1 /* Copyright 2008, Google Inc.
2 * All rights reserved.
3 *
4 * Code released into the public domain.
5 *
6 * curve25519-donna: Curve25519 elliptic curve, public key function
7 *
8 * http://code.google.com/p/curve25519-donna/
9 *
10 * Adam Langley <agl@imperialviolet.org>
11 * Parts optimised by floodyberry
12 * Derived from public domain C code by Daniel J. Bernstein <djb@cr.yp.to>
13 *
14 * More information about curve25519 can be found here
15 * http://cr.yp.to/ecdh.html
16 *
17 * djb's sample implementation of curve25519 is written in a special assembly
18 * language called qhasm and uses the floating point registers.
19 *
20 * This is, almost, a clean room reimplementation from the curve25519 paper. It
21 * uses many of the tricks described therein. Only the crecip function is taken
22 * from the sample implementation.
23 */
24
25 #include <string.h>
26 #include <stdint.h>
27 #include "crypto_scalarmult.h"
28
29 typedef uint8_t u8;
30 typedef uint64_t limb;
31 typedef limb felem[5];
32 // This is a special gcc mode for 128-bit integers. It's implemented on 64-bit
33 // platforms only as far as I know.
34 typedef unsigned uint128_t __attribute__((mode(TI)));
35
36 #undef force_inline
37 #define force_inline inline __attribute__((always_inline))
38
39 /* Sum two numbers: output += in */
40 static void force_inline
fsum(limb * output,const limb * in)41 fsum(limb *output, const limb *in) {
42 output[0] += in[0];
43 output[1] += in[1];
44 output[2] += in[2];
45 output[3] += in[3];
46 output[4] += in[4];
47 }
48
49 /* Find the difference of two numbers: output = in - output
50 * (note the order of the arguments!)
51 *
52 * Assumes that out[i] < 2**52
53 * On return, out[i] < 2**55
54 */
55 static void force_inline
fdifference_backwards(felem out,const felem in)56 fdifference_backwards(felem out, const felem in) {
57 /* 152 is 19 << 3 */
58 static const limb two54m152 = (((limb)1) << 54) - 152;
59 static const limb two54m8 = (((limb)1) << 54) - 8;
60
61 out[0] = in[0] + two54m152 - out[0];
62 out[1] = in[1] + two54m8 - out[1];
63 out[2] = in[2] + two54m8 - out[2];
64 out[3] = in[3] + two54m8 - out[3];
65 out[4] = in[4] + two54m8 - out[4];
66 }
67
68 /* Multiply a number by a scalar: output = in * scalar */
69 static void force_inline
fscalar_product(felem output,const felem in,const limb scalar)70 fscalar_product(felem output, const felem in, const limb scalar) {
71 uint128_t a;
72
73 a = ((uint128_t) in[0]) * scalar;
74 output[0] = ((limb)a) & 0x7ffffffffffff;
75
76 a = ((uint128_t) in[1]) * scalar + ((limb) (a >> 51));
77 output[1] = ((limb)a) & 0x7ffffffffffff;
78
79 a = ((uint128_t) in[2]) * scalar + ((limb) (a >> 51));
80 output[2] = ((limb)a) & 0x7ffffffffffff;
81
82 a = ((uint128_t) in[3]) * scalar + ((limb) (a >> 51));
83 output[3] = ((limb)a) & 0x7ffffffffffff;
84
85 a = ((uint128_t) in[4]) * scalar + ((limb) (a >> 51));
86 output[4] = ((limb)a) & 0x7ffffffffffff;
87
88 output[0] += (a >> 51) * 19;
89 }
90
91 /* Multiply two numbers: output = in2 * in
92 *
93 * output must be distinct to both inputs. The inputs are reduced coefficient
94 * form, the output is not.
95 *
96 * Assumes that in[i] < 2**55 and likewise for in2.
