1/* Copyright 2008, Google Inc.
2 * All rights reserved.
3 *
4 * Redistribution and use in source and binary forms, with or without
5 * modification, are permitted provided that the following conditions are
6 * met:
7 *
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
11 * copyright notice, this list of conditions and the following disclaimer
12 * in the documentation and/or other materials provided with the
13 * distribution.
14 * * Neither the name of Google Inc. nor the names of its
15 * contributors may be used to endorse or promote products derived from
16 * this software without specific prior written permission.
17 *
18 * THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
19 * "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
20 * LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR
21 * A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT
22 * OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL,
23 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT
24 * LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE,
25 * DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY
26 * THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT
27 * (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE
28 * OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.
29 *
30 * curve25519-donna: Curve25519 elliptic curve, public key function
31 *
32 * http://code.google.com/p/curve25519-donna/
33 *
34 * Adam Langley <agl@imperialviolet.org>
35 *
36 * Derived from public domain C code by Daniel J. Bernstein <djb@cr.yp.to>
37 *
38 * More information about curve25519 can be found here
39 * http://cr.yp.to/ecdh.html
40 *
41 * djb's sample implementation of curve25519 is written in a special assembly
42 * language called qhasm and uses the floating point registers.
43 *
44 * This is, almost, a clean room reimplementation from the curve25519 paper. It
45 * uses many of the tricks described therein. Only the crecip function is taken
46 * from the sample implementation. */
47
48#include <string.h>
49#include <stdint.h>
50
51#ifdef _MSC_VER
52#define inline __inline
53#endif
54
55typedef uint8_t u8;
56typedef int32_t s32;
57typedef int64_t limb;
58
59/* Field element representation:
60 *
61 * Field elements are written as an array of signed, 64-bit limbs, least
62 * significant first. The value of the field element is:
63 * x[0] + 2^26·x[1] + x^51·x[2] + 2^102·x[3] + ...
64 *
65 * i.e. the limbs are 26, 25, 26, 25, ... bits wide. */
66
67/* Sum two numbers: output += in */
68static void fsum(limb *output, const limb *in) {
69 unsigned i;
70 for (i = 0; i < 10; i += 2) {
71 output[0+i] = output[0+i] + in[0+i];
72 output[1+i] = output[1+i] + in[1+i];
73 }
74}
75
76/* Find the difference of two numbers: output = in - output
77 * (note the order of the arguments!). */
78static void fdifference(limb *output, const limb *in) {
79 unsigned i;
80 for (i = 0; i < 10; ++i) {
81 output[i] = in[i] - output[i];
82 }
83}
84
85/* Multiply a number by a scalar: output = in * scalar */
86static void fscalar_product(limb *output, const limb *in, const limb scalar) {
87 unsigned i;
88 for (i = 0; i < 10; ++i) {
89 output[i] = in[i] * scalar;
90 }
91}
92
93/* Multiply two numbers: output = in2 * in
94 *
95 * output must be distinct to both inputs. The inputs are reduced coefficient
96 * form, the output is not.
97 *
98 * output[x] <= 14 * the largest product of the input limbs. */
99static void fproduct(limb *output, const limb *in2, const limb *in) {
100 output[0] = ((limb) ((s32) in2[0])) * ((s32) in[0]);
101 output[1] = ((limb) ((s32) in2[0])) * ((s32) in[1]) +
102 ((limb) ((s32) in2[1])) * ((s32) in[0]);
103 output[2] = 2 * ((limb) ((s32) in2[1])) * ((s32) in[1]) +
104 ((limb) ((s32) in2[0])) * ((s32) in[2]) +
105 ((limb) ((s32) in2[2])) * ((s32) in[0]);
106 output[3] = ((limb) ((s32) in2[1])) * ((s32) in[2]) +
107 ((limb) ((s32) in2[2])) * ((s32) in[1]) +
108 ((limb) ((s32) in2[0])) * ((s32) in[3]) +
109 ((limb) ((s32) in2[3])) * ((s32) in[0]);
110 output[4] = ((limb) ((s32) in2[2])) * ((s32) in[2]) +
111 2 * (((limb) ((s32) in2[1])) * ((s32) in[3]) +
112 ((limb) ((s32) in2[3])) * ((s32) in[1])) +
113 ((limb) ((s32) in2[0])) * ((s32) in[4]) +
