Tor  0.4.6.0-alpha-dev
curve25519-donna-c64.c
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 "orconfig.h"
26 
27 #include <string.h>
28 #include "lib/cc/torint.h"
29 
30 typedef uint8_t u8;
31 typedef uint64_t limb;
32 typedef limb felem[5];
33 // This is a special gcc mode for 128-bit integers. It's implemented on 64-bit
34 // platforms only as far as I know.
35 typedef unsigned uint128_t __attribute__((mode(TI)));
36 
37 #undef force_inline
38 #define force_inline __attribute__((always_inline))
39 
40 /* Sum two numbers: output += in */
41 static inline void force_inline
42 fsum(limb *output, const limb *in) {
43  output[0] += in[0];
44  output[1] += in[1];
45  output[2] += in[2];
46  output[3] += in[3];
47  output[4] += in[4];
48 }
49 
50 /* Find the difference of two numbers: output = in - output
51  * (note the order of the arguments!)
52  *
53  * Assumes that out[i] < 2**52
54  * On return, out[i] < 2**55
55  */
56 static inline void force_inline
57 fdifference_backwards(felem out, const felem in) {
58  /* 152 is 19 << 3 */
59  static const limb two54m152 = (((limb)1) << 54) - 152;
60  static const limb two54m8 = (((limb)1) << 54) - 8;
61 
62  out[0] = in[0] + two54m152 - out[0];
63  out[1] = in[1] + two54m8 - out[1];
64  out[2] = in[2] + two54m8 - out[2];
65  out[3] = in[3] + two54m8 - out[3];
66  out[4] = in[4] + two54m8 - out[4];
67 }
68 
69 /* Multiply a number by a scalar: output = in * scalar */
70 static inline void force_inline
71 fscalar_product(felem output, const felem in, const limb scalar) {
72  uint128_t a;
73 
74  a = ((uint128_t) in[0]) * scalar;
75  output[0] = ((limb)a) & 0x7ffffffffffff;
76 
77  a = ((uint128_t) in[1]) * scalar + ((limb) (a >> 51));
78  output[1] = ((limb)a) & 0x7ffffffffffff;
79 
80  a = ((uint128_t) in[2]) * scalar + ((limb) (a >> 51));
81  output[2] = ((limb)a) & 0x7ffffffffffff;
82 
83  a = ((uint128_t) in[3]) * scalar + ((limb) (a >> 51));
84  output[3] = ((limb)a) & 0x7ffffffffffff;
85 
86  a = ((uint128_t) in[4]) * scalar + ((limb) (a >> 51));
87  output[4] = ((limb)a) & 0x7ffffffffffff;
88 
89  output[0] += (a >> 51) * 19;
90 }
91 
92 /* Multiply two numbers: output = in2 * in
93  *
94  * output must be distinct to both inputs. The inputs are reduced coefficient
95  * form, the output is not.
96  *
97  * Assumes that in[i] < 2**55 and likewise for in2.
98  * On return, output[i] < 2**52
99  */
100 static inline void force_inline
101 fmul(felem output, const felem in2, const felem in) {
102  uint128_t t[5];
103  limb r0,r1,r2,r3,r4,s0,s1,s2,s3,s4,c;
104 
105  r0 = in[0];
106  r1 = in[1];
107  r2 = in[2];
108  r3 = in[3];
109  r4 = in[4];
110 
111  s0 = in2[0];
112  s1 = in2[1];
113  s2 = in2[2];
114  s3 = in2[3];
115  s4 = in2[4];
116 
117  t[0] = ((uint128_t) r0) * s0;
118  t[1] = ((uint128_t) r0) * s1 + ((uint128_t) r1) * s0;
119  t[2] = ((uint128_t) r0) * s2 + ((uint128_t) r2) * s0 + ((uint128_t) r1) * s1;
120  t[3] = ((uint128_t) r0) * s3 + ((uint128_t) r3) * s0 + ((uint128_t) r1) * s2 + ((uint128_t) r2) * s1;
121  t[4] = ((uint128_t) r0) * s4 + ((uint128_t) r4) * s0 + ((uint128_t) r3) * s1 + ((uint128_t) r1) * s3 + ((uint128_t) r2) * s2;
122 
123  r4 *= 19;
124  r1 *= 19;
125  r2 *= 19;
126  r3 *= 19;
127 
128  t[0] += ((uint128_t) r4) * s1 + ((uint128_t) r1) * s4 + ((uint128_t) r2) * s3 + ((uint128_t) r3) * s2;
129  t[1] += ((uint128_t) r4) * s2 + ((uint128_t) r2) * s4 + ((uint128_t) r3) * s3;
130  t[2] += ((uint128_t) r4) * s3 + ((uint128_t) r3) * s4;
