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1 | /* |
2 | * Copyright (c) 2013, Kenneth MacKay | |
3 | * All rights reserved. | |
4 | * | |
5 | * Redistribution and use in source and binary forms, with or without | |
6 | * modification, are permitted provided that the following conditions are | |
7 | * met: | |
8 | * * Redistributions of source code must retain the above copyright | |
9 | * notice, this list of conditions and the following disclaimer. | |
10 | * * Redistributions in binary form must reproduce the above copyright | |
11 | * notice, this list of conditions and the following disclaimer in the | |
12 | * documentation and/or other materials provided with the distribution. | |
13 | * | |
14 | * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS | |
15 | * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT | |
16 | * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR | |
17 | * A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT | |
18 | * HOLDER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, | |
19 | * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT | |
20 | * LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, | |
21 | * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY | |
22 | * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT | |
23 | * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE | |
24 | * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. | |
25 | */ | |
26 | ||
27 | #include <linux/random.h> | |
28 | ||
29 | #include "ecc.h" | |
30 | ||
31 | /* 256-bit curve */ | |
32 | #define ECC_BYTES 32 | |
33 | ||
34 | #define MAX_TRIES 16 | |
35 | ||
36 | /* Number of u64's needed */ | |
37 | #define NUM_ECC_DIGITS (ECC_BYTES / 8) | |
38 | ||
39 | struct ecc_point { | |
40 | u64 x[NUM_ECC_DIGITS]; | |
41 | u64 y[NUM_ECC_DIGITS]; | |
42 | }; | |
43 | ||
44 | typedef struct { | |
45 | u64 m_low; | |
46 | u64 m_high; | |
47 | } uint128_t; | |
48 | ||
49 | #define CURVE_P_32 { 0xFFFFFFFFFFFFFFFFull, 0x00000000FFFFFFFFull, \ | |
50 | 0x0000000000000000ull, 0xFFFFFFFF00000001ull } | |
51 | ||
52 | #define CURVE_G_32 { \ | |
53 | { 0xF4A13945D898C296ull, 0x77037D812DEB33A0ull, \ | |
54 | 0xF8BCE6E563A440F2ull, 0x6B17D1F2E12C4247ull }, \ | |
55 | { 0xCBB6406837BF51F5ull, 0x2BCE33576B315ECEull, \ | |
56 | 0x8EE7EB4A7C0F9E16ull, 0x4FE342E2FE1A7F9Bull } \ | |
57 | } | |
58 | ||
59 | #define CURVE_N_32 { 0xF3B9CAC2FC632551ull, 0xBCE6FAADA7179E84ull, \ | |
60 | 0xFFFFFFFFFFFFFFFFull, 0xFFFFFFFF00000000ull } | |
61 | ||
62 | static u64 curve_p[NUM_ECC_DIGITS] = CURVE_P_32; | |
63 | static struct ecc_point curve_g = CURVE_G_32; | |
64 | static u64 curve_n[NUM_ECC_DIGITS] = CURVE_N_32; | |
65 | ||
66 | static void vli_clear(u64 *vli) | |
67 | { | |
68 | int i; | |
69 | ||
70 | for (i = 0; i < NUM_ECC_DIGITS; i++) | |
71 | vli[i] = 0; | |
72 | } | |
73 | ||
74 | /* Returns true if vli == 0, false otherwise. */ | |
75 | static bool vli_is_zero(const u64 *vli) | |
76 | { | |
77 | int i; | |
78 | ||
79 | for (i = 0; i < NUM_ECC_DIGITS; i++) { | |
80 | if (vli[i]) | |
81 | return false; | |
82 | } | |
83 | ||
84 | return true; | |
85 | } | |
86 | ||
87 | /* Returns nonzero if bit bit of vli is set. */ | |
88 | static u64 vli_test_bit(const u64 *vli, unsigned int bit) | |
89 | { | |
90 | return (vli[bit / 64] & ((u64) 1 << (bit % 64))); | |
91 | } | |
92 | ||