114 ((limb) ((s32) in2[4])) * ((s32) in[0]);
115 output[5] = ((limb) ((s32) in2[2])) * ((s32) in[3]) +
116 ((limb) ((s32) in2[3])) * ((s32) in[2]) +
117 ((limb) ((s32) in2[1])) * ((s32) in[4]) +
118 ((limb) ((s32) in2[4])) * ((s32) in[1]) +
119 ((limb) ((s32) in2[0])) * ((s32) in[5]) +
120 ((limb) ((s32) in2[5])) * ((s32) in[0]);
121 output[6] = 2 * (((limb) ((s32) in2[3])) * ((s32) in[3]) +
122 ((limb) ((s32) in2[1])) * ((s32) in[5]) +
123 ((limb) ((s32) in2[5])) * ((s32) in[1])) +
124 ((limb) ((s32) in2[2])) * ((s32) in[4]) +
125 ((limb) ((s32) in2[4])) * ((s32) in[2]) +
126 ((limb) ((s32) in2[0])) * ((s32) in[6]) +
127 ((limb) ((s32) in2[6])) * ((s32) in[0]);
128 output[7] = ((limb) ((s32) in2[3])) * ((s32) in[4]) +
129 ((limb) ((s32) in2[4])) * ((s32) in[3]) +
130 ((limb) ((s32) in2[2])) * ((s32) in[5]) +
131 ((limb) ((s32) in2[5])) * ((s32) in[2]) +
132 ((limb) ((s32) in2[1])) * ((s32) in[6]) +
133 ((limb) ((s32) in2[6])) * ((s32) in[1]) +
134 ((limb) ((s32) in2[0])) * ((s32) in[7]) +
135 ((limb) ((s32) in2[7])) * ((s32) in[0]);
136 output[8] = ((limb) ((s32) in2[4])) * ((s32) in[4]) +
137 2 * (((limb) ((s32) in2[3])) * ((s32) in[5]) +
138 ((limb) ((s32) in2[5])) * ((s32) in[3]) +
139 ((limb) ((s32) in2[1])) * ((s32) in[7]) +
140 ((limb) ((s32) in2[7])) * ((s32) in[1])) +
141 ((limb) ((s32) in2[2])) * ((s32) in[6]) +
142 ((limb) ((s32) in2[6])) * ((s32) in[2]) +
143 ((limb) ((s32) in2[0])) * ((s32) in[8]) +
144 ((limb) ((s32) in2[8])) * ((s32) in[0]);
145 output[9] = ((limb) ((s32) in2[4])) * ((s32) in[5]) +
146 ((limb) ((s32) in2[5])) * ((s32) in[4]) +
147 ((limb) ((s32) in2[3])) * ((s32) in[6]) +
148 ((limb) ((s32) in2[6])) * ((s32) in[3]) +
149 ((limb) ((s32) in2[2])) * ((s32) in[7]) +
150 ((limb) ((s32) in2[7])) * ((s32) in[2]) +
151 ((limb) ((s32) in2[1])) * ((s32) in[8]) +
152 ((limb) ((s32) in2[8])) * ((s32) in[1]) +
153 ((limb) ((s32) in2[0])) * ((s32) in[9]) +
154 ((limb) ((s32) in2[9])) * ((s32) in[0]);
155 output[10] = 2 * (((limb) ((s32) in2[5])) * ((s32) in[5]) +
156 ((limb) ((s32) in2[3])) * ((s32) in[7]) +
157 ((limb) ((s32) in2[7])) * ((s32) in[3]) +
158 ((limb) ((s32) in2[1])) * ((s32) in[9]) +
159 ((limb) ((s32) in2[9])) * ((s32) in[1])) +
160 ((limb) ((s32) in2[4])) * ((s32) in[6]) +
161 ((limb) ((s32) in2[6])) * ((s32) in[4]) +
162 ((limb) ((s32) in2[2])) * ((s32) in[8]) +
163 ((limb) ((s32) in2[8])) * ((s32) in[2]);
164 output[11] = ((limb) ((s32) in2[5])) * ((s32) in[6]) +
165 ((limb) ((s32) in2[6])) * ((s32) in[5]) +
166 ((limb) ((s32) in2[4])) * ((s32) in[7]) +
167 ((limb) ((s32) in2[7])) * ((s32) in[4]) +
168 ((limb) ((s32) in2[3])) * ((s32) in[8]) +
169 ((limb) ((s32) in2[8])) * ((s32) in[3]) +
170 ((limb) ((s32) in2[2])) * ((s32) in[9]) +
171 ((limb) ((s32) in2[9])) * ((s32) in[2]);
172 output[12] = ((limb) ((s32) in2[6])) * ((s32) in[6]) +
173 2 * (((limb) ((s32) in2[5])) * ((s32) in[7]) +
174 ((limb) ((s32) in2[7])) * ((s32) in[5]) +
175 ((limb) ((s32) in2[3])) * ((s32) in[9]) +
176 ((limb) ((s32) in2[9])) * ((s32) in[3])) +
177 ((limb) ((s32) in2[4])) * ((s32) in[8]) +
178 ((limb) ((s32) in2[8])) * ((s32) in[4]);
179 output[13] = ((limb) ((s32) in2[6])) * ((s32) in[7]) +
180 ((limb) ((s32) in2[7])) * ((s32) in[6]) +
181 ((limb) ((s32) in2[5])) * ((s32) in[8]) +
182 ((limb) ((s32) in2[8])) * ((s32) in[5]) +
183 ((limb) ((s32) in2[4])) * ((s32) in[9]) +
184 ((limb) ((s32) in2[9])) * ((s32) in[4]);
185 output[14] = 2 * (((limb) ((s32) in2[7])) * ((s32) in[7]) +
186 ((limb) ((s32) in2[5])) * ((s32) in[9]) +
187 ((limb) ((s32) in2[9])) * ((s32) in[5])) +
188 ((limb) ((s32) in2[6])) * ((s32) in[8]) +
189 ((limb) ((s32) in2[8])) * ((s32) in[6]);
190 output[15] = ((limb) ((s32) in2[7])) * ((s32) in[8]) +
191 ((limb) ((s32) in2[8])) * ((s32) in[7]) +
192 ((limb) ((s32) in2[6])) * ((s32) in[9]) +
193 ((limb) ((s32) in2[9])) * ((s32) in[6]);
194 output[16] = ((limb) ((s32) in2[8])) * ((s32) in[8]) +
195 2 * (((limb) ((s32) in2[7])) * ((s32) in[9]) +
196 ((limb) ((s32) in2[9])) * ((s32) in[7]));
197 output[17] = ((limb) ((s32) in2[8])) * ((s32) in[9]) +
198 ((limb) ((s32) in2[9])) * ((s32) in[8]);
199 output[18] = 2 * ((limb) ((s32) in2[9])) * ((s32) in[9]);
200}
201
202/* Reduce a long form to a short form by taking the input mod 2^255 - 19.