97 * On return, output[i] < 2**52
98 */
99 static void force_inline
fmul(felem output,const felem in2,const felem in)100 fmul(felem output, const felem in2, const felem in) {
101 uint128_t t[5];
102 limb r0,r1,r2,r3,r4,s0,s1,s2,s3,s4,c;
103
104 r0 = in[0];
105 r1 = in[1];
106 r2 = in[2];
107 r3 = in[3];
108 r4 = in[4];
109
110 s0 = in2[0];
111 s1 = in2[1];
112 s2 = in2[2];
113 s3 = in2[3];
114 s4 = in2[4];
115
116 t[0] = ((uint128_t) r0) * s0;
117 t[1] = ((uint128_t) r0) * s1 + ((uint128_t) r1) * s0;
118 t[2] = ((uint128_t) r0) * s2 + ((uint128_t) r2) * s0 + ((uint128_t) r1) * s1;
119 t[3] = ((uint128_t) r0) * s3 + ((uint128_t) r3) * s0 + ((uint128_t) r1) * s2 + ((uint128_t) r2) * s1;
120 t[4] = ((uint128_t) r0) * s4 + ((uint128_t) r4) * s0 + ((uint128_t) r3) * s1 + ((uint128_t) r1) * s3 + ((uint128_t) r2) * s2;
121
122 r4 *= 19;
123 r1 *= 19;
124 r2 *= 19;
125 r3 *= 19;
126
127 t[0] += ((uint128_t) r4) * s1 + ((uint128_t) r1) * s4 + ((uint128_t) r2) * s3 + ((uint128_t) r3) * s2;
128 t[1] += ((uint128_t) r4) * s2 + ((uint128_t) r2) * s4 + ((uint128_t) r3) * s3;
129 t[2] += ((uint128_t) r4) * s3 + ((uint128_t) r3) * s4;
130 t[3] += ((uint128_t) r4) * s4;
131
132 r0 = (limb)t[0] & 0x7ffffffffffff; c = (limb)(t[0] >> 51);
133 t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffff; c = (limb)(t[1] >> 51);
134 t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffff; c = (limb)(t[2] >> 51);
135 t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffff; c = (limb)(t[3] >> 51);
136 t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffff; c = (limb)(t[4] >> 51);
137 r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff;
138 r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff;
139 r2 += c;
140
141 output[0] = r0;
142 output[1] = r1;
143 output[2] = r2;
144 output[3] = r3;
145 output[4] = r4;
146 }
147
148 static void force_inline
fsquare_times(felem output,const felem in,limb count)149 fsquare_times(felem output, const felem in, limb count) {
150 uint128_t t[5];
151 limb r0,r1,r2,r3,r4,c;
152 limb d0,d1,d2,d4,d419;
153
154 r0 = in[0];
155 r1 = in[1];
156 r2 = in[2];
157 r3 = in[3];
158 r4 = in[4];
159
160 do {
161 d0 = r0 * 2;
162 d1 = r1 * 2;
163 d2 = r2 * 2 * 19;
164 d419 = r4 * 19;
165 d4 = d419 * 2;
166
167 t[0] = ((uint128_t) r0) * r0 + ((uint128_t) d4) * r1 + (((uint128_t) d2) * (r3 ));
168 t[1] = ((uint128_t) d0) * r1 + ((uint128_t) d4) * r2 + (((uint128_t) r3) * (r3 * 19));
169 t[2] = ((uint128_t) d0) * r2 + ((uint128_t) r1) * r1 + (((uint128_t) d4) * (r3 ));
170 t[3] = ((uint128_t) d0) * r3 + ((uint128_t) d1) * r2 + (((uint128_t) r4) * (d419 ));
171 t[4] = ((uint128_t) d0) * r4 + ((uint128_t) d1) * r3 + (((uint128_t) r2) * (r2 ));
172
173 r0 = (limb)t[0] & 0x7ffffffffffff; c = (limb)(t[0] >> 51);
174 t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffff; c = (limb)(t[1] >> 51);
175 t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffff; c = (limb)(t[2] >> 51);
176 t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffff; c = (limb)(t[3] >> 51);
177 t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffff; c = (limb)(t[4] >> 51);
178 r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff;
179 r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff;
180 r2 += c;
181 } while(--count);
182
183 output[0] = r0;
184 output[1] = r1;
185 output[2] = r2;
186 output[3] = r3;
187 output[4] = r4;
188 }
189
190 /* Take a little-endian, 32-byte number and expand it into polynomial form */
191 static void
fexpand(limb * output,const u8 * in)192 fexpand(limb *output, const u8 *in) {
193 output[0] = *((const uint64_t *)(in)) & 0x7ffffffffffff;
194 output[1] = (*((const uint64_t *)(in+6)) >> 3) & 0x7ffffffffffff;
195 output[2] = (*((const uint64_t *)(in+12)) >> 6) & 0x7ffffffffffff;
196 output[3] = (*((const uint64_t *)(in+19)) >> 1) & 0x7ffffffffffff;
197 output[4] = (*((const uint64_t *)(in+25)) >> 4) & 0x7ffffffffffff;
198 }
199
200 /* Take a fully reduced polynomial form number and contract it into a