93 | /* Counts the number of 64-bit "digits" in vli. */ | |
94 | static unsigned int vli_num_digits(const u64 *vli) | |
95 | { | |
96 | int i; | |
97 | ||
98 | /* Search from the end until we find a non-zero digit. | |
99 | * We do it in reverse because we expect that most digits will | |
100 | * be nonzero. | |
101 | */ | |
102 | for (i = NUM_ECC_DIGITS - 1; i >= 0 && vli[i] == 0; i--); | |
103 | ||
104 | return (i + 1); | |
105 | } | |
106 | ||
107 | /* Counts the number of bits required for vli. */ | |
108 | static unsigned int vli_num_bits(const u64 *vli) | |
109 | { | |
110 | unsigned int i, num_digits; | |
111 | u64 digit; | |
112 | ||
113 | num_digits = vli_num_digits(vli); | |
114 | if (num_digits == 0) | |
115 | return 0; | |
116 | ||
117 | digit = vli[num_digits - 1]; | |
118 | for (i = 0; digit; i++) | |
119 | digit >>= 1; | |
120 | ||
121 | return ((num_digits - 1) * 64 + i); | |
122 | } | |
123 | ||
124 | /* Sets dest = src. */ | |
125 | static void vli_set(u64 *dest, const u64 *src) | |
126 | { | |
127 | int i; | |
128 | ||
129 | for (i = 0; i < NUM_ECC_DIGITS; i++) | |
130 | dest[i] = src[i]; | |
131 | } | |
132 | ||
133 | /* Returns sign of left - right. */ | |
134 | static int vli_cmp(const u64 *left, const u64 *right) | |
135 | { | |
136 | int i; | |
137 | ||
138 | for (i = NUM_ECC_DIGITS - 1; i >= 0; i--) { | |
139 | if (left[i] > right[i]) | |
140 | return 1; | |
141 | else if (left[i] < right[i]) | |
142 | return -1; | |
143 | } | |
144 | ||
145 | return 0; | |
146 | } | |
147 | ||
148 | /* Computes result = in << c, returning carry. Can modify in place | |
149 | * (if result == in). 0 < shift < 64. | |
150 | */ | |
151 | static u64 vli_lshift(u64 *result, const u64 *in, | |
152 | unsigned int shift) | |
153 | { | |
154 | u64 carry = 0; | |
155 | int i; | |
156 | ||
157 | for (i = 0; i < NUM_ECC_DIGITS; i++) { | |
158 | u64 temp = in[i]; | |
159 | ||
160 | result[i] = (temp << shift) | carry; | |
161 | carry = temp >> (64 - shift); | |
162 | } | |
163 | ||
164 | return carry; | |
165 | } | |
166 | ||
167 | /* Computes vli = vli >> 1. */ | |
168 | static void vli_rshift1(u64 *vli) | |
169 | { | |
170 | u64 *end = vli; | |
171 | u64 carry = 0; | |
172 | ||
173 | vli += NUM_ECC_DIGITS; | |
174 | ||
175 | while (vli-- > end) { | |
176 | u64 temp = *vli; | |
177 | *vli = (temp >> 1) | carry; | |
178 | carry = temp << 63; | |
179 | } | |
180 | } | |
181 | ||
182 | /* Computes result = left + right, returning carry. Can modify in place. */ | |
183 | static u64 vli_add(u64 *result, const u64 *left, | |
184 | const u64 *right) | |
185 | { | |
186 | u64 carry = 0; | |
187 | int i; | |
188 | ||
189 | for (i = 0; i < NUM_ECC_DIGITS; i++) { | |
190 | u64 sum; | |
191 | ||
192 | sum = left[i] + right[i] + carry; | |
193 | if (sum != left[i]) | |
194 | carry = (sum < left[i]); | |
195 | ||
196 | result[i] = sum; | |
197 | } | |
198 | ||
199 | return carry; | |
200 | } | |
201 | ||
202 | /* Computes result = left - right, returning borrow. Can modify in place. */ | |
203 | static u64 vli_sub(u64 *result, const u64 *left, const u64 *right) | |
204 | { | |
205 | u64 borrow = 0; | |
206 | int i; | |
207 | ||
208 | for (i = 0; i < NUM_ECC_DIGITS; i++) { | |
209 | u64 diff; | |
210 | ||
211 | diff = left[i] - right[i] - borrow; | |
212 | if (diff != left[i]) | |
213 | borrow = (diff > left[i]); | |
214 | ||