203 *
204 * On entry: |output[i]| < 14*2^54
205 * On exit: |output[0..8]| < 280*2^54 */
206static void freduce_degree(limb *output) {
207 /* Each of these shifts and adds ends up multiplying the value by 19.
208 *
209 * For output[0..8], the absolute entry value is < 14*2^54 and we add, at
210 * most, 19*14*2^54 thus, on exit, |output[0..8]| < 280*2^54. */
211 output[8] += output[18] << 4;
212 output[8] += output[18] << 1;
213 output[8] += output[18];
214 output[7] += output[17] << 4;
215 output[7] += output[17] << 1;
216 output[7] += output[17];
217 output[6] += output[16] << 4;
218 output[6] += output[16] << 1;
219 output[6] += output[16];
220 output[5] += output[15] << 4;
221 output[5] += output[15] << 1;
222 output[5] += output[15];
223 output[4] += output[14] << 4;
224 output[4] += output[14] << 1;
225 output[4] += output[14];
226 output[3] += output[13] << 4;
227 output[3] += output[13] << 1;
228 output[3] += output[13];
229 output[2] += output[12] << 4;
230 output[2] += output[12] << 1;
231 output[2] += output[12];
232 output[1] += output[11] << 4;
233 output[1] += output[11] << 1;
234 output[1] += output[11];
235 output[0] += output[10] << 4;
236 output[0] += output[10] << 1;
237 output[0] += output[10];
238}
239
240#if (-1 & 3) != 3
241#error "This code only works on a two's complement system"
242#endif
243
244/* return v / 2^26, using only shifts and adds.
245 *
246 * On entry: v can take any value. */
247static inline limb
248div_by_2_26(const limb v)
249{
250 /* High word of v; no shift needed. */
251 const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
252 /* Set to all 1s if v was negative; else set to 0s. */
253 const int32_t sign = ((int32_t) highword) >> 31;
254 /* Set to 0x3ffffff if v was negative; else set to 0. */
255 const int32_t roundoff = ((uint32_t) sign) >> 6;
256 /* Should return v / (1<<26) */
257 return (v + roundoff) >> 26;
258}
259
260/* return v / (2^25), using only shifts and adds.
261 *
262 * On entry: v can take any value. */
263static inline limb
264div_by_2_25(const limb v)
265{
266 /* High word of v; no shift needed*/
267 const uint32_t highword = (uint32_t) (((uint64_t) v) >> 32);
268 /* Set to all 1s if v was negative; else set to 0s. */
269 const int32_t sign = ((int32_t) highword) >> 31;
270 /* Set to 0x1ffffff if v was negative; else set to 0. */
271 const int32_t roundoff = ((uint32_t) sign) >> 7;
272 /* Should return v / (1<<25) */
273 return (v + roundoff) >> 25;
274}
275
276/* Reduce all coefficients of the short form input so that |x| < 2^26.
277 *
278 * On entry: |output[i]| < 280*2^54 */
279static void freduce_coefficients(limb *output) {
280 unsigned i;
281
282 output[10] = 0;
283
284 for (i = 0; i < 10; i += 2) {
285 limb over = div_by_2_26(v: output[i]);
286 /* The entry condition (that |output[i]| < 280*2^54) means that over is, at
287 * most, 280*2^28 in the first iteration of this loop. This is added to the
288 * next limb and we can approximate the resulting bound of that limb by
289 * 281*2^54. */
290 output[i] -= over << 26;
291 output[i+1] += over;
292
293 /* For the first iteration, |output[i+1]| < 281*2^54, thus |over| <
294 * 281*2^29. When this is added to the next limb, the resulting bound can
295 * be approximated as 281*2^54.
296 *
297 * For subsequent iterations of the loop, 281*2^54 remains a conservative
298 * bound and no overflow occurs. */
299 over = div_by_2_25(v: output[i+1]);
300 output[i+1] -= over << 25;
301 output[i+2] += over;
302 }
303 /* Now |output[10]| < 281*2^29 and all other coefficients are reduced. */
304 output[0] += output[10] << 4;
305 output[0] += output[10] << 1;
306 output[0] += output[10];
307
308 output[10] = 0;
309
310 /* Now output[1..9] are reduced, and |output[0]| < 2^26 + 19*281*2^29
311 * So |over| will be no more than 2^16. */
312 {
313 limb over = div_by_2_26(v: output[0]);
314 output[0] -= over << 26;
315 output[1] += over;
316 }
317
318 /* Now output[0,2..9] are reduced, and |output[1]| < 2^25 + 2^16 < 2^26. The
319 * bound on |output[1]| is sufficient to meet our needs. */
320}
321
322/* A helpful wrapper around fproduct: output = in * in2.
