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 *
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
28 typedef uint8_t u8;
29 typedef uint64_t limb;
30 typedef limb felem[5];
31 // This is a special gcc mode for 128-bit integers. It's implemented on 64-bit
32 // platforms only as far as I know.
33 typedef unsigned uint128_t __attribute__((mode(TI)));
34
35 #undef force_inline
36 #define force_inline __attribute__((always_inline))
37
38 /* Sum two numbers: output += in */
39 static inline void force_inline
fsum(limb * output,const limb * in)40 fsum(limb *output, const limb *in) {
41 output[0] += in[0];
42 output[1] += in[1];
43 output[2] += in[2];
44 output[3] += in[3];
45 output[4] += in[4];
46 }
47
48 /* Find the difference of two numbers: output = in - output
49 * (note the order of the arguments!)
50 *
51 * Assumes that out[i] < 2**52
52 * On return, out[i] < 2**55
53 */
54 static inline void force_inline
fdifference_backwards(felem out,const felem in)55 fdifference_backwards(felem out, const felem in) {
56 /* 152 is 19 << 3 */
57 static const limb two54m152 = (((limb)1) << 54) - 152;
58 static const limb two54m8 = (((limb)1) << 54) - 8;
59
60 out[0] = in[0] + two54m152 - out[0];
61 out[1] = in[1] + two54m8 - out[1];
62 out[2] = in[2] + two54m8 - out[2];
63 out[3] = in[3] + two54m8 - out[3];
64 out[4] = in[4] + two54m8 - out[4];
65 }
66
67 /* Multiply a number by a scalar: output = in * scalar */
68 static inline void force_inline
fscalar_product(felem output,const felem in,const limb scalar)69 fscalar_product(felem output, const felem in, const limb scalar) {
70 uint128_t a;
71
72 a = ((uint128_t) in[0]) * scalar;
73 output[0] = ((limb)a) & 0x7ffffffffffff;
74
75 a = ((uint128_t) in[1]) * scalar + ((limb) (a >> 51));
76 output[1] = ((limb)a) & 0x7ffffffffffff;
77
78 a = ((uint128_t) in[2]) * scalar + ((limb) (a >> 51));
79 output[2] = ((limb)a) & 0x7ffffffffffff;
80
81 a = ((uint128_t) in[3]) * scalar + ((limb) (a >> 51));
82 output[3] = ((limb)a) & 0x7ffffffffffff;
83
84 a = ((uint128_t) in[4]) * scalar + ((limb) (a >> 51));
85 output[4] = ((limb)a) & 0x7ffffffffffff;
86
87 output[0] += (a >> 51) * 19;
88 }
89
90 /* Multiply two numbers: output = in2 * in
91 *
92 * output must be distinct to both inputs. The inputs are reduced coefficient
93 * form, the output is not.
94 *
95 * Assumes that in[i] < 2**55 and likewise for in2.
96 * On return, output[i] < 2**52
97 */
98 static inline void force_inline
fmul(felem output,const felem in2,const felem in)99 fmul(felem output, const felem in2, const felem in) {
100 uint128_t t[5];
101 limb r0,r1,r2,r3,r4,s0,s1,s2,s3,s4,c;
102
103 r0 = in[0];
104 r1 = in[1];
105 r2 = in[2];
106 r3 = in[3];
107 r4 = in[4];
108
109 s0 = in2[0];
110 s1 = in2[1];
111 s2 = in2[2];
112 s3 = in2[3];
113 s4 = in2[4];
114
115 t[0] = ((uint128_t) r0) * s0;
116 t[1] = ((uint128_t) r0) * s1 + ((uint128_t) r1) * s0;
117 t[2] = ((uint128_t) r0) * s2 + ((uint128_t) r2) * s0 + ((uint128_t) r1) * s1;
118 t[3] = ((uint128_t) r0) * s3 + ((uint128_t) r3) * s0 + ((uint128_t) r1) * s2 + ((uint128_t) r2) * s1;
119 t[4] = ((uint128_t) r0) * s4 + ((uint128_t) r4) * s0 + ((uint128_t) r3) * s1 + ((uint128_t) r1) * s3 + ((uint128_t) r2) * s2;
120
121 r4 *= 19;
122 r1 *= 19;
123 r2 *= 19;
124 r3 *= 19;
125
126 t[0] += ((uint128_t) r4) * s1 + ((uint128_t) r1) * s4 + ((uint128_t) r2) * s3 + ((uint128_t) r3) * s2;
127 t[1] += ((uint128_t) r4) * s2 + ((uint128_t) r2) * s4 + ((uint128_t) r3) * s3;
128 t[2] += ((uint128_t) r4) * s3 + ((uint128_t) r3) * s4;
129 t[3] += ((uint128_t) r4) * s4;
130
131 r0 = (limb)t[0] & 0x7ffffffffffff; c = (limb)(t[0] >> 51);
