Botan  2.9.0
Crypto and TLS for C++11
donna.cpp
Go to the documentation of this file.
1 /*
2 * Based on curve25519-donna-c64.c from github.com/agl/curve25519-donna
3 * revision 80ad9b9930c9baef5829dd2a235b6b7646d32a8e
4 *
5 * Further changes
6 * (C) 2014,2018 Jack Lloyd
7 *
8 * Botan is released under the Simplified BSD License (see license.txt)
9 */
10 
11 /* Copyright 2008, Google Inc.
12 * All rights reserved.
13 *
14 * Code released into the public domain.
15 *
16 * curve25519-donna: Curve25519 elliptic curve, public key function
17 *
18 * https://code.google.com/p/curve25519-donna/
19 *
20 * Adam Langley <agl@imperialviolet.org>
21 *
22 * Derived from public domain C code by Daniel J. Bernstein <djb@cr.yp.to>
23 *
24 * More information about curve25519 can be found here
25 * https://cr.yp.to/ecdh.html
26 *
27 * djb's sample implementation of curve25519 is written in a special assembly
28 * language called qhasm and uses the floating point registers.
29 *
30 * This is, almost, a clean room reimplementation from the curve25519 paper. It
31 * uses many of the tricks described therein. Only the crecip function is taken
32 * from the sample implementation.
33 */
34 
35 #include <botan/curve25519.h>
36 #include <botan/mul128.h>
37 #include <botan/internal/ct_utils.h>
38 #include <botan/internal/donna128.h>
39 #include <botan/loadstor.h>
40 
41 namespace Botan {
42 
43 namespace {
44 
45 #if !defined(BOTAN_TARGET_HAS_NATIVE_UINT128)
46 typedef donna128 uint128_t;
47 #endif
48 
49 /* Sum two numbers: output += in */
50 inline void fsum(uint64_t out[5], const uint64_t in[5])
51  {
52  out[0] += in[0];
53  out[1] += in[1];
54  out[2] += in[2];
55  out[3] += in[3];
56  out[4] += in[4];
57  }
58 
59 /* Find the difference of two numbers: out = in - out
60 * (note the order of the arguments!)
61 *
62 * Assumes that out[i] < 2**52
63 * On return, out[i] < 2**55
64 */
65 inline void fdifference_backwards(uint64_t out[5], const uint64_t in[5])
66  {
67  /* 152 is 19 << 3 */
68  const uint64_t two54m152 = (static_cast<uint64_t>(1) << 54) - 152;
69  const uint64_t two54m8 = (static_cast<uint64_t>(1) << 54) - 8;
70 
71  out[0] = in[0] + two54m152 - out[0];
72  out[1] = in[1] + two54m8 - out[1];
73  out[2] = in[2] + two54m8 - out[2];
74  out[3] = in[3] + two54m8 - out[3];
75  out[4] = in[4] + two54m8 - out[4];
76  }
77 
78 inline void fadd_sub(uint64_t x[5],
79  uint64_t y[5])
80  {
81  // TODO merge these and avoid the tmp array
82  uint64_t tmp[5];
83  copy_mem(tmp, y, 5);
84  fsum(y, x);
85  fdifference_backwards(x, tmp); // does x - z
86  }
87 
88 /* Multiply a number by a scalar: out = in * scalar */
89 inline void fscalar_product(uint64_t out[5], const uint64_t in[5], const uint64_t scalar)
90  {
91  uint128_t a = uint128_t(in[0]) * scalar;
92  out[0] = a & 0x7ffffffffffff;
93 
94  a = uint128_t(in[1]) * scalar + carry_shift(a, 51);
95  out[1] = a & 0x7ffffffffffff;
96 
97  a = uint128_t(in[2]) * scalar + carry_shift(a, 51);
98  out[2] = a & 0x7ffffffffffff;
99 
100  a = uint128_t(in[3]) * scalar + carry_shift(a, 51);
101  out[3] = a & 0x7ffffffffffff;
102 
103  a = uint128_t(in[4]) * scalar + carry_shift(a, 51);
104  out[4] = a & 0x7ffffffffffff;
105 
106  out[0] += carry_shift(a, 51) * 19;
107  }
108 
109 /* Multiply two numbers: out = in2 * in
110 *
111 * out must be distinct to both inputs. The inputs are reduced coefficient
112 * form, the output is not.