215 | result[i] = diff; | |
216 | } | |
217 | ||
218 | return borrow; | |
219 | } | |
220 | ||
221 | static uint128_t mul_64_64(u64 left, u64 right) | |
222 | { | |
223 | u64 a0 = left & 0xffffffffull; | |
224 | u64 a1 = left >> 32; | |
225 | u64 b0 = right & 0xffffffffull; | |
226 | u64 b1 = right >> 32; | |
227 | u64 m0 = a0 * b0; | |
228 | u64 m1 = a0 * b1; | |
229 | u64 m2 = a1 * b0; | |
230 | u64 m3 = a1 * b1; | |
231 | uint128_t result; | |
232 | ||
233 | m2 += (m0 >> 32); | |
234 | m2 += m1; | |
235 | ||
236 | /* Overflow */ | |
237 | if (m2 < m1) | |
238 | m3 += 0x100000000ull; | |
239 | ||
240 | result.m_low = (m0 & 0xffffffffull) | (m2 << 32); | |
241 | result.m_high = m3 + (m2 >> 32); | |
242 | ||
243 | return result; | |
244 | } | |
245 | ||
246 | static uint128_t add_128_128(uint128_t a, uint128_t b) | |
247 | { | |
248 | uint128_t result; | |
249 | ||
250 | result.m_low = a.m_low + b.m_low; | |
251 | result.m_high = a.m_high + b.m_high + (result.m_low < a.m_low); | |
252 | ||
253 | return result; | |
254 | } | |
255 | ||
256 | static void vli_mult(u64 *result, const u64 *left, const u64 *right) | |
257 | { | |
258 | uint128_t r01 = { 0, 0 }; | |
259 | u64 r2 = 0; | |
260 | unsigned int i, k; | |
261 | ||
262 | /* Compute each digit of result in sequence, maintaining the | |
263 | * carries. | |
264 | */ | |
265 | for (k = 0; k < NUM_ECC_DIGITS * 2 - 1; k++) { | |
266 | unsigned int min; | |
267 | ||
268 | if (k < NUM_ECC_DIGITS) | |
269 | min = 0; | |
270 | else | |
271 | min = (k + 1) - NUM_ECC_DIGITS; | |
272 | ||
273 | for (i = min; i <= k && i < NUM_ECC_DIGITS; i++) { | |
274 | uint128_t product; | |
275 | ||
276 | product = mul_64_64(left[i], right[k - i]); | |
277 | ||
278 | r01 = add_128_128(r01, product); | |
279 | r2 += (r01.m_high < product.m_high); | |
280 | } | |
281 | ||
282 | result[k] = r01.m_low; | |
283 | r01.m_low = r01.m_high; | |
284 | r01.m_high = r2; | |
285 | r2 = 0; | |
286 | } | |
287 | ||
288 | result[NUM_ECC_DIGITS * 2 - 1] = r01.m_low; | |
289 | } | |
290 | ||
291 | static void vli_square(u64 *result, const u64 *left) | |
292 | { | |
293 | uint128_t r01 = { 0, 0 }; | |
294 | u64 r2 = 0; | |
295 | int i, k; | |
296 | ||
297 | for (k = 0; k < NUM_ECC_DIGITS * 2 - 1; k++) { | |
298 | unsigned int min; | |
299 | ||
300 | if (k < NUM_ECC_DIGITS) | |
301 | min = 0; | |
302 | else | |
303 | min = (k + 1) - NUM_ECC_DIGITS; | |
304 | ||
305 | for (i = min; i <= k && i <= k - i; i++) { | |
306 | uint128_t product; | |
307 | ||
308 | product = mul_64_64(left[i], left[k - i]); | |
309 | ||
310 | if (i < k - i) { | |
311 | r2 += product.m_high >> 63; | |
312 | product.m_high = (product.m_high << 1) | | |
313 | (product.m_low >> 63); | |
314 | product.m_low <<= 1; | |
315 | } | |
316 | ||
317 | r01 = add_128_128(r01, product); | |
318 | r2 += (r01.m_high < product.m_high); | |
319 | } | |
320 | ||
321 | result[k] = r01.m_low; | |
322 | r01.m_low = r01.m_high; | |
323 | r01.m_high = r2; | |
324 | r2 = 0; | |
325 | } | |
326 | ||
327 | result[NUM_ECC_DIGITS * 2 - 1] = r01.m_low; | |
328 | } | |
329 | ||
330 | /* Computes result = (left + right) % mod. | |
331 | * Assumes that left < mod and right < mod, result != mod. | |
332 | */ | |
333 | static void vli_mod_add(u64 *result, const u64 *left, const u64 *right, | |
334 | const u64 *mod) | |
335 | { | |
336 | u64 carry; | |
337 | ||