323 *
324 * On entry: |in[i]| < 2^27 and |in2[i]| < 2^27.
325 *
326 * output must be distinct to both inputs. The output is reduced degree
327 * (indeed, one need only provide storage for 10 limbs) and |output[i]| < 2^26. */
328static void
329fmul(limb *output, const limb *in, const limb *in2) {
330 limb t[19];
331 fproduct(output: t, in2: in, in: in2);
332 /* |t[i]| < 14*2^54 */
333 freduce_degree(output: t);
334 freduce_coefficients(output: t);
335 /* |t[i]| < 2^26 */
336 memcpy(dest: output, src: t, n: sizeof(limb) * 10);
337}
338
339/* Square a number: output = in**2
340 *
341 * output must be distinct from the input. The inputs are reduced coefficient
342 * form, the output is not.
343 *
344 * output[x] <= 14 * the largest product of the input limbs. */
345static void fsquare_inner(limb *output, const limb *in) {
346 output[0] = ((limb) ((s32) in[0])) * ((s32) in[0]);
347 output[1] = 2 * ((limb) ((s32) in[0])) * ((s32) in[1]);
348 output[2] = 2 * (((limb) ((s32) in[1])) * ((s32) in[1]) +
349 ((limb) ((s32) in[0])) * ((s32) in[2]));
350 output[3] = 2 * (((limb) ((s32) in[1])) * ((s32) in[2]) +
351 ((limb) ((s32) in[0])) * ((s32) in[3]));
352 output[4] = ((limb) ((s32) in[2])) * ((s32) in[2]) +
353 4 * ((limb) ((s32) in[1])) * ((s32) in[3]) +
354 2 * ((limb) ((s32) in[0])) * ((s32) in[4]);
355 output[5] = 2 * (((limb) ((s32) in[2])) * ((s32) in[3]) +
356 ((limb) ((s32) in[1])) * ((s32) in[4]) +
357 ((limb) ((s32) in[0])) * ((s32) in[5]));
358 output[6] = 2 * (((limb) ((s32) in[3])) * ((s32) in[3]) +
359 ((limb) ((s32) in[2])) * ((s32) in[4]) +
360 ((limb) ((s32) in[0])) * ((s32) in[6]) +
361 2 * ((limb) ((s32) in[1])) * ((s32) in[5]));
362 output[7] = 2 * (((limb) ((s32) in[3])) * ((s32) in[4]) +
363 ((limb) ((s32) in[2])) * ((s32) in[5]) +
364 ((limb) ((s32) in[1])) * ((s32) in[6]) +
365 ((limb) ((s32) in[0])) * ((s32) in[7]));
366 output[8] = ((limb) ((s32) in[4])) * ((s32) in[4]) +
367 2 * (((limb) ((s32) in[2])) * ((s32) in[6]) +
368 ((limb) ((s32) in[0])) * ((s32) in[8]) +
369 2 * (((limb) ((s32) in[1])) * ((s32) in[7]) +
370 ((limb) ((s32) in[3])) * ((s32) in[5])));
371 output[9] = 2 * (((limb) ((s32) in[4])) * ((s32) in[5]) +
372 ((limb) ((s32) in[3])) * ((s32) in[6]) +
373 ((limb) ((s32) in[2])) * ((s32) in[7]) +
374 ((limb) ((s32) in[1])) * ((s32) in[8]) +
375 ((limb) ((s32) in[0])) * ((s32) in[9]));
376 output[10] = 2 * (((limb) ((s32) in[5])) * ((s32) in[5]) +
377 ((limb) ((s32) in[4])) * ((s32) in[6]) +
378 ((limb) ((s32) in[2])) * ((s32) in[8]) +
379 2 * (((limb) ((s32) in[3])) * ((s32) in[7]) +
380 ((limb) ((s32) in[1])) * ((s32) in[9])));
381 output[11] = 2 * (((limb) ((s32) in[5])) * ((s32) in[6]) +
382 ((limb) ((s32) in[4])) * ((s32) in[7]) +
383 ((limb) ((s32) in[3])) * ((s32) in[8]) +
384 ((limb) ((s32) in[2])) * ((s32) in[9]));
385 output[12] = ((limb) ((s32) in[6])) * ((s32) in[6]) +
386 2 * (((limb) ((s32) in[4])) * ((s32) in[8]) +
387 2 * (((limb) ((s32) in[5])) * ((s32) in[7]) +
388 ((limb) ((s32) in[3])) * ((s32) in[9])));
389 output[13] = 2 * (((limb) ((s32) in[6])) * ((s32) in[7]) +
390 ((limb) ((s32) in[5])) * ((s32) in[8]) +
391 ((limb) ((s32) in[4])) * ((s32) in[9]));
392 output[14] = 2 * (((limb) ((s32) in[7])) * ((s32) in[7]) +
393 ((limb) ((s32) in[6])) * ((s32) in[8]) +
394 2 * ((limb) ((s32) in[5])) * ((s32) in[9]));
395 output[15] = 2 * (((limb) ((s32) in[7])) * ((s32) in[8]) +
396 ((limb) ((s32) in[6])) * ((s32) in[9]));
397 output[16] = ((limb) ((s32) in[8])) * ((s32) in[8]) +
398 4 * ((limb) ((s32) in[7])) * ((s32) in[9]);
399 output[17] = 2 * ((limb) ((s32) in[8])) * ((s32) in[9]);
400 output[18] = 2 * ((limb) ((s32) in[9])) * ((s32) in[9]);
401}
402
403/* fsquare sets output = in^2.