132 t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffff; c = (limb)(t[1] >> 51);
133 t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffff; c = (limb)(t[2] >> 51);
134 t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffff; c = (limb)(t[3] >> 51);
135 t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffff; c = (limb)(t[4] >> 51);
136 r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff;
137 r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff;
138 r2 += c;
139
140 output[0] = r0;
141 output[1] = r1;
142 output[2] = r2;
143 output[3] = r3;
144 output[4] = r4;
145 }
146
147 static inline void force_inline
fsquare_times(felem output,const felem in,limb count)148 fsquare_times(felem output, const felem in, limb count) {
149 uint128_t t[5];
150 limb r0,r1,r2,r3,r4,c;
151 limb d0,d1,d2,d4,d419;
152
153 r0 = in[0];
154 r1 = in[1];
155 r2 = in[2];
156 r3 = in[3];
157 r4 = in[4];
158
159 do {
160 d0 = r0 * 2;
161 d1 = r1 * 2;
162 d2 = r2 * 2 * 19;
163 d419 = r4 * 19;
164 d4 = d419 * 2;
165
166 t[0] = ((uint128_t) r0) * r0 + ((uint128_t) d4) * r1 + (((uint128_t) d2) * (r3 ));
167 t[1] = ((uint128_t) d0) * r1 + ((uint128_t) d4) * r2 + (((uint128_t) r3) * (r3 * 19));
168 t[2] = ((uint128_t) d0) * r2 + ((uint128_t) r1) * r1 + (((uint128_t) d4) * (r3 ));
169 t[3] = ((uint128_t) d0) * r3 + ((uint128_t) d1) * r2 + (((uint128_t) r4) * (d419 ));
170 t[4] = ((uint128_t) d0) * r4 + ((uint128_t) d1) * r3 + (((uint128_t) r2) * (r2 ));
171
172 r0 = (limb)t[0] & 0x7ffffffffffff; c = (limb)(t[0] >> 51);
173 t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffff; c = (limb)(t[1] >> 51);
174 t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffff; c = (limb)(t[2] >> 51);
175 t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffff; c = (limb)(t[3] >> 51);
176 t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffff; c = (limb)(t[4] >> 51);
177 r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff;
178 r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff;
179 r2 += c;
180 } while(--count);
181
182 output[0] = r0;
183 output[1] = r1;
184 output[2] = r2;
185 output[3] = r3;
186 output[4] = r4;
187 }
188
189 /* Load a little-endian 64-bit number */
190 static limb
load_limb(const u8 * in)191 load_limb(const u8 *in) {
192 return
193 ((limb)in[0]) |
194 (((limb)in[1]) << 8) |
195 (((limb)in[2]) << 16) |
196 (((limb)in[3]) << 24) |
197 (((limb)in[4]) << 32) |
198 (((limb)in[5]) << 40) |
199 (((limb)in[6]) << 48) |
200 (((limb)in[7]) << 56);
201 }
202
203 static void
store_limb(u8 * out,limb in)204 store_limb(u8 *out, limb in) {
205 out[0] = in & 0xff;
206 out[1] = (in >> 8) & 0xff;
207 out[2] = (in >> 16) & 0xff;
208 out[3] = (in >> 24) & 0xff;
209 out[4] = (in >> 32) & 0xff;
210 out[5] = (in >> 40) & 0xff;
211 out[6] = (in >> 48) & 0xff;
212 out[7] = (in >> 56) & 0xff;
213 }
214
215 /* Take a little-endian, 32-byte number and expand it into polynomial form */
216 static void
fexpand(limb * output,const u8 * in)217 fexpand(limb *output, const u8 *in) {
218 output[0] = load_limb(in) & 0x7ffffffffffff;
219 output[1] = (load_limb(in+6) >> 3) & 0x7ffffffffffff;
220 output[2] = (load_limb(in+12) >> 6) & 0x7ffffffffffff;
221 output[3] = (load_limb(in+19) >> 1) & 0x7ffffffffffff;
222 output[4] = (load_limb(in+24) >> 12) & 0xfffffffffffff;
223 }
224
225 /* Take a fully reduced polynomial form number and contract it into a
226 * little-endian, 32-byte array
227 */
228 static void
fcontract(u8 * output,const felem input)229 fcontract(u8 *output, const felem input) {
230 uint128_t t[5];
231
232 t[0] = input[0];
233 t[1] = input[1];
234 t[2] = input[2];
235 t[3] = input[3];
236 t[4] = input[4];
237