338 | carry = vli_add(result, left, right); | |
339 | ||
340 | /* result > mod (result = mod + remainder), so subtract mod to | |
341 | * get remainder. | |
342 | */ | |
343 | if (carry || vli_cmp(result, mod) >= 0) | |
344 | vli_sub(result, result, mod); | |
345 | } | |
346 | ||
347 | /* Computes result = (left - right) % mod. | |
348 | * Assumes that left < mod and right < mod, result != mod. | |
349 | */ | |
350 | static void vli_mod_sub(u64 *result, const u64 *left, const u64 *right, | |
351 | const u64 *mod) | |
352 | { | |
353 | u64 borrow = vli_sub(result, left, right); | |
354 | ||
355 | /* In this case, p_result == -diff == (max int) - diff. | |
356 | * Since -x % d == d - x, we can get the correct result from | |
357 | * result + mod (with overflow). | |
358 | */ | |
359 | if (borrow) | |
360 | vli_add(result, result, mod); | |
361 | } | |
362 | ||
363 | /* Computes result = product % curve_p | |
364 | from http://www.nsa.gov/ia/_files/nist-routines.pdf */ | |
365 | static void vli_mmod_fast(u64 *result, const u64 *product) | |
366 | { | |
367 | u64 tmp[NUM_ECC_DIGITS]; | |
368 | int carry; | |
369 | ||
370 | /* t */ | |
371 | vli_set(result, product); | |
372 | ||
373 | /* s1 */ | |
374 | tmp[0] = 0; | |
375 | tmp[1] = product[5] & 0xffffffff00000000ull; | |
376 | tmp[2] = product[6]; | |
377 | tmp[3] = product[7]; | |
378 | carry = vli_lshift(tmp, tmp, 1); | |
379 | carry += vli_add(result, result, tmp); | |
380 | ||
381 | /* s2 */ | |
382 | tmp[1] = product[6] << 32; | |
383 | tmp[2] = (product[6] >> 32) | (product[7] << 32); | |
384 | tmp[3] = product[7] >> 32; | |
385 | carry += vli_lshift(tmp, tmp, 1); | |
386 | carry += vli_add(result, result, tmp); | |
387 | ||
388 | /* s3 */ | |
389 | tmp[0] = product[4]; | |
390 | tmp[1] = product[5] & 0xffffffff; | |
391 | tmp[2] = 0; | |
392 | tmp[3] = product[7]; | |
393 | carry += vli_add(result, result, tmp); | |
394 | ||
395 | /* s4 */ | |
396 | tmp[0] = (product[4] >> 32) | (product[5] << 32); | |
397 | tmp[1] = (product[5] >> 32) | (product[6] & 0xffffffff00000000ull); | |
398 | tmp[2] = product[7]; | |
399 | tmp[3] = (product[6] >> 32) | (product[4] << 32); | |
400 | carry += vli_add(result, result, tmp); | |
401 | ||
402 | /* d1 */ | |
403 | tmp[0] = (product[5] >> 32) | (product[6] << 32); | |
404 | tmp[1] = (product[6] >> 32); | |
405 | tmp[2] = 0; | |
406 | tmp[3] = (product[4] & 0xffffffff) | (product[5] << 32); | |
407 | carry -= vli_sub(result, result, tmp); | |
408 | ||
409 | /* d2 */ | |
410 | tmp[0] = product[6]; | |
411 | tmp[1] = product[7]; | |
412 | tmp[2] = 0; | |
413 | tmp[3] = (product[4] >> 32) | (product[5] & 0xffffffff00000000ull); | |
414 | carry -= vli_sub(result, result, tmp); | |
415 | ||
416 | /* d3 */ | |
417 | tmp[0] = (product[6] >> 32) | (product[7] << 32); | |
418 | tmp[1] = (product[7] >> 32) | (product[4] << 32); | |
419 | tmp[2] = (product[4] >> 32) | (product[5] << 32); | |
420 | tmp[3] = (product[6] << 32); | |
421 | carry -= vli_sub(result, result, tmp); | |
422 | ||
423 | /* d4 */ | |
424 | tmp[0] = product[7]; | |
425 | tmp[1] = product[4] & 0xffffffff00000000ull; | |
426 | tmp[2] = product[5]; | |
427 | tmp[3] = product[6] & 0xffffffff00000000ull; | |
428 | carry -= vli_sub(result, result, tmp); | |
429 | ||
430 | if (carry < 0) { | |
431 | do { | |
432 | carry += vli_add(result, result, curve_p); | |
433 | } while (carry < 0); | |