113 *
114 * Assumes that in[i] < 2**55 and likewise for in2.
115 * On return, out[i] < 2**52
116 */
117 inline void fmul(uint64_t out[5], const uint64_t in[5], const uint64_t in2[5])
118  {
119  const uint128_t s0 = in2[0];
120  const uint128_t s1 = in2[1];
121  const uint128_t s2 = in2[2];
122  const uint128_t s3 = in2[3];
123  const uint128_t s4 = in2[4];
124 
125  uint64_t r0 = in[0];
126  uint64_t r1 = in[1];
127  uint64_t r2 = in[2];
128  uint64_t r3 = in[3];
129  uint64_t r4 = in[4];
130 
131  uint128_t t0 = r0 * s0;
132  uint128_t t1 = r0 * s1 + r1 * s0;
133  uint128_t t2 = r0 * s2 + r2 * s0 + r1 * s1;
134  uint128_t t3 = r0 * s3 + r3 * s0 + r1 * s2 + r2 * s1;
135  uint128_t t4 = r0 * s4 + r4 * s0 + r3 * s1 + r1 * s3 + r2 * s2;
136 
137  r4 *= 19;
138  r1 *= 19;
139  r2 *= 19;
140  r3 *= 19;
141 
142  t0 += r4 * s1 + r1 * s4 + r2 * s3 + r3 * s2;
143  t1 += r4 * s2 + r2 * s4 + r3 * s3;
144  t2 += r4 * s3 + r3 * s4;
145  t3 += r4 * s4;
146 
147  r0 = t0 & 0x7ffffffffffff; t1 += carry_shift(t0, 51);
148  r1 = t1 & 0x7ffffffffffff; t2 += carry_shift(t1, 51);
149  r2 = t2 & 0x7ffffffffffff; t3 += carry_shift(t2, 51);
150  r3 = t3 & 0x7ffffffffffff; t4 += carry_shift(t3, 51);
151  r4 = t4 & 0x7ffffffffffff; uint64_t c = carry_shift(t4, 51);
152 
153  r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff;
154  r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff;
155  r2 += c;
156 
157  out[0] = r0;
158  out[1] = r1;
159  out[2] = r2;
160  out[3] = r3;
161  out[4] = r4;
162  }
163 
164 inline void fsquare(uint64_t out[5], const uint64_t in[5], size_t count = 1)
165  {
166  uint64_t r0 = in[0];
167  uint64_t r1 = in[1];
168  uint64_t r2 = in[2];
169  uint64_t r3 = in[3];
170  uint64_t r4 = in[4];
171 
172  for(size_t i = 0; i != count; ++i)
173  {
174  const uint64_t d0 = r0 * 2;
175  const uint64_t d1 = r1 * 2;
176  const uint64_t d2 = r2 * 2 * 19;
177  const uint64_t d419 = r4 * 19;
178  const uint64_t d4 = d419 * 2;
179 
180  uint128_t t0 = uint128_t(r0) * r0 + uint128_t(d4) * r1 + uint128_t(d2) * (r3 );
181  uint128_t t1 = uint128_t(d0) * r1 + uint128_t(d4) * r2 + uint128_t(r3) * (r3 * 19);
182  uint128_t t2 = uint128_t(d0) * r2 + uint128_t(r1) * r1 + uint128_t(d4) * (r3 );
183  uint128_t t3 = uint128_t(d0) * r3 + uint128_t(d1) * r2 + uint128_t(r4) * (d419 );
184  uint128_t t4 = uint128_t(d0) * r4 + uint128_t(d1) * r3 + uint128_t(r2) * (r2 );
185 
186  r0 = t0 & 0x7ffffffffffff; t1 += carry_shift(t0, 51);
187  r1 = t1 & 0x7ffffffffffff; t2 += carry_shift(t1, 51);
188  r2 = t2 & 0x7ffffffffffff; t3 += carry_shift(t2, 51);
189  r3 = t3 & 0x7ffffffffffff; t4 += carry_shift(t3, 51);
190  r4 = t4 & 0x7ffffffffffff; uint64_t c = carry_shift(t4, 51);
191 
192  r0 += c * 19; c = r0 >> 51; r0 = r0 & 0x7ffffffffffff;
193  r1 += c; c = r1 >> 51; r1 = r1 & 0x7ffffffffffff;
194  r2 += c;
195  }
196 
197  out[0] = r0;
198  out[1] = r1;