238 t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff;
239 t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff;
240 t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff;
241 t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff;
242 t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff;
243
244 t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff;
245 t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff;
246 t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff;
247 t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff;
248 t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff;
249
250 /* now t is between 0 and 2^255-1, properly carried. */
251 /* case 1: between 0 and 2^255-20. case 2: between 2^255-19 and 2^255-1. */
252
253 t[0] += 19;
254
255 t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff;
256 t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff;
257 t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff;
258 t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff;
259 t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff;
260
261 /* now between 19 and 2^255-1 in both cases, and offset by 19. */
262
263 t[0] += 0x8000000000000 - 19;
264 t[1] += 0x8000000000000 - 1;
265 t[2] += 0x8000000000000 - 1;
266 t[3] += 0x8000000000000 - 1;
267 t[4] += 0x8000000000000 - 1;
268
269 /* now between 2^255 and 2^256-20, and offset by 2^255. */
270
271 t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff;
272 t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff;
273 t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff;
274 t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff;
275 t[4] &= 0x7ffffffffffff;
276
277 store_limb(output, t[0] | (t[1] << 51));
278 store_limb(output+8, (t[1] >> 13) | (t[2] << 38));
279 store_limb(output+16, (t[2] >> 26) | (t[3] << 25));
280 store_limb(output+24, (t[3] >> 39) | (t[4] << 12));
281 }
282
283 /* Input: Q, Q', Q-Q'
284 * Output: 2Q, Q+Q'
285 *
286 * x2 z3: long form
287 * x3 z3: long form
288 * x z: short form, destroyed
289 * xprime zprime: short form, destroyed
290 * qmqp: short form, preserved
291 */
292 static void
fmonty(limb * x2,limb * z2,limb * x3,limb * z3,limb * x,limb * z,limb * xprime,limb * zprime,const limb * qmqp)293 fmonty(limb *x2, limb *z2, /* output 2Q */
294 limb *x3, limb *z3, /* output Q + Q' */
295 limb *x, limb *z, /* input Q */
296 limb *xprime, limb *zprime, /* input Q' */
297 const limb *qmqp /* input Q - Q' */) {
298 limb origx[5], origxprime[5], zzz[5], xx[5], zz[5], xxprime[5],
299 zzprime[5], zzzprime[5];
300
301 memcpy(origx, x, 5 * sizeof(limb));
302 fsum(x, z);
303 fdifference_backwards(z, origx); // does x - z
304
305 memcpy(origxprime, xprime, sizeof(limb) * 5);
306 fsum(xprime, zprime);
307 fdifference_backwards(zprime, origxprime);
308 fmul(xxprime, xprime, z);
309 fmul(zzprime, x, zprime);
310 memcpy(origxprime, xxprime, sizeof(limb) * 5);
311 fsum(xxprime, zzprime);
312 fdifference_backwards(zzprime, origxprime);
313 fsquare_times(x3, xxprime, 1);
314 fsquare_times(zzzprime, zzprime, 1);
315 fmul(z3, zzzprime, qmqp);
316
317 fsquare_times(xx, x, 1);
318 fsquare_times(zz, z, 1);
319 fmul(x2, xx, zz);
320 fdifference_backwards(zz, xx); // does zz = xx - zz
321 fscalar_product(zzz, zz, 121665);
322 fsum(zzz, xx);
323 fmul(z2, zz, zzz);
324 }
325
326 // -----------------------------------------------------------------------------
327 // Maybe swap the contents of two limb arrays (@a and @b), each @len elements
328 // long. Perform the swap iff @swap is non-zero.
329 //
330 // This function performs the swap without leaking any side-channel
331 // information.