131  t[3] += ((uint128_t) r4) * s4;
132 
133  r0 = (limb)t[0] & 0x7ffffffffffff; c = (limb)(t[0] >> 51);
134  t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffff; c = (limb)(t[1] >> 51);
135  t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffff; c = (limb)(t[2] >> 51);
136  t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffff; c = (limb)(t[3] >> 51);
137  t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffff; c = (limb)(t[4] >> 51);
138  r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff;
139  r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff;
140  r2 += c;
141 
142  output[0] = r0;
143  output[1] = r1;
144  output[2] = r2;
145  output[3] = r3;
146  output[4] = r4;
147 }
148 
149 static inline void force_inline
150 fsquare_times(felem output, const felem in, limb count) {
151  uint128_t t[5];
152  limb r0,r1,r2,r3,r4,c;
153  limb d0,d1,d2,d4,d419;
154 
155  r0 = in[0];
156  r1 = in[1];
157  r2 = in[2];
158  r3 = in[3];
159  r4 = in[4];
160 
161  do {
162  d0 = r0 * 2;
163  d1 = r1 * 2;
164  d2 = r2 * 2 * 19;
165  d419 = r4 * 19;
166  d4 = d419 * 2;
167 
168  t[0] = ((uint128_t) r0) * r0 + ((uint128_t) d4) * r1 + (((uint128_t) d2) * (r3 ));
169  t[1] = ((uint128_t) d0) * r1 + ((uint128_t) d4) * r2 + (((uint128_t) r3) * (r3 * 19));
170  t[2] = ((uint128_t) d0) * r2 + ((uint128_t) r1) * r1 + (((uint128_t) d4) * (r3 ));
171  t[3] = ((uint128_t) d0) * r3 + ((uint128_t) d1) * r2 + (((uint128_t) r4) * (d419 ));
172  t[4] = ((uint128_t) d0) * r4 + ((uint128_t) d1) * r3 + (((uint128_t) r2) * (r2 ));
173 
174  r0 = (limb)t[0] & 0x7ffffffffffff; c = (limb)(t[0] >> 51);
175  t[1] += c; r1 = (limb)t[1] & 0x7ffffffffffff; c = (limb)(t[1] >> 51);
176  t[2] += c; r2 = (limb)t[2] & 0x7ffffffffffff; c = (limb)(t[2] >> 51);
177  t[3] += c; r3 = (limb)t[3] & 0x7ffffffffffff; c = (limb)(t[3] >> 51);
178  t[4] += c; r4 = (limb)t[4] & 0x7ffffffffffff; c = (limb)(t[4] >> 51);
179  r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff;
180  r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff;
181  r2 += c;
182  } while(--count);
183 
184  output[0] = r0;
185  output[1] = r1;
186  output[2] = r2;
187  output[3] = r3;
188  output[4] = r4;
189 }
190 
191 /* Load a little-endian 64-bit number */
192 static limb
193 load_limb(const u8 *in) {
194  return
195  ((limb)in[0]) |
196  (((limb)in[1]) << 8) |
197  (((limb)in[2]) << 16) |
198  (((limb)in[3]) << 24) |
199  (((limb)in[4]) << 32) |
200  (((limb)in[5]) << 40) |
201  (((limb)in[6]) << 48) |
202  (((limb)in[7]) << 56);
203 }
204 
205 static void
206 store_limb(u8 *out, limb in) {
207  out[0] = in & 0xff;
208  out[1] = (in >> 8) & 0xff;
209  out[2] = (in >> 16) & 0xff;
210  out[3] = (in >> 24) & 0xff;
211  out[4] = (in >> 32) & 0xff;
212  out[5] = (in >> 40) & 0xff;
213  out[6] = (in >> 48) & 0xff;
214  out[7] = (in >> 56) & 0xff;
215 }
216 
217 /* Take a little-endian, 32-byte number and expand it into polynomial form */
218 static void
219 fexpand(limb *output, const u8 *in) {
220  output[0] = load_limb(in) & 0x7ffffffffffff;
221  output[1] = (load_limb(in+6) >> 3) & 0x7ffffffffffff;
222  output[2] = (load_limb(in+12) >> 6) & 0x7ffffffffffff;
223  output[3] = (load_limb(in+19) >> 1) & 0x7ffffffffffff;
224  output[4] = (load_limb(in+24) >> 12) & 0x7ffffffffffff;
225 }
226 
227 /* Take a fully reduced polynomial form number and contract it into a
228  * little-endian, 32-byte array
229  */
230 static void
231 fcontract(u8 *output, const felem input) {
232  uint128_t t[5];