404 *
405 * On entry: The |in| argument is in reduced coefficients form and |in[i]| <
406 * 2^27.
407 *
408 * On exit: The |output| argument is in reduced coefficients form (indeed, one
409 * need only provide storage for 10 limbs) and |out[i]| < 2^26. */
410static void
411fsquare(limb *output, const limb *in) {
412 limb t[19];
413 fsquare_inner(output: t, in);
414 /* |t[i]| < 14*2^54 because the largest product of two limbs will be <
415 * 2^(27+27) and fsquare_inner adds together, at most, 14 of those
416 * products. */
417 freduce_degree(output: t);
418 freduce_coefficients(output: t);
419 /* |t[i]| < 2^26 */
420 memcpy(dest: output, src: t, n: sizeof(limb) * 10);
421}
422
423/* Take a little-endian, 32-byte number and expand it into polynomial form */
424static void
425fexpand(limb *output, const u8 *input) {
426#define F(n,start,shift,mask) \
427 output[n] = ((((limb) input[start + 0]) | \
428 ((limb) input[start + 1]) << 8 | \
429 ((limb) input[start + 2]) << 16 | \
430 ((limb) input[start + 3]) << 24) >> shift) & mask;
431 F(0, 0, 0, 0x3ffffff);
432 F(1, 3, 2, 0x1ffffff);
433 F(2, 6, 3, 0x3ffffff);
434 F(3, 9, 5, 0x1ffffff);
435 F(4, 12, 6, 0x3ffffff);
436 F(5, 16, 0, 0x1ffffff);
437 F(6, 19, 1, 0x3ffffff);
438 F(7, 22, 3, 0x1ffffff);
439 F(8, 25, 4, 0x3ffffff);
440 F(9, 28, 6, 0x1ffffff);
441#undef F
442}
443
444#if (-32 >> 1) != -16
445#error "This code only works when >> does sign-extension on negative numbers"
446#endif
447
448/* s32_eq returns 0xffffffff iff a == b and zero otherwise. */
449static s32 s32_eq(s32 a, s32 b) {
450 a = ~(a ^ b);
451 a &= a << 16;
452 a &= a << 8;
453 a &= a << 4;
454 a &= a << 2;
455 a &= a << 1;
456 return a >> 31;
457}
458
459/* s32_gte returns 0xffffffff if a >= b and zero otherwise, where a and b are
460 * both non-negative. */
461static s32 s32_gte(s32 a, s32 b) {
462 a -= b;
463 /* a >= 0 iff a >= b. */
464 return ~(a >> 31);
465}
466
467/* Take a fully reduced polynomial form number and contract it into a
468 * little-endian, 32-byte array.
469 *
470 * On entry: |input_limbs[i]| < 2^26 */
471static void
472fcontract(u8 *output, limb *input_limbs) {
473 int i;
474 int j;
475 s32 input[10];
476 s32 mask;
477
478 /* |input_limbs[i]| < 2^26, so it's valid to convert to an s32. */
479 for (i = 0; i < 10; i++) {
480 input[i] = input_limbs[i];
481 }
482
483 for (j = 0; j < 2; ++j) {
484 for (i = 0; i < 9; ++i) {
485 if ((i & 1) == 1) {
486 /* This calculation is a time-invariant way to make input[i]
487 * non-negative by borrowing from the next-larger limb. */
488 const s32 mask = input[i] >> 31;
489 const s32 carry = -((input[i] & mask) >> 25);
490 input[i] = input[i] + (carry << 25);
491 input[i+1] = input[i+1] - carry;
492 } else {
493 const s32 mask = input[i] >> 31;
494 const s32 carry = -((input[i] & mask) >> 26);
495 input[i] = input[i] + (carry << 26);
496 input[i+1] = input[i+1] - carry;
497 }
498 }
499
500 /* There's no greater limb for input[9] to borrow from, but we can multiply
501 * by 19 and borrow from input[0], which is valid mod 2^255-19. */
502 {
503 const s32 mask = input[9] >> 31;
504 const s32 carry = -((input[9] & mask) >> 25);
505 input[9] = input[9] + (carry << 25);
506 input[0] = input[0] - (carry * 19);
507 }
508
509 /* After the first iteration, input[1..9] are non-negative and fit within
510 * 25 or 26 bits, depending on position. However, input[0] may be
511 * negative. */
512 }
513
514 /* The first borrow-propagation pass above ended with every limb
515 except (possibly) input[0] non-negative.
516
517 If input[0] was negative after the first pass, then it was because of a
518 carry from input[9]. On entry, input[9] < 2^26 so the carry was, at most,
519 one, since (2**26-1) >> 25 = 1. Thus input[0] >= -19.