332 // -----------------------------------------------------------------------------
333 static void
swap_conditional(limb a[5],limb b[5],limb iswap)334 swap_conditional(limb a[5], limb b[5], limb iswap) {
335 unsigned i;
336 const limb swap = -iswap;
337
338 for (i = 0; i < 5; ++i) {
339 const limb x = swap & (a[i] ^ b[i]);
340 a[i] ^= x;
341 b[i] ^= x;
342 }
343 }
344
345 /* Calculates nQ where Q is the x-coordinate of a point on the curve
346 *
347 * resultx/resultz: the x coordinate of the resulting curve point (short form)
348 * n: a little endian, 32-byte number
349 * q: a point of the curve (short form)
350 */
351 static void
cmult(limb * resultx,limb * resultz,const u8 * n,const limb * q)352 cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) {
353 limb a[5] = {0}, b[5] = {1}, c[5] = {1}, d[5] = {0};
354 limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;
355 limb e[5] = {0}, f[5] = {1}, g[5] = {0}, h[5] = {1};
356 limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;
357
358 unsigned i, j;
359
360 memcpy(nqpqx, q, sizeof(limb) * 5);
361
362 for (i = 0; i < 32; ++i) {
363 u8 byte = n[31 - i];
364 for (j = 0; j < 8; ++j) {
365 const limb bit = byte >> 7;
366
367 swap_conditional(nqx, nqpqx, bit);
368 swap_conditional(nqz, nqpqz, bit);
369 fmonty(nqx2, nqz2,
370 nqpqx2, nqpqz2,
371 nqx, nqz,
372 nqpqx, nqpqz,
373 q);
374 swap_conditional(nqx2, nqpqx2, bit);
375 swap_conditional(nqz2, nqpqz2, bit);
376
377 t = nqx;
378 nqx = nqx2;
379 nqx2 = t;
380 t = nqz;
381 nqz = nqz2;
382 nqz2 = t;
383 t = nqpqx;
384 nqpqx = nqpqx2;
385 nqpqx2 = t;
386 t = nqpqz;
387 nqpqz = nqpqz2;
388 nqpqz2 = t;
389
390 byte <<= 1;
391 }
392 }
393
394 memcpy(resultx, nqx, sizeof(limb) * 5);
395 memcpy(resultz, nqz, sizeof(limb) * 5);
396 }
397
398
399 // -----------------------------------------------------------------------------
400 // Shamelessly copied from djb's code, tightened a little
401 // -----------------------------------------------------------------------------
402 static void
crecip(felem out,const felem z)403 crecip(felem out, const felem z) {
404 felem a,t0,b,c;
405
406 /* 2 */ fsquare_times(a, z, 1); // a = 2
407 /* 8 */ fsquare_times(t0, a, 2);
408 /* 9 */ fmul(b, t0, z); // b = 9
409 /* 11 */ fmul(a, b, a); // a = 11
410 /* 22 */ fsquare_times(t0, a, 1);
411 /* 2^5 - 2^0 = 31 */ fmul(b, t0, b);
412 /* 2^10 - 2^5 */ fsquare_times(t0, b, 5);
413 /* 2^10 - 2^0 */ fmul(b, t0, b);
414 /* 2^20 - 2^10 */ fsquare_times(t0, b, 10);
415 /* 2^20 - 2^0 */ fmul(c, t0, b);
416 /* 2^40 - 2^20 */ fsquare_times(t0, c, 20);
417 /* 2^40 - 2^0 */ fmul(t0, t0, c);
418 /* 2^50 - 2^10 */ fsquare_times(t0, t0, 10);
419 /* 2^50 - 2^0 */ fmul(b, t0, b);
420 /* 2^100 - 2^50 */ fsquare_times(t0, b, 50);
421 /* 2^100 - 2^0 */ fmul(c, t0, b);
422 /* 2^200 - 2^100 */ fsquare_times(t0, c, 100);
423 /* 2^200 - 2^0 */ fmul(t0, t0, c);
424 /* 2^250 - 2^50 */ fsquare_times(t0, t0, 50);
425 /* 2^250 - 2^0 */ fmul(t0, t0, b);
426 /* 2^255 - 2^5 */ fsquare_times(t0, t0, 5);
427 /* 2^255 - 21 */ fmul(out, t0, a);
428 }
429
430 int curve25519_donna(u8 *, const u8 *, const u8 *);
431
432 int
curve25519_donna(u8 * mypublic,const u8 * secret,const u8 * basepoint)433 curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) {
434 limb bp[5], x[5], z[5], zmone[5];
435 uint8_t e[32];
436 int i;
437
438 for (i = 0;i < 32;++i) e[i] = secret[i];
439 e[0] &= 248;
440 e[31] &= 127;
441 e[31] |= 64;
442
443 fexpand(bp, basepoint);
444 cmult(x, z, e, bp);
445 crecip(zmone, z);
446 fmul(z, x, zmone);
447 fcontract(mypublic, z);
448 return 0;
449 }
450