233 
234  t[0] = input[0];
235  t[1] = input[1];
236  t[2] = input[2];
237  t[3] = input[3];
238  t[4] = input[4];
239 
240  t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff;
241  t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff;
242  t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff;
243  t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff;
244  t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff;
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[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff;
251 
252  /* now t is between 0 and 2^255-1, properly carried. */
253  /* case 1: between 0 and 2^255-20. case 2: between 2^255-19 and 2^255-1. */
254 
255  t[0] += 19;
256 
257  t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff;
258  t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff;
259  t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff;
260  t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff;
261  t[0] += 19 * (t[4] >> 51); t[4] &= 0x7ffffffffffff;
262 
263  /* now between 19 and 2^255-1 in both cases, and offset by 19. */
264 
265  t[0] += 0x8000000000000 - 19;
266  t[1] += 0x8000000000000 - 1;
267  t[2] += 0x8000000000000 - 1;
268  t[3] += 0x8000000000000 - 1;
269  t[4] += 0x8000000000000 - 1;
270 
271  /* now between 2^255 and 2^256-20, and offset by 2^255. */
272 
273  t[1] += t[0] >> 51; t[0] &= 0x7ffffffffffff;
274  t[2] += t[1] >> 51; t[1] &= 0x7ffffffffffff;
275  t[3] += t[2] >> 51; t[2] &= 0x7ffffffffffff;
276  t[4] += t[3] >> 51; t[3] &= 0x7ffffffffffff;
277  t[4] &= 0x7ffffffffffff;
278 
279  store_limb(output, t[0] | (t[1] << 51));
280  store_limb(output+8, (t[1] >> 13) | (t[2] << 38));
281  store_limb(output+16, (t[2] >> 26) | (t[3] << 25));
282  store_limb(output+24, (t[3] >> 39) | (t[4] << 12));
283 }
284 
285 /* Input: Q, Q', Q-Q'
286  * Output: 2Q, Q+Q'
287  *
288  * x2 z3: long form
289  * x3 z3: long form
290  * x z: short form, destroyed
291  * xprime zprime: short form, destroyed
292  * qmqp: short form, preserved
293  */
294 static void
295 fmonty(limb *x2, limb *z2, /* output 2Q */
296  limb *x3, limb *z3, /* output Q + Q' */
297  limb *x, limb *z, /* input Q */
298  limb *xprime, limb *zprime, /* input Q' */
299  const limb *qmqp /* input Q - Q' */) {
300  limb origx[5], origxprime[5], zzz[5], xx[5], zz[5], xxprime[5],
301  zzprime[5], zzzprime[5];
302 
303  memcpy(origx, x, 5 * sizeof(limb));
304  fsum(x, z);
305  fdifference_backwards(z, origx); // does x - z
306 
307  memcpy(origxprime, xprime, sizeof(limb) * 5);
308  fsum(xprime, zprime);
309  fdifference_backwards(zprime, origxprime);
310  fmul(xxprime, xprime, z);
311  fmul(zzprime, x, zprime);
312  memcpy(origxprime, xxprime, sizeof(limb) * 5);
313  fsum(xxprime, zzprime);
314  fdifference_backwards(zzprime, origxprime);
315  fsquare_times(x3, xxprime, 1);
316  fsquare_times(zzzprime, zzprime, 1);
317  fmul(z3, zzzprime, qmqp);
318 
319  fsquare_times(xx, x, 1);
320  fsquare_times(zz, z, 1);
321  fmul(x2, xx, zz);
322  fdifference_backwards(zz, xx); // does zz = xx - zz
323  fscalar_product(zzz, zz, 121665);
324  fsum(zzz, xx);
325  fmul(z2, zz, zzz);
326 }
327 
328 // -----------------------------------------------------------------------------
329 // Maybe swap the contents of two limb arrays (@a and @b), each @len elements
330 // long. Perform the swap iff @swap is non-zero.
331 //
332 // This function performs the swap without leaking any side-channel
333 // information.