520
521 In the second pass, each limb is decreased by at most one. Thus the second
522 borrow-propagation pass could only have wrapped around to decrease
523 input[0] again if the first pass left input[0] negative *and* input[1]
524 through input[9] were all zero. In that case, input[1] is now 2^25 - 1,
525 and this last borrow-propagation step will leave input[1] non-negative. */
526 {
527 const s32 mask = input[0] >> 31;
528 const s32 carry = -((input[0] & mask) >> 26);
529 input[0] = input[0] + (carry << 26);
530 input[1] = input[1] - carry;
531 }
532
533 /* All input[i] are now non-negative. However, there might be values between
534 * 2^25 and 2^26 in a limb which is, nominally, 25 bits wide. */
535 for (j = 0; j < 2; j++) {
536 for (i = 0; i < 9; i++) {
537 if ((i & 1) == 1) {
538 const s32 carry = input[i] >> 25;
539 input[i] &= 0x1ffffff;
540 input[i+1] += carry;
541 } else {
542 const s32 carry = input[i] >> 26;
543 input[i] &= 0x3ffffff;
544 input[i+1] += carry;
545 }
546 }
547
548 {
549 const s32 carry = input[9] >> 25;
550 input[9] &= 0x1ffffff;
551 input[0] += 19*carry;
552 }
553 }
554
555 /* If the first carry-chain pass, just above, ended up with a carry from
556 * input[9], and that caused input[0] to be out-of-bounds, then input[0] was
557 * < 2^26 + 2*19, because the carry was, at most, two.
558 *
559 * If the second pass carried from input[9] again then input[0] is < 2*19 and
560 * the input[9] -> input[0] carry didn't push input[0] out of bounds. */
561
562 /* It still remains the case that input might be between 2^255-19 and 2^255.
563 * In this case, input[1..9] must take their maximum value and input[0] must
564 * be >= (2^255-19) & 0x3ffffff, which is 0x3ffffed. */
565 mask = s32_gte(a: input[0], b: 0x3ffffed);
566 for (i = 1; i < 10; i++) {
567 if ((i & 1) == 1) {
568 mask &= s32_eq(a: input[i], b: 0x1ffffff);
569 } else {
570 mask &= s32_eq(a: input[i], b: 0x3ffffff);
571 }
572 }
573
574 /* mask is either 0xffffffff (if input >= 2^255-19) and zero otherwise. Thus
575 * this conditionally subtracts 2^255-19. */
576 input[0] -= mask & 0x3ffffed;
577
578 for (i = 1; i < 10; i++) {
579 if ((i & 1) == 1) {
580 input[i] -= mask & 0x1ffffff;
581 } else {
582 input[i] -= mask & 0x3ffffff;
583 }
584 }
585
586 input[1] <<= 2;
587 input[2] <<= 3;
588 input[3] <<= 5;
589 input[4] <<= 6;
590 input[6] <<= 1;
591 input[7] <<= 3;
592 input[8] <<= 4;
593 input[9] <<= 6;
594#define F(i, s) \
595 output[s+0] |= input[i] & 0xff; \
596 output[s+1] = (input[i] >> 8) & 0xff; \
597 output[s+2] = (input[i] >> 16) & 0xff; \
598 output[s+3] = (input[i] >> 24) & 0xff;
599 output[0] = 0;
600 output[16] = 0;
601 F(0,0);
602 F(1,3);
603 F(2,6);
604 F(3,9);
605 F(4,12);
606 F(5,16);
607 F(6,19);
608 F(7,22);
609 F(8,25);
610 F(9,28);
611#undef F
612}
613
614/* Input: Q, Q', Q-Q'
615 * Output: 2Q, Q+Q'
616 *
617 * x2 z3: long form
618 * x3 z3: long form
619 * x z: short form, destroyed
620 * xprime zprime: short form, destroyed
621 * qmqp: short form, preserved
622 *
623 * On entry and exit, the absolute value of the limbs of all inputs and outputs
624 * are < 2^26. */
625static void fmonty(limb *x2, limb *z2, /* output 2Q */
626 limb *x3, limb *z3, /* output Q + Q' */
627 limb *x, limb *z, /* input Q */
628 limb *xprime, limb *zprime, /* input Q' */
629 const limb *qmqp /* input Q - Q' */) {
630 limb origx[10], origxprime[10], zzz[19], xx[19], zz[19], xxprime[19],
631 zzprime[19], zzzprime[19], xxxprime[19];
632
633 memcpy(dest: origx, src: x, n: 10 * sizeof(limb));
634 fsum(output: x, in: z);
635 /* |x[i]| < 2^27 */
636 fdifference(output: z, in: origx); /* does x - z */
637 /* |z[i]| < 2^27 */
638
639 memcpy(dest: origxprime, src: xprime, n: sizeof(limb) * 10);
640 fsum(output: xprime, in: zprime);
641 /* |xprime[i]| < 2^27 */
642 fdifference(output: zprime, in: origxprime);
643 /* |zprime[i]| < 2^27 */
644 fproduct(output: xxprime, in2: xprime, in: z);
645 /* |xxprime[i]| < 14*2^54: the largest product of two limbs will be <
646 * 2^(27+27) and fproduct adds together, at most, 14 of those products.