199  out[2] = r2;
200  out[3] = r3;
201  out[4] = r4;
202  }
203 
204 /* Take a little-endian, 32-byte number and expand it into polynomial form */
205 inline void fexpand(uint64_t *out, const uint8_t *in)
206  {
207  out[0] = load_le<uint64_t>(in, 0) & 0x7ffffffffffff;
208  out[1] = (load_le<uint64_t>(in+6, 0) >> 3) & 0x7ffffffffffff;
209  out[2] = (load_le<uint64_t>(in+12, 0) >> 6) & 0x7ffffffffffff;
210  out[3] = (load_le<uint64_t>(in+19, 0) >> 1) & 0x7ffffffffffff;
211  out[4] = (load_le<uint64_t>(in+24, 0) >> 12) & 0x7ffffffffffff;
212  }
213 
214 /* Take a fully reduced polynomial form number and contract it into a
215 * little-endian, 32-byte array
216 */
217 inline void fcontract(uint8_t *out, const uint64_t input[5])
218  {
219  uint128_t t0 = input[0];
220  uint128_t t1 = input[1];
221  uint128_t t2 = input[2];
222  uint128_t t3 = input[3];
223  uint128_t t4 = input[4];
224 
225  for(size_t i = 0; i != 2; ++i)
226  {
227  t1 += t0 >> 51; t0 &= 0x7ffffffffffff;
228  t2 += t1 >> 51; t1 &= 0x7ffffffffffff;
229  t3 += t2 >> 51; t2 &= 0x7ffffffffffff;
230  t4 += t3 >> 51; t3 &= 0x7ffffffffffff;
231  t0 += (t4 >> 51) * 19; t4 &= 0x7ffffffffffff;
232  }
233 
234  /* now t is between 0 and 2^255-1, properly carried. */
235  /* case 1: between 0 and 2^255-20. case 2: between 2^255-19 and 2^255-1. */
236 
237  t0 += 19;
238 
239  t1 += t0 >> 51; t0 &= 0x7ffffffffffff;
240  t2 += t1 >> 51; t1 &= 0x7ffffffffffff;
241  t3 += t2 >> 51; t2 &= 0x7ffffffffffff;
242  t4 += t3 >> 51; t3 &= 0x7ffffffffffff;
243  t0 += (t4 >> 51) * 19; t4 &= 0x7ffffffffffff;
244 
245  /* now between 19 and 2^255-1 in both cases, and offset by 19. */
246 
247  t0 += 0x8000000000000 - 19;
248  t1 += 0x8000000000000 - 1;
249  t2 += 0x8000000000000 - 1;
250  t3 += 0x8000000000000 - 1;
251  t4 += 0x8000000000000 - 1;
252 
253  /* now between 2^255 and 2^256-20, and offset by 2^255. */
254 
255  t1 += t0 >> 51; t0 &= 0x7ffffffffffff;
256  t2 += t1 >> 51; t1 &= 0x7ffffffffffff;
257  t3 += t2 >> 51; t2 &= 0x7ffffffffffff;
258  t4 += t3 >> 51; t3 &= 0x7ffffffffffff;
259  t4 &= 0x7ffffffffffff;
260 
261  store_le(out,
262  combine_lower(t0, 0, t1, 51),
263  combine_lower(t1, 13, t2, 38),
264  combine_lower(t2, 26, t3, 25),
265  combine_lower(t3, 39, t4, 12));
266  }
267 
268 /* Input: Q, Q', Q-Q'
269 * Out: 2Q, Q+Q'
270 *
271 * result.two_q (2*Q): long form
272 * result.q_plus_q_dash (Q + Q): long form
273 * in_q: short form, destroyed
274 * in_q_dash: short form, destroyed
275 * in_q_minus_q_dash: short form, preserved
276 */
277 void fmonty(uint64_t result_two_q_x[5],
278  uint64_t result_two_q_z[5],
279  uint64_t result_q_plus_q_dash_x[5],
280  uint64_t result_q_plus_q_dash_z[5],
281  uint64_t in_q_x[5],
282  uint64_t in_q_z[5],
283  uint64_t in_q_dash_x[5],
284  uint64_t in_q_dash_z[5],
285  const uint64_t q_minus_q_dash[5])
286  {
287  uint64_t zzz[5];
288  uint64_t xx[5];