434 | } else { | |
435 | while (carry || vli_cmp(curve_p, result) != 1) | |
436 | carry -= vli_sub(result, result, curve_p); | |
437 | } | |
438 | } | |
439 | ||
440 | /* Computes result = (left * right) % curve_p. */ | |
441 | static void vli_mod_mult_fast(u64 *result, const u64 *left, const u64 *right) | |
442 | { | |
443 | u64 product[2 * NUM_ECC_DIGITS]; | |
444 | ||
445 | vli_mult(product, left, right); | |
446 | vli_mmod_fast(result, product); | |
447 | } | |
448 | ||
449 | /* Computes result = left^2 % curve_p. */ | |
450 | static void vli_mod_square_fast(u64 *result, const u64 *left) | |
451 | { | |
452 | u64 product[2 * NUM_ECC_DIGITS]; | |
453 | ||
454 | vli_square(product, left); | |
455 | vli_mmod_fast(result, product); | |
456 | } | |
457 | ||
458 | #define EVEN(vli) (!(vli[0] & 1)) | |
459 | /* Computes result = (1 / p_input) % mod. All VLIs are the same size. | |
460 | * See "From Euclid's GCD to Montgomery Multiplication to the Great Divide" | |
461 | * https://labs.oracle.com/techrep/2001/smli_tr-2001-95.pdf | |
462 | */ | |
463 | static void vli_mod_inv(u64 *result, const u64 *input, const u64 *mod) | |
464 | { | |
465 | u64 a[NUM_ECC_DIGITS], b[NUM_ECC_DIGITS]; | |
466 | u64 u[NUM_ECC_DIGITS], v[NUM_ECC_DIGITS]; | |
467 | u64 carry; | |
468 | int cmp_result; | |
469 | ||
470 | if (vli_is_zero(input)) { | |
471 | vli_clear(result); | |
472 | return; | |
473 | } | |
474 | ||
475 | vli_set(a, input); | |
476 | vli_set(b, mod); | |
477 | vli_clear(u); | |
478 | u[0] = 1; | |
479 | vli_clear(v); | |
480 | ||
481 | while ((cmp_result = vli_cmp(a, b)) != 0) { | |
482 | carry = 0; | |
483 | ||
484 | if (EVEN(a)) { | |
485 | vli_rshift1(a); | |
486 | ||
487 | if (!EVEN(u)) | |
488 | carry = vli_add(u, u, mod); | |
489 | ||
490 | vli_rshift1(u); | |
491 | if (carry) | |
492 | u[NUM_ECC_DIGITS - 1] |= 0x8000000000000000ull; | |
493 | } else if (EVEN(b)) { | |
494 | vli_rshift1(b); | |
495 | ||
496 | if (!EVEN(v)) | |
497 | carry = vli_add(v, v, mod); | |
498 | ||
499 | vli_rshift1(v); | |
500 | if (carry) | |
501 | v[NUM_ECC_DIGITS - 1] |= 0x8000000000000000ull; | |
502 | } else if (cmp_result > 0) { | |
503 | vli_sub(a, a, b); | |
504 | vli_rshift1(a); | |
505 | ||
506 | if (vli_cmp(u, v) < 0) | |
507 | vli_add(u, u, mod); | |
508 | ||
509 | vli_sub(u, u, v); | |
510 | if (!EVEN(u)) | |
511 | carry = vli_add(u, u, mod); | |
512 | ||
513 | vli_rshift1(u); | |
514 | if (carry) | |
515 | u[NUM_ECC_DIGITS - 1] |= 0x8000000000000000ull; | |
516 | } else { | |
517 | vli_sub(b, b, a); | |
518 | vli_rshift1(b); | |
519 | ||
520 | if (vli_cmp(v, u) < 0) | |
521 | vli_add(v, v, mod); | |
522 | ||
523 | vli_sub(v, v, u); | |
524 | if (!EVEN(v)) | |
525 | carry = vli_add(v, v, mod); | |
526 | ||
527 | vli_rshift1(v); | |
528 | if (carry) | |
529 | v[NUM_ECC_DIGITS - 1] |= 0x8000000000000000ull; | |
530 | } | |
531 | } | |
532 | ||
533 | vli_set(result, u); | |
534 | } | |
535 | ||
536 | /* ------ Point operations ------ */ | |
537 | ||
538 | /* Returns true if p_point is the point at infinity, false otherwise. */ | |
539 | static bool ecc_point_is_zero(const struct ecc_point *point) | |
540 | { | |
541 | return (vli_is_zero(point->x) && vli_is_zero(point->y)); | |
542 | } | |
543 | ||
544 | /* Point multiplication algorithm using Montgomery's ladder with co-Z | |
545 | * coordinates. From http://eprint.iacr.org/2011/338.pdf | |