201 * little-endian, 32-byte array
202 */
203 static void
fcontract(u8 * output,const felem input)204 fcontract(u8 *output, const felem input) {
205 uint128_t t[5];
206
207 t[0] = input[0];
208 t[1] = input[1];
209 t[2] = input[2];
210 t[3] = input[3];
211 t[4] = input[4];
212
213 t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff;
214 t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff;
215 t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff;
216 t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff;
217 t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff;
218
219 t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff;
220 t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff;
221 t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff;
222 t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff;
223 t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff;
224
225 /* now t is between 0 and 2^255-1, properly carried. */
226 /* case 1: between 0 and 2^255-20. case 2: between 2^255-19 and 2^255-1. */
227
228 t[0] += 19;
229
230 t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff;
231 t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff;
232 t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff;
233 t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff;
234 t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff;
235
236 /* now between 19 and 2^255-1 in both cases, and offset by 19. */
237
238 t[0] += 0x8000000000000 - 19;
239 t[1] += 0x8000000000000 - 1;
240 t[2] += 0x8000000000000 - 1;
241 t[3] += 0x8000000000000 - 1;
242 t[4] += 0x8000000000000 - 1;
243
244 /* now between 2^255 and 2^256-20, and offset by 2^255. */
245
246 t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff;
247 t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff;
248 t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff;
249 t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff;
250 t[4] &= 0x7ffffffffffff;
251
252 *((uint64_t *)(output)) = t[0] | (t[1] << 51);
253 *((uint64_t *)(output+8)) = (t[1] >> 13) | (t[2] << 38);
254 *((uint64_t *)(output+16)) = (t[2] >> 26) | (t[3] << 25);
255 *((uint64_t *)(output+24)) = (t[3] >> 39) | (t[4] << 12);
256 }
257
258 /* Input: Q, Q', Q-Q'
259 * Output: 2Q, Q+Q'
260 *
261 * x2 z3: long form
262 * x3 z3: long form
263 * x z: short form, destroyed
264 * xprime zprime: short form, destroyed
265 * qmqp: short form, preserved
266 */
267 static void
fmonty(limb * x2,limb * z2,limb * x3,limb * z3,limb * x,limb * z,limb * xprime,limb * zprime,const limb * qmqp)268 fmonty(limb *x2, limb *z2, /* output 2Q */
269 limb *x3, limb *z3, /* output Q + Q' */
270 limb *x, limb *z, /* input Q */
271 limb *xprime, limb *zprime, /* input Q' */
272 const limb *qmqp /* input Q - Q' */) {
273 limb origx[5], origxprime[5], zzz[5], xx[5], zz[5], xxprime[5],
274 zzprime[5], zzzprime[5];
275
276 memcpy(origx, x, 5 * sizeof(limb));
277 fsum(x, z);
278 fdifference_backwards(z, origx); // does x - z
279
280 memcpy(origxprime, xprime, sizeof(limb) * 5);
281 fsum(xprime, zprime);
282 fdifference_backwards(zprime, origxprime);
283 fmul(xxprime, xprime, z);
284 fmul(zzprime, x, zprime);
285 memcpy(origxprime, xxprime, sizeof(limb) * 5);
286 fsum(xxprime, zzprime);
287 fdifference_backwards(zzprime, origxprime);
288 fsquare_times(x3, xxprime, 1);
289 fsquare_times(zzzprime, zzprime, 1);
290 fmul(z3, zzzprime, qmqp);
291
292 fsquare_times(xx, x, 1);
293 fsquare_times(zz, z, 1);
294 fmul(x2, xx, zz);
295 fdifference_backwards(zz, xx); // does zz = xx - zz
296 fscalar_product(zzz, zz, 121665);
297 fsum(zzz, xx);
298 fmul(z2, zz, zzz);
299 }
300
301 // -----------------------------------------------------------------------------
302 // Maybe swap the contents of two limb arrays (@a and @b), each @len elements
303 // long. Perform the swap iff @swap is non-zero.
304 //
305 // This function performs the swap without leaking any side-channel
306 // information.