289  uint64_t zz[5];
290  uint64_t xxprime[5];
291  uint64_t zzprime[5];
292  uint64_t zzzprime[5];
293 
294  fadd_sub(in_q_z, in_q_x);
295  fadd_sub(in_q_dash_z, in_q_dash_x);
296 
297  fmul(xxprime, in_q_dash_x, in_q_z);
298  fmul(zzprime, in_q_dash_z, in_q_x);
299 
300  fadd_sub(zzprime, xxprime);
301 
302  fsquare(result_q_plus_q_dash_x, xxprime);
303  fsquare(zzzprime, zzprime);
304  fmul(result_q_plus_q_dash_z, zzzprime, q_minus_q_dash);
305 
306  fsquare(xx, in_q_x);
307  fsquare(zz, in_q_z);
308  fmul(result_two_q_x, xx, zz);
309 
310  fdifference_backwards(zz, xx); // does zz = xx - zz
311  fscalar_product(zzz, zz, 121665);
312  fsum(zzz, xx);
313 
314  fmul(result_two_q_z, zz, zzz);
315  }
316 
317 /*
318 * Maybe swap the contents of two uint64_t arrays (@a and @b),
319 * Param @iswap is assumed to be either 0 or 1
320 *
321 * This function performs the swap without leaking any side-channel
322 * information.
323 */
324 inline void swap_conditional(uint64_t a[5], uint64_t b[5],
325  uint64_t c[5], uint64_t d[5],
326  uint64_t iswap)
327  {
328  const uint64_t swap = 0 - iswap;
329 
330  for(size_t i = 0; i < 5; ++i)
331  {
332  const uint64_t x0 = swap & (a[i] ^ b[i]);
333  const uint64_t x1 = swap & (c[i] ^ d[i]);
334  a[i] ^= x0;
335  b[i] ^= x0;
336  c[i] ^= x1;
337  d[i] ^= x1;
338  }
339  }
340 
341 /* Calculates nQ where Q is the x-coordinate of a point on the curve
342 *
343 * resultx/resultz: the x/z coordinate of the resulting curve point (short form)
344 * n: a little endian, 32-byte number
345 * q: a point of the curve (short form)
346 */
347 void cmult(uint64_t resultx[5], uint64_t resultz[5], const uint8_t n[32], const uint64_t q[5])
348  {
349  uint64_t a[5] = {0}; // nqpqx
350  uint64_t b[5] = {1}; // npqpz
351  uint64_t c[5] = {1}; // nqx
352  uint64_t d[5] = {0}; // nqz
353  uint64_t e[5] = {0}; // npqqx2
354  uint64_t f[5] = {1}; // npqqz2
355  uint64_t g[5] = {0}; // nqx2
356  uint64_t h[5] = {1}; // nqz2
357 
358  copy_mem(a, q, 5);
359 
360  for(size_t i = 0; i < 32; ++i)
361  {
362  const uint64_t bit0 = (n[31 - i] >> 7) & 1;
363  const uint64_t bit1 = (n[31 - i] >> 6) & 1;
364  const uint64_t bit2 = (n[31 - i] >> 5) & 1;
365  const uint64_t bit3 = (n[31 - i] >> 4) & 1;
366  const uint64_t bit4 = (n[31 - i] >> 3) & 1;
367  const uint64_t bit5 = (n[31 - i] >> 2) & 1;
368  const uint64_t bit6 = (n[31 - i] >> 1) & 1;
369  const uint64_t bit7 = (n[31 - i] >> 0) & 1;
370 
371  swap_conditional(c, a, d, b, bit0);
372  fmonty(g, h, e, f, c, d, a, b, q);
373 
374  swap_conditional(g, e, h, f, bit0 ^ bit1);
375  fmonty(c, d, a, b, g, h, e, f, q);
376 
377  swap_conditional(c, a, d, b, bit1 ^ bit2);
378  fmonty(g, h, e, f, c, d, a, b, q);
379 
380  swap_conditional(g, e, h, f, bit2 ^ bit3);
381  fmonty(c, d, a, b, g, h, e, f, q);
382 
383  swap_conditional(c, a, d, b, bit3 ^ bit4);
384  fmonty(g, h, e, f, c, d, a, b, q);