546 | */ | |
547 | ||
548 | /* Double in place */ | |
549 | static void ecc_point_double_jacobian(u64 *x1, u64 *y1, u64 *z1) | |
550 | { | |
551 | /* t1 = x, t2 = y, t3 = z */ | |
552 | u64 t4[NUM_ECC_DIGITS]; | |
553 | u64 t5[NUM_ECC_DIGITS]; | |
554 | ||
555 | if (vli_is_zero(z1)) | |
556 | return; | |
557 | ||
558 | vli_mod_square_fast(t4, y1); /* t4 = y1^2 */ | |
559 | vli_mod_mult_fast(t5, x1, t4); /* t5 = x1*y1^2 = A */ | |
560 | vli_mod_square_fast(t4, t4); /* t4 = y1^4 */ | |
561 | vli_mod_mult_fast(y1, y1, z1); /* t2 = y1*z1 = z3 */ | |
562 | vli_mod_square_fast(z1, z1); /* t3 = z1^2 */ | |
563 | ||
564 | vli_mod_add(x1, x1, z1, curve_p); /* t1 = x1 + z1^2 */ | |
565 | vli_mod_add(z1, z1, z1, curve_p); /* t3 = 2*z1^2 */ | |
566 | vli_mod_sub(z1, x1, z1, curve_p); /* t3 = x1 - z1^2 */ | |
567 | vli_mod_mult_fast(x1, x1, z1); /* t1 = x1^2 - z1^4 */ | |
568 | ||
569 | vli_mod_add(z1, x1, x1, curve_p); /* t3 = 2*(x1^2 - z1^4) */ | |
570 | vli_mod_add(x1, x1, z1, curve_p); /* t1 = 3*(x1^2 - z1^4) */ | |
571 | if (vli_test_bit(x1, 0)) { | |
572 | u64 carry = vli_add(x1, x1, curve_p); | |
573 | vli_rshift1(x1); | |
574 | x1[NUM_ECC_DIGITS - 1] |= carry << 63; | |
575 | } else { | |
576 | vli_rshift1(x1); | |
577 | } | |
578 | /* t1 = 3/2*(x1^2 - z1^4) = B */ | |
579 | ||
580 | vli_mod_square_fast(z1, x1); /* t3 = B^2 */ | |
581 | vli_mod_sub(z1, z1, t5, curve_p); /* t3 = B^2 - A */ | |
582 | vli_mod_sub(z1, z1, t5, curve_p); /* t3 = B^2 - 2A = x3 */ | |
583 | vli_mod_sub(t5, t5, z1, curve_p); /* t5 = A - x3 */ | |
584 | vli_mod_mult_fast(x1, x1, t5); /* t1 = B * (A - x3) */ | |
585 | vli_mod_sub(t4, x1, t4, curve_p); /* t4 = B * (A - x3) - y1^4 = y3 */ | |
586 | ||
587 | vli_set(x1, z1); | |
588 | vli_set(z1, y1); | |
589 | vli_set(y1, t4); | |
590 | } | |
591 | ||
592 | /* Modify (x1, y1) => (x1 * z^2, y1 * z^3) */ | |
593 | static void apply_z(u64 *x1, u64 *y1, u64 *z) | |
594 | { | |
595 | u64 t1[NUM_ECC_DIGITS]; | |
596 | ||
597 | vli_mod_square_fast(t1, z); /* z^2 */ | |
598 | vli_mod_mult_fast(x1, x1, t1); /* x1 * z^2 */ | |
599 | vli_mod_mult_fast(t1, t1, z); /* z^3 */ | |
600 | vli_mod_mult_fast(y1, y1, t1); /* y1 * z^3 */ | |
601 | } | |
602 | ||
603 | /* P = (x1, y1) => 2P, (x2, y2) => P' */ | |
604 | static void xycz_initial_double(u64 *x1, u64 *y1, u64 *x2, u64 *y2, | |
605 | u64 *p_initial_z) | |
606 | { | |
607 | u64 z[NUM_ECC_DIGITS]; | |
608 | ||
609 | vli_set(x2, x1); | |
610 | vli_set(y2, y1); | |
611 | ||
612 | vli_clear(z); | |
613 | z[0] = 1; | |
614 | ||
615 | if (p_initial_z) | |
616 | vli_set(z, p_initial_z); | |
617 | ||
618 | apply_z(x1, y1, z); | |
619 | ||
620 | ecc_point_double_jacobian(x1, y1, z); | |
621 | ||
622 | apply_z(x2, y2, z); | |
623 | } | |
624 | ||
625 | /* Input P = (x1, y1, Z), Q = (x2, y2, Z) | |
626 | * Output P' = (x1', y1', Z3), P + Q = (x3, y3, Z3) | |
627 | * or P => P', Q => P + Q | |
628 | */ | |
629 | static void xycz_add(u64 *x1, u64 *y1, u64 *x2, u64 *y2) | |
630 | { | |
631 | /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */ | |
632 | u64 t5[NUM_ECC_DIGITS]; | |
633 | ||
634 | vli_mod_sub(t5, x2, x1, curve_p); /* t5 = x2 - x1 */ | |
635 | vli_mod_square_fast(t5, t5); /* t5 = (x2 - x1)^2 = A */ | |
636 | vli_mod_mult_fast(x1, x1, t5); /* t1 = x1*A = B */ | |
637 | vli_mod_mult_fast(x2, x2, t5); /* t3 = x2*A = C */ | |
