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mp_karat.cpp
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1 /*
2 * Multiplication and Squaring
3 * (C) 1999-2010,2018 Jack Lloyd
4 * 2016 Matthias Gierlings
5 *
6 * Botan is released under the Simplified BSD License (see license.txt)
7 */
8
9 #include <botan/internal/mp_core.h>
10 #include <botan/internal/mp_asmi.h>
11 #include <botan/internal/ct_utils.h>
12 #include <botan/mem_ops.h>
13 #include <botan/exceptn.h>
14
15 namespace Botan {
16
17 namespace {
18
19 const size_t KARATSUBA_MULTIPLY_THRESHOLD = 32;
20 const size_t KARATSUBA_SQUARE_THRESHOLD = 32;
21
22 /*
23 * Simple O(N^2) Multiplication
24 */
25 void basecase_mul(word z[], size_t z_size,
26 const word x[], size_t x_size,
27 const word y[], size_t y_size)
28 {
29 if(z_size < x_size + y_size)
30 throw Invalid_Argument("basecase_mul z_size too small");
31
32 const size_t x_size_8 = x_size - (x_size % 8);
33
34 clear_mem(z, z_size);
35
36 for(size_t i = 0; i != y_size; ++i)
37 {
38 const word y_i = y[i];
39
40 word carry = 0;
41
42 for(size_t j = 0; j != x_size_8; j += 8)
43 carry = word8_madd3(z + i + j, x + j, y_i, carry);
44
45 for(size_t j = x_size_8; j != x_size; ++j)
46 z[i+j] = word_madd3(x[j], y_i, z[i+j], &carry);
47
48 z[x_size+i] = carry;
49 }
50 }
51
52 void basecase_sqr(word z[], size_t z_size,
53 const word x[], size_t x_size)
54 {
55 if(z_size < 2*x_size)
56 throw Invalid_Argument("basecase_sqr z_size too small");
57
58 const size_t x_size_8 = x_size - (x_size % 8);
59
60 clear_mem(z, z_size);
61
62 for(size_t i = 0; i != x_size; ++i)
63 {
64 const word x_i = x[i];
65
66 word carry = 0;
67
68 for(size_t j = 0; j != x_size_8; j += 8)
69 carry = word8_madd3(z + i + j, x + j, x_i, carry);
70
71 for(size_t j = x_size_8; j != x_size; ++j)
72 z[i+j] = word_madd3(x[j], x_i, z[i+j], &carry);
73
74 z[x_size+i] = carry;
75 }
76 }
77
78 /*
79 * Karatsuba Multiplication Operation
80 */
81 void karatsuba_mul(word z[], const word x[], const word y[], size_t N,
82 word workspace[])
83 {
84 if(N < KARATSUBA_MULTIPLY_THRESHOLD || N % 2)
85 {
86 switch(N)
87 {
88 case 6:
89 return bigint_comba_mul6(z, x, y);
90 case 8:
91 return bigint_comba_mul8(z, x, y);
92 case 9:
93 return bigint_comba_mul9(z, x, y);
94 case 16:
95 return bigint_comba_mul16(z, x, y);
96 case 24:
97 return bigint_comba_mul24(z, x, y);
98 default:
99 return basecase_mul(z, 2*N, x, N, y, N);
100 }
101 }
102
103 const size_t N2 = N / 2;
104
105 const word* x0 = x;
106 const word* x1 = x + N2;
107 const word* y0 = y;
108 const word* y1 = y + N2;
109 word* z0 = z;
110 word* z1 = z + N;
111
112 word* ws0 = workspace;
113 word* ws1 = workspace + N;
114
115 clear_mem(workspace, 2*N);
116
117 /*
118 * If either of cmp0 or cmp1 is zero then z0 or z1 resp is zero here,
119 * resulting in a no-op - z0*z1 will be equal to zero so we don't need to do
120 * anything, clear_mem above already set the correct result.
121 *
122 * However we ignore the result of the comparisons and always perform the
123 * subtractions and recursively multiply to avoid the timing channel.
124 */
125
126 // First compute (X_lo - X_hi)*(Y_hi - Y_lo)
127 const auto cmp0 = bigint_sub_abs(z0, x0, x1, N2, workspace);
128 const auto cmp1 = bigint_sub_abs(z1, y1, y0, N2, workspace);
129 const auto neg_mask = ~(cmp0 ^ cmp1);
130
131 karatsuba_mul(ws0, z0, z1, N2, ws1);
132
133 // Compute X_lo * Y_lo
134 karatsuba_mul(z0, x0, y0, N2, ws1);
135
136 // Compute X_hi * Y_hi
137 karatsuba_mul(z1, x1, y1, N2, ws1);
138
139 const word ws_carry = bigint_add3_nc(ws1, z0, N, z1, N);
140 word z_carry = bigint_add2_nc(z + N2, N, ws1, N);
141
142 z_carry += bigint_add2_nc(z + N + N2, N2, &ws_carry, 1);
143 bigint_add2_nc(z + N + N2, N2, &z_carry, 1);
144
145 clear_mem(workspace + N, N2);
146
147 bigint_cnd_add_or_sub(neg_mask, z + N2, workspace, 2*N-N2);
148 }
149
150 /*
151 * Karatsuba Squaring Operation
152 */
153 void karatsuba_sqr(word z[], const word x[], size_t N, word workspace[])
154 {
155 if(N < KARATSUBA_SQUARE_THRESHOLD || N % 2)
156 {
157 switch(N)
158 {
159 case 6:
160 return bigint_comba_sqr6(z, x);
161 case 8:
162 return bigint_comba_sqr8(z, x);
163 case 9:
164 return bigint_comba_sqr9(z, x);
165 case 16:
166 return bigint_comba_sqr16(z, x);
167 case 24:
168 return bigint_comba_sqr24(z, x);
169 default:
170 return basecase_sqr(z, 2