647 * (Approximating that to 2^58 doesn't work out.) */
648 fproduct(output: zzprime, in2: x, in: zprime);
649 /* |zzprime[i]| < 14*2^54 */
650 freduce_degree(output: xxprime);
651 freduce_coefficients(output: xxprime);
652 /* |xxprime[i]| < 2^26 */
653 freduce_degree(output: zzprime);
654 freduce_coefficients(output: zzprime);
655 /* |zzprime[i]| < 2^26 */
656 memcpy(dest: origxprime, src: xxprime, n: sizeof(limb) * 10);
657 fsum(output: xxprime, in: zzprime);
658 /* |xxprime[i]| < 2^27 */
659 fdifference(output: zzprime, in: origxprime);
660 /* |zzprime[i]| < 2^27 */
661 fsquare(output: xxxprime, in: xxprime);
662 /* |xxxprime[i]| < 2^26 */
663 fsquare(output: zzzprime, in: zzprime);
664 /* |zzzprime[i]| < 2^26 */
665 fproduct(output: zzprime, in2: zzzprime, in: qmqp);
666 /* |zzprime[i]| < 14*2^52 */
667 freduce_degree(output: zzprime);
668 freduce_coefficients(output: zzprime);
669 /* |zzprime[i]| < 2^26 */
670 memcpy(dest: x3, src: xxxprime, n: sizeof(limb) * 10);
671 memcpy(dest: z3, src: zzprime, n: sizeof(limb) * 10);
672
673 fsquare(output: xx, in: x);
674 /* |xx[i]| < 2^26 */
675 fsquare(output: zz, in: z);
676 /* |zz[i]| < 2^26 */
677 fproduct(output: x2, in2: xx, in: zz);
678 /* |x2[i]| < 14*2^52 */
679 freduce_degree(output: x2);
680 freduce_coefficients(output: x2);
681 /* |x2[i]| < 2^26 */
682 fdifference(output: zz, in: xx); // does zz = xx - zz
683 /* |zz[i]| < 2^27 */
684 memset(s: zzz + 10, c: 0, n: sizeof(limb) * 9);
685 fscalar_product(output: zzz, in: zz, scalar: 121665);
686 /* |zzz[i]| < 2^(27+17) */
687 /* No need to call freduce_degree here:
688 fscalar_product doesn't increase the degree of its input. */
689 freduce_coefficients(output: zzz);
690 /* |zzz[i]| < 2^26 */
691 fsum(output: zzz, in: xx);
692 /* |zzz[i]| < 2^27 */
693 fproduct(output: z2, in2: zz, in: zzz);
694 /* |z2[i]| < 14*2^(26+27) */
695 freduce_degree(output: z2);
696 freduce_coefficients(output: z2);
697 /* |z2|i| < 2^26 */
698}
699
700/* Conditionally swap two reduced-form limb arrays if 'iswap' is 1, but leave
701 * them unchanged if 'iswap' is 0. Runs in data-invariant time to avoid
702 * side-channel attacks.
703 *
704 * NOTE that this function requires that 'iswap' be 1 or 0; other values give
705 * wrong results. Also, the two limb arrays must be in reduced-coefficient,
706 * reduced-degree form: the values in a[10..19] or b[10..19] aren't swapped,
707 * and all all values in a[0..9],b[0..9] must have magnitude less than
708 * INT32_MAX. */
709static void
710swap_conditional(limb a[19], limb b[19], limb iswap) {
711 unsigned i;
712 const s32 swap = (s32) -iswap;
713
714 for (i = 0; i < 10; ++i) {
715 const s32 x = swap & ( ((s32)a[i]) ^ ((s32)b[i]) );
716 a[i] = ((s32)a[i]) ^ x;
717 b[i] = ((s32)b[i]) ^ x;
718 }
719}
720
721/* Calculates nQ where Q is the x-coordinate of a point on the curve
722 *
723 * resultx/resultz: the x coordinate of the resulting curve point (short form)
724 * n: a little endian, 32-byte number
725 * q: a point of the curve (short form) */
726static void
727cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) {
728 limb a[19] = {0}, b[19] = {1}, c[19] = {1}, d[19] = {0};
729 limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;
730 limb e[19] = {0}, f[19] = {1}, g[19] = {0}, h[19] = {1};
731 limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;
732
733 unsigned i, j;
734
735 memcpy(dest: nqpqx, src: q, n: sizeof(limb) * 10);
736
737 for (i = 0; i < 32; ++i) {
738 u8 byte = n[31 - i];
739 for (j = 0; j < 8; ++j) {
740 const limb bit = byte >> 7;
741
742 swap_conditional(a: nqx, b: nqpqx, iswap: bit);
743 swap_conditional(a: nqz, b: nqpqz, iswap: bit);
744 fmonty(x2: nqx2, z2: nqz2,
745 x3: nqpqx2, z3: nqpqz2,
746 x: nqx, z: nqz,
747 xprime: nqpqx, zprime: nqpqz,
748 qmqp: q);
749 swap_conditional(a: nqx2, b: nqpqx2, iswap: bit);
750 swap_conditional(a: nqz2, b: nqpqz2, iswap: bit);
751
752 t = nqx;