638 | vli_mod_sub(y2, y2, y1, curve_p); /* t4 = y2 - y1 */ | |
639 | vli_mod_square_fast(t5, y2); /* t5 = (y2 - y1)^2 = D */ | |
640 | ||
641 | vli_mod_sub(t5, t5, x1, curve_p); /* t5 = D - B */ | |
642 | vli_mod_sub(t5, t5, x2, curve_p); /* t5 = D - B - C = x3 */ | |
643 | vli_mod_sub(x2, x2, x1, curve_p); /* t3 = C - B */ | |
644 | vli_mod_mult_fast(y1, y1, x2); /* t2 = y1*(C - B) */ | |
645 | vli_mod_sub(x2, x1, t5, curve_p); /* t3 = B - x3 */ | |
646 | vli_mod_mult_fast(y2, y2, x2); /* t4 = (y2 - y1)*(B - x3) */ | |
647 | vli_mod_sub(y2, y2, y1, curve_p); /* t4 = y3 */ | |
648 | ||
649 | vli_set(x2, t5); | |
650 | } | |
651 | ||
652 | /* Input P = (x1, y1, Z), Q = (x2, y2, Z) | |
653 | * Output P + Q = (x3, y3, Z3), P - Q = (x3', y3', Z3) | |
654 | * or P => P - Q, Q => P + Q | |
655 | */ | |
656 | static void xycz_add_c(u64 *x1, u64 *y1, u64 *x2, u64 *y2) | |
657 | { | |
658 | /* t1 = X1, t2 = Y1, t3 = X2, t4 = Y2 */ | |
659 | u64 t5[NUM_ECC_DIGITS]; | |
660 | u64 t6[NUM_ECC_DIGITS]; | |
661 | u64 t7[NUM_ECC_DIGITS]; | |
662 | ||
663 | vli_mod_sub(t5, x2, x1, curve_p); /* t5 = x2 - x1 */ | |
664 | vli_mod_square_fast(t5, t5); /* t5 = (x2 - x1)^2 = A */ | |
665 | vli_mod_mult_fast(x1, x1, t5); /* t1 = x1*A = B */ | |
666 | vli_mod_mult_fast(x2, x2, t5); /* t3 = x2*A = C */ | |
667 | vli_mod_add(t5, y2, y1, curve_p); /* t4 = y2 + y1 */ | |
668 | vli_mod_sub(y2, y2, y1, curve_p); /* t4 = y2 - y1 */ | |
669 | ||
670 | vli_mod_sub(t6, x2, x1, curve_p); /* t6 = C - B */ | |
671 | vli_mod_mult_fast(y1, y1, t6); /* t2 = y1 * (C - B) */ | |
672 | vli_mod_add(t6, x1, x2, curve_p); /* t6 = B + C */ | |
673 | vli_mod_square_fast(x2, y2); /* t3 = (y2 - y1)^2 */ | |
674 | vli_mod_sub(x2, x2, t6, curve_p); /* t3 = x3 */ | |
675 | ||
676 | vli_mod_sub(t7, x1, x2, curve_p); /* t7 = B - x3 */ | |
677 | vli_mod_mult_fast(y2, y2, t7); /* t4 = (y2 - y1)*(B - x3) */ | |
678 | vli_mod_sub(y2, y2, y1, curve_p); /* t4 = y3 */ | |
679 | ||
680 | vli_mod_square_fast(t7, t5); /* t7 = (y2 + y1)^2 = F */ | |
681 | vli_mod_sub(t7, t7, t6, curve_p); /* t7 = x3' */ | |
682 | vli_mod_sub(t6, t7, x1, curve_p); /* t6 = x3' - B */ | |
683 | vli_mod_mult_fast(t6, t6, t5); /* t6 = (y2 + y1)*(x3' - B) */ | |
684 | vli_mod_sub(y1, t6, y1, curve_p); /* t2 = y3' */ | |
685 | ||
686 | vli_set(x1, t7); | |
687 | } | |
688 | ||
689 | static void ecc_point_mult(struct ecc_point *result, | |
690 | const struct ecc_point *point, u64 *scalar, | |
691 | u64 *initial_z, int num_bits) | |
692 | { | |
693 | /* R0 and R1 */ | |
694 | u64 rx[2][NUM_ECC_DIGITS]; | |
695 | u64 ry[2][NUM_ECC_DIGITS]; | |
696 | u64 z[NUM_ECC_DIGITS]; | |
697 | int i, nb; | |
698 | ||
699 | vli_set(rx[1], point->x); | |
700 | vli_set(ry[1], point->y); | |
701 | ||
702 | xycz_initial_double(rx[1], ry[1], rx[0], ry[0], initial_z); | |
703 | ||
704 | for (i = num_bits - 2; i > 0; i--) { | |
705 | nb = !vli_test_bit(scalar, i); | |
706 | xycz_add_c(rx[1 - nb], ry[1 - nb], rx[nb], ry[nb]); | |
707 | xycz_add(rx[nb], ry[nb], rx[1 - nb], ry[1 - nb]); | |
708 | } | |
709 | ||
710 | nb = !vli_test_bit(scalar, 0); | |
711 | xycz_add_c(rx[1 - nb], ry[1 - nb], rx[nb], ry[nb]); | |
712 | ||
713 | /* Find final 1/Z value. */ | |
714 | vli_mod_sub(z, rx[1], rx[0], curve_p); /* X1 - X0 */ | |
715 | vli_mod_mult_fast(z, z, ry[1 - nb]); /* Yb * (X1 - X0) */ | |
716 | vli_mod_mult_fast(z, z, point->x); /* xP * Yb * (X1 - X0) */ | |