385 
386  swap_conditional(g, e, h, f, bit4 ^ bit5);
387  fmonty(c, d, a, b, g, h, e, f, q);
388 
389  swap_conditional(c, a, d, b, bit5 ^ bit6);
390  fmonty(g, h, e, f, c, d, a, b, q);
391 
392  swap_conditional(g, e, h, f, bit6 ^ bit7);
393  fmonty(c, d, a, b, g, h, e, f, q);
394 
395  swap_conditional(c, a, d, b, bit7);
396  }
397 
398  copy_mem(resultx, c, 5);
399  copy_mem(resultz, d, 5);
400  }
401 
402 
403 // -----------------------------------------------------------------------------
404 // Shamelessly copied from djb's code, tightened a little
405 // -----------------------------------------------------------------------------
406 void crecip(uint64_t out[5], const uint64_t z[5])
407  {
408  uint64_t a[5];
409  uint64_t b[5];
410  uint64_t c[5];
411  uint64_t t0[5];
412 
413  fsquare(a, z); // 2
414  fsquare(t0, a, 2); // 8
415  fmul(b, t0, z); // 9
416  fmul(a, b, a); // 11
417  fsquare(t0, a); // 22
418  fmul(b, t0, b); // 2^5 - 2^0 = 31
419  fsquare(t0, b, 5); // 2^10 - 2^5
420  fmul(b, t0, b); // 2^10 - 2^0
421  fsquare(t0, b, 10); // 2^20 - 2^10
422  fmul(c, t0, b); // 2^20 - 2^0
423  fsquare(t0, c, 20); // 2^40 - 2^20
424  fmul(t0, t0, c); // 2^40 - 2^0
425  fsquare(t0, t0, 10); // 2^50 - 2^10
426  fmul(b, t0, b); // 2^50 - 2^0
427  fsquare(t0, b, 50); // 2^100 - 2^50
428  fmul(c, t0, b); // 2^100 - 2^0
429  fsquare(t0, c, 100); // 2^200 - 2^100
430  fmul(t0, t0, c); // 2^200 - 2^0
431  fsquare(t0, t0, 50); // 2^250 - 2^50
432  fmul(t0, t0, b); // 2^250 - 2^0
433  fsquare(t0, t0, 5); // 2^255 - 2^5
434  fmul(out, t0, a); // 2^255 - 21
435  }
436 
437 }
438 
439 void
440 curve25519_donna(uint8_t mypublic[32], const uint8_t secret[32], const uint8_t basepoint[32])
441  {
442  CT::poison(secret, 32);
443  CT::poison(basepoint, 32);
444 
445  uint64_t bp[5], x[5], z[5], zmone[5];
446  uint8_t e[32];
447 
448  copy_mem(e, secret, 32);
449  e[ 0] &= 248;
450  e[31] &= 127;
451  e[31] |= 64;
452 
453  fexpand(bp, basepoint);
454  cmult(x, z, e, bp);
455  crecip(zmone, z);
456  fmul(z, x, zmone);
457  fcontract(mypublic, z);
458 
459  CT::unpoison(secret, 32);
460  CT::unpoison(basepoint, 32);
461  CT::unpoison(mypublic, 32);
462  }
463 
464 }
void curve25519_donna(uint8_t mypublic[32], const uint8_t secret[32], const uint8_t basepoint[32])
Definition: donna.cpp:440
void poison(const T *p, size_t n)
Definition: ct_utils.h:48
uint64_t carry_shift(const donna128 &a, size_t shift)
Definition: donna128.h:116
uint64_t combine_lower(const donna128 &a, size_t s1, const donna128 &b, size_t s2)
Definition: donna128.h:121
uint64_t load_le< uint64_t >(const uint8_t in[], size_t off)
Definition: loadstor.h:235
void copy_mem(T *out, const T *in, size_t n)
Definition: mem_ops.h:122
Definition: alg_id.cpp:13
void unpoison(const T *p, size_t n)
Definition: ct_utils.h:59
void store_le(uint16_t in, uint8_t out[2])
Definition: loadstor.h:452