->g_pre_comp[4][1],
1446 pre->g_pre_comp[4][2]);
1447 /* 2^56*G + 2^112*G + 2^168*G */
1448 point_add(pre->g_pre_comp[14][0], pre->g_pre_comp[14][1],
1449 pre->g_pre_comp[14][2], pre->g_pre_comp[12][0],
1450 pre->g_pre_comp[12][1], pre->g_pre_comp[12][2],
1451 pre->g_pre_comp[2][0], pre->g_pre_comp[2][1],
1452 pre->g_pre_comp[2][2]);
1453 for (i = 1; i < 8; ++i)
1455 /* odd multiples: add G */
1456 point_add(pre->g_pre_comp[2*i+1][0], pre->g_pre_comp[2*i+1][1],
1457 pre->g_pre_comp[2*i+1][2], pre->g_pre_comp[2*i][0],
1458 pre->g_pre_comp[2*i][1], pre->g_pre_comp[2*i][2],
1459 pre->g_pre_comp[1][0], pre->g_pre_comp[1][1],
1460 pre->g_pre_comp[1][2]);
1463 if (!EC_EX_DATA_set_data(&group->extra_data, pre, nistp224_pre_comp_dup,
1464 nistp224_pre_comp_free, nistp224_pre_comp_clear_free))
1470 if (generator != NULL)
1471 EC_POINT_free(generator);
1472 if (new_ctx != NULL)
1473 BN_CTX_free(new_ctx);
1475 nistp224_pre_comp_free(pre);
1479 int ec_GFp_nistp224_have_precompute_mult(const EC_GROUP *group)
1481 if (EC_EX_DATA_get_data(group->extra_data, nistp224_pre_comp_dup,
1482 nistp224_pre_comp_free, nistp224_pre_comp_clear_free)
1490 static void *dummy=&dummy;