307 // -----------------------------------------------------------------------------
308 static void
swap_conditional(limb a[5],limb b[5],limb iswap)309 swap_conditional(limb a[5], limb b[5], limb iswap) {
310 unsigned i;
311 const limb swap = -iswap;
312
313 for (i = 0; i < 5; ++i) {
314 const limb x = swap & (a[i] ^ b[i]);
315 a[i] ^= x;
316 b[i] ^= x;
317 }
318 }
319
320 /* Calculates nQ where Q is the x-coordinate of a point on the curve
321 *
322 * resultx/resultz: the x coordinate of the resulting curve point (short form)
323 * n: a little endian, 32-byte number
324 * q: a point of the curve (short form)
325 */
326 static void
cmult(limb * resultx,limb * resultz,const u8 * n,const limb * q)327 cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) {
328 limb a[5] = {0}, b[5] = {1}, c[5] = {1}, d[5] = {0};
329 limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;
330 limb e[5] = {0}, f[5] = {1}, g[5] = {0}, h[5] = {1};
331 limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;
332
333 unsigned i, j;
334
335 memcpy(nqpqx, q, sizeof(limb) * 5);
336
337 for (i = 0; i < 32; ++i) {
338 u8 byte = n[31 - i];
339 for (j = 0; j < 8; ++j) {
340 const limb bit = byte >> 7;
341
342 swap_conditional(nqx, nqpqx, bit);
343 swap_conditional(nqz, nqpqz, bit);
344 fmonty(nqx2, nqz2,
345 nqpqx2, nqpqz2,
346 nqx, nqz,
347 nqpqx, nqpqz,
348 q);
349 swap_conditional(nqx2, nqpqx2, bit);
350 swap_conditional(nqz2, nqpqz2, bit);
351
352 t = nqx;
353 nqx = nqx2;
354 nqx2 = t;
355 t = nqz;
356 nqz = nqz2;
357 nqz2 = t;
358 t = nqpqx;
359 nqpqx = nqpqx2;
360 nqpqx2 = t;
361 t = nqpqz;
362 nqpqz = nqpqz2;
363 nqpqz2 = t;
364
365 byte <<= 1;
366 }
367 }
368
369 memcpy(resultx, nqx, sizeof(limb) * 5);
370 memcpy(resultz, nqz, sizeof(limb) * 5);
371 }
372
373
374 // -----------------------------------------------------------------------------
375 // Shamelessly copied from djb's code, tightened a little
376 // -----------------------------------------------------------------------------
377 static void
crecip(felem out,const felem z)378 crecip(felem out, const felem z) {
379 felem a,t0,b,c;
380
381 /* 2 */ fsquare_times(a, z, 1); // a = 2
382 /* 8 */ fsquare_times(t0, a, 2);
383 /* 9 */ fmul(b, t0, z); // b = 9
384 /* 11 */ fmul(a, b, a); // a = 11
385 /* 22 */ fsquare_times(t0, a, 1);
386 /* 2^5 - 2^0 = 31 */ fmul(b, t0, b);
387 /* 2^10 - 2^5 */ fsquare_times(t0, b, 5);
388 /* 2^10 - 2^0 */ fmul(b, t0, b);
389 /* 2^20 - 2^10 */ fsquare_times(t0, b, 10);
390 /* 2^20 - 2^0 */ fmul(c, t0, b);
391 /* 2^40 - 2^20 */ fsquare_times(t0, c, 20);
392 /* 2^40 - 2^0 */ fmul(t0, t0, c);
393 /* 2^50 - 2^10 */ fsquare_times(t0, t0, 10);
394 /* 2^50 - 2^0 */ fmul(b, t0, b);
395 /* 2^100 - 2^50 */ fsquare_times(t0, b, 50);
396 /* 2^100 - 2^0 */ fmul(c, t0, b);
397 /* 2^200 - 2^100 */ fsquare_times(t0, c, 100);
398 /* 2^200 - 2^0 */ fmul(t0, t0, c);
399 /* 2^250 - 2^50 */ fsquare_times(t0, t0, 50);
400 /* 2^250 - 2^0 */ fmul(t0, t0, b);
401 /* 2^255 - 2^5 */ fsquare_times(t0, t0, 5);
402 /* 2^255 - 21 */ fmul(out, t0, a);
403 }
404
405 int
crypto_scalarmult(u8 * mypublic,const u8 * secret,const u8 * basepoint)406 crypto_scalarmult(u8 *mypublic, const u8 *secret, const u8 *basepoint) {
407 limb bp[5], x[5], z[5], zmone[5];
408 uint8_t e[32];
409 int i;
410
411 for (i = 0;i < 32;++i) e[i] = secret[i];
412 e[0] &= 248;
413 e[31] &= 127;
414 e[31] |= 64;
415
416 fexpand(bp, basepoint);
417 cmult(x, z, e, bp);
418 crecip(zmone, z);
419 fmul(z, x, zmone);
420 fcontract(mypublic, z);
421 return 0;
422 }
423