334 // -----------------------------------------------------------------------------
335 static void
336 swap_conditional(limb a[5], limb b[5], limb iswap) {
337  unsigned i;
338  const limb swap = -iswap;
339 
340  for (i = 0; i < 5; ++i) {
341  const limb x = swap & (a[i] ^ b[i]);
342  a[i] ^= x;
343  b[i] ^= x;
344  }
345 }
346 
347 /* Calculates nQ where Q is the x-coordinate of a point on the curve
348  *
349  * resultx/resultz: the x coordinate of the resulting curve point (short form)
350  * n: a little endian, 32-byte number
351  * q: a point of the curve (short form)
352  */
353 static void
354 cmult(limb *resultx, limb *resultz, const u8 *n, const limb *q) {
355  limb a[5] = {0}, b[5] = {1}, c[5] = {1}, d[5] = {0};
356  limb *nqpqx = a, *nqpqz = b, *nqx = c, *nqz = d, *t;
357  limb e[5] = {0}, f[5] = {1}, g[5] = {0}, h[5] = {1};
358  limb *nqpqx2 = e, *nqpqz2 = f, *nqx2 = g, *nqz2 = h;
359 
360  unsigned i, j;
361 
362  memcpy(nqpqx, q, sizeof(limb) * 5);
363 
364  for (i = 0; i < 32; ++i) {
365  u8 byte = n[31 - i];
366  for (j = 0; j < 8; ++j) {
367  const limb bit = byte >> 7;
368 
369  swap_conditional(nqx, nqpqx, bit);
370  swap_conditional(nqz, nqpqz, bit);
371  fmonty(nqx2, nqz2,
372  nqpqx2, nqpqz2,
373  nqx, nqz,
374  nqpqx, nqpqz,
375  q);
376  swap_conditional(nqx2, nqpqx2, bit);
377  swap_conditional(nqz2, nqpqz2, bit);
378 
379  t = nqx;
380  nqx = nqx2;
381  nqx2 = t;
382  t = nqz;
383  nqz = nqz2;
384  nqz2 = t;
385  t = nqpqx;
386  nqpqx = nqpqx2;
387  nqpqx2 = t;
388  t = nqpqz;
389  nqpqz = nqpqz2;
390  nqpqz2 = t;
391 
392  byte <<= 1;
393  }
394  }
395 
396  memcpy(resultx, nqx, sizeof(limb) * 5);
397  memcpy(resultz, nqz, sizeof(limb) * 5);
398 }
399 
400 
401 // -----------------------------------------------------------------------------
402 // Shamelessly copied from djb's code, tightened a little
403 // -----------------------------------------------------------------------------
404 static void
405 crecip(felem out, const felem z) {
406  felem a,t0,b,c;
407 
408  /* 2 */ fsquare_times(a, z, 1); // a = 2
409  /* 8 */ fsquare_times(t0, a, 2);
410  /* 9 */ fmul(b, t0, z); // b = 9
411  /* 11 */ fmul(a, b, a); // a = 11
412  /* 22 */ fsquare_times(t0, a, 1);
413  /* 2^5 - 2^0 = 31 */ fmul(b, t0, b);
414  /* 2^10 - 2^5 */ fsquare_times(t0, b, 5);
415  /* 2^10 - 2^0 */ fmul(b, t0, b);
416  /* 2^20 - 2^10 */ fsquare_times(t0, b, 10);
417  /* 2^20 - 2^0 */ fmul(c, t0, b);
418  /* 2^40 - 2^20 */ fsquare_times(t0, c, 20);
419  /* 2^40 - 2^0 */ fmul(t0, t0, c);
420  /* 2^50 - 2^10 */ fsquare_times(t0, t0, 10);
421  /* 2^50 - 2^0 */ fmul(b, t0, b);
422  /* 2^100 - 2^50 */ fsquare_times(t0, b, 50);
423  /* 2^100 - 2^0 */ fmul(c, t0, b);
424  /* 2^200 - 2^100 */ fsquare_times(t0, c, 100);
425  /* 2^200 - 2^0 */ fmul(t0, t0, c);
426  /* 2^250 - 2^50 */ fsquare_times(t0, t0, 50);
427  /* 2^250 - 2^0 */ fmul(t0, t0, b);
428  /* 2^255 - 2^5 */ fsquare_times(t0, t0, 5);
429  /* 2^255 - 21 */ fmul(out, t0, a);
430 }
431 
432 int curve25519_donna(u8 *, const u8 *, const u8 *);
433 
434 int
435 curve25519_donna(u8 *mypublic, const u8 *secret, const u8 *basepoint) {
436  limb bp[5], x[5], z[5], zmone[5];
437  uint8_t e[32];
438  int i;
439 
440  for (i = 0;i < 32;++i) e[i] = secret[i];
441  e[0] &= 248;
442  e[31] &= 127;
443  e[31] |= 64;
444 
445  fexpand(bp, basepoint);
446  cmult(x, z, e, bp);
447  crecip(zmone, z);
448  fmul(z, x, zmone);
449  fcontract(mypublic, z);
450  return 0;
451 }
torint.h
Integer definitions used throughout Tor.