753 nqx = nqx2;
754 nqx2 = t;
755 t = nqz;
756 nqz = nqz2;
757 nqz2 = t;
758 t = nqpqx;
759 nqpqx = nqpqx2;
760 nqpqx2 = t;
761 t = nqpqz;
762 nqpqz = nqpqz2;
763 nqpqz2 = t;
764
765 byte <<= 1;
766 }
767 }
768
769 memcpy(dest: resultx, src: nqx, n: sizeof(limb) * 10);
770 memcpy(dest: resultz, src: nqz, n: sizeof(limb) * 10);
771}
772
773// -----------------------------------------------------------------------------
774// Shamelessly copied from djb's code
775// -----------------------------------------------------------------------------
776static void
777crecip(limb *out, const limb *z) {
778 limb z2[10];
779 limb z9[10];
780 limb z11[10];
781 limb z2_5_0[10];
782 limb z2_10_0[10];
783 limb z2_20_0[10];
784 limb z2_50_0[10];
785 limb z2_100_0[10];
786 limb t0[10];
787 limb t1[10];
788 int i;
789
790 /* 2 */ fsquare(output: z2,in: z);
791 /* 4 */ fsquare(output: t1,in: z2);
792 /* 8 */ fsquare(output: t0,in: t1);
793 /* 9 */ fmul(output: z9,in: t0,in2: z);
794 /* 11 */ fmul(output: z11,in: z9,in2: z2);
795 /* 22 */ fsquare(output: t0,in: z11);
796 /* 2^5 - 2^0 = 31 */ fmul(output: z2_5_0,in: t0,in2: z9);
797
798 /* 2^6 - 2^1 */ fsquare(output: t0,in: z2_5_0);
799 /* 2^7 - 2^2 */ fsquare(output: t1,in: t0);
800 /* 2^8 - 2^3 */ fsquare(output: t0,in: t1);
801 /* 2^9 - 2^4 */ fsquare(output: t1,in: t0);
802 /* 2^10 - 2^5 */ fsquare(output: t0,in: t1);
803 /* 2^10 - 2^0 */ fmul(output: z2_10_0,in: t0,in2: z2_5_0);
804
805 /* 2^11 - 2^1 */ fsquare(output: t0,in: z2_10_0);
806 /* 2^12 - 2^2 */ fsquare(output: t1,in: t0);
807 /* 2^20 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(output: t0,in: t1); fsquare(output: t1,in: t0); }
808 /* 2^20 - 2^0 */ fmul(output: z2_20_0,in: t1,in2: z2_10_0);
809
810 /* 2^21 - 2^1 */ fsquare(output: t0,in: z2_20_0);
811 /* 2^22 - 2^2 */ fsquare(output: t1,in: t0);
812 /* 2^40 - 2^20 */ for (i = 2;i < 20;i += 2) { fsquare(output: t0,in: t1); fsquare(output: t1,in: t0); }
813 /* 2^40 - 2^0 */ fmul(output: t0,in: t1,in2: z2_20_0);
814
815 /* 2^41 - 2^1 */ fsquare(output: t1,in: t0);
816 /* 2^42 - 2^2 */ fsquare(output: t0,in: t1);
817 /* 2^50 - 2^10 */ for (i = 2;i < 10;i += 2) { fsquare(output: t1,in: t0); fsquare(output: t0,in: t1); }
818 /* 2^50 - 2^0 */ fmul(output: z2_50_0,in: t0,in2: z2_10_0);
819
820 /* 2^51 - 2^1 */ fsquare(output: t0,in: z2_50_0);
821 /* 2^52 - 2^2 */ fsquare(output: t1,in: t0);
822 /* 2^100 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(output: t0,in: t1); fsquare(output: t1,in: t0); }
823 /* 2^100 - 2^0 */ fmul(output: z2_100_0,in: t1,in2: z2_50_0);
824
825 /* 2^101 - 2^1 */ fsquare(output: t1,in: z2_100_0);
826 /* 2^102 - 2^2 */ fsquare(output: t0,in: t1);
827 /* 2^200 - 2^100 */ for (i = 2;i < 100;i += 2) { fsquare(output: t1,in: t0); fsquare(output: t0,in: t1); }
828 /* 2^200 - 2^0 */ fmul(output: t1,in: t0,in2: z2_100_0);
829
830 /* 2^201 - 2^1 */ fsquare(output: t0,in: t1);
831 /* 2^202 - 2^2 */ fsquare(output: t1,in: t0);
832 /* 2^250 - 2^50 */ for (i = 2;i < 50;i += 2) { fsquare(output: t0,in: t1); fsquare(output: t1,in: t0); }
833 /* 2^250 - 2^0 */ fmul(output: t0,in: t1,in2: z2_50_0);
834
835 /* 2^251 - 2^1 */ fsquare(output: t1,in: t0);
836 /* 2^252 - 2^2 */ fsquare(output: t0,in: t1);
837 /* 2^253 - 2^3 */ fsquare(output: t1,in: t0);
838 /* 2^254 - 2^4 */ fsquare(output: t0,in: t1);
839 /* 2^255 - 2^5 */ fsquare(output: t1,in: t0);
840 /* 2^255 - 21 */ fmul(output: out,in: t1,in2: z11);
841}
842
843int
844curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) {
845 limb bp[10], x[10], z[11], zmone[10];
846 uint8_t e[32];
847 int i;
848
849 for (i = 0; i < 32; ++i) e[i] = secret[i];
850 e[0] &= 248;
851 e[31] &= 127;
852 e[31] |= 64;
853
854 fexpand(output: bp, input: basepoint);
855 cmult(resultx: x, resultz: z, n: e, q: bp);
856 crecip(out: zmone, z);
857 fmul(output: z, in: x, in2: zmone);
858 fcontract(output: mypublic, input_limbs: z);
859 return 0;
860}
861