717 | vli_mod_inv(z, z, curve_p); /* 1 / (xP * Yb * (X1 - X0)) */ | |
718 | vli_mod_mult_fast(z, z, point->y); /* yP / (xP * Yb * (X1 - X0)) */ | |
719 | vli_mod_mult_fast(z, z, rx[1 - nb]); /* Xb * yP / (xP * Yb * (X1 - X0)) */ | |
720 | /* End 1/Z calculation */ | |
721 | ||
722 | xycz_add(rx[nb], ry[nb], rx[1 - nb], ry[1 - nb]); | |
723 | ||
724 | apply_z(rx[0], ry[0], z); | |
725 | ||
726 | vli_set(result->x, rx[0]); | |
727 | vli_set(result->y, ry[0]); | |
728 | } | |
729 | ||
730 | static void ecc_bytes2native(const u8 bytes[ECC_BYTES], | |
731 | u64 native[NUM_ECC_DIGITS]) | |
732 | { | |
733 | int i; | |
734 | ||
735 | for (i = 0; i < NUM_ECC_DIGITS; i++) { | |
736 | const u8 *digit = bytes + 8 * (NUM_ECC_DIGITS - 1 - i); | |
737 | ||
738 | native[NUM_ECC_DIGITS - 1 - i] = | |
739 | ((u64) digit[0] << 0) | | |
740 | ((u64) digit[1] << 8) | | |
741 | ((u64) digit[2] << 16) | | |
742 | ((u64) digit[3] << 24) | | |
743 | ((u64) digit[4] << 32) | | |
744 | ((u64) digit[5] << 40) | | |
745 | ((u64) digit[6] << 48) | | |
746 | ((u64) digit[7] << 56); | |
747 | } | |
748 | } | |
749 | ||
750 | static void ecc_native2bytes(const u64 native[NUM_ECC_DIGITS], | |
751 | u8 bytes[ECC_BYTES]) | |
752 | { | |
753 | int i; | |
754 | ||
755 | for (i = 0; i < NUM_ECC_DIGITS; i++) { | |
756 | u8 *digit = bytes + 8 * (NUM_ECC_DIGITS - 1 - i); | |
757 | ||
758 | digit[0] = native[NUM_ECC_DIGITS - 1 - i] >> 0; | |
759 | digit[1] = native[NUM_ECC_DIGITS - 1 - i] >> 8; | |
760 | digit[2] = native[NUM_ECC_DIGITS - 1 - i] >> 16; | |
761 | digit[3] = native[NUM_ECC_DIGITS - 1 - i] >> 24; | |
762 | digit[4] = native[NUM_ECC_DIGITS - 1 - i] >> 32; | |
763 | digit[5] = native[NUM_ECC_DIGITS - 1 - i] >> 40; | |
764 | digit[6] = native[NUM_ECC_DIGITS - 1 - i] >> 48; | |
765 | digit[7] = native[NUM_ECC_DIGITS - 1 - i] >> 56; | |
766 | } | |
767 | } | |
768 | ||
769 | bool ecc_make_key(u8 public_key[64], u8 private_key[32]) | |
770 | { | |
771 | struct ecc_point pk; | |
772 | u64 priv[NUM_ECC_DIGITS]; | |
773 | unsigned int tries = 0; | |
774 | ||
775 | do { | |
776 | if (tries++ >= MAX_TRIES) | |
777 | return false; | |
778 | ||
779 | get_random_bytes(priv, ECC_BYTES); | |
780 | ||
781 | if (vli_is_zero(priv)) | |
782 | continue; | |
783 | ||
784 | /* Make sure the private key is in the range [1, n-1]. */ | |
785 | if (vli_cmp(curve_n, priv) != 1) | |
786 | continue; | |
787 | ||
788 | ecc_point_mult(&pk, &curve_g, priv, NULL, vli_num_bits(priv)); | |
789 | } while (ecc_point_is_zero(&pk)); | |
790 | ||
791 | ecc_native2bytes(priv, private_key); | |
792 | ecc_native2bytes(pk.x, public_key); | |
793 | ecc_native2bytes(pk.y, &public_key[32]); | |
794 | ||
795 | return true; | |
796 | } | |
797 | ||
798 | bool ecdh_shared_secret(const u8 public_key[64], const u8 private_key[32], | |
799 | u8 secret[32]) | |
800 | { | |
801 | u64 priv[NUM_ECC_DIGITS]; | |
802 | u64 rand[NUM_ECC_DIGITS]; | |
803 | struct ecc_point product, pk; | |
804 | ||
805 | get_random_bytes(rand, ECC_BYTES); | |
806 | ||
807 | ecc_bytes2native(public_key, pk.x); | |
808 | ecc_bytes2native(&public_key[32], pk.y); | |
809 | ecc_bytes2native(private_key, priv); | |
810 | ||
811 | ecc_point_mult(&product, &pk, priv, rand, vli_num_bits(priv)); | |
812 | ||
813 | ecc_native2bytes(product.x, secret); | |
814 | ||
815 | return !ecc_point_is_zero(&product); | |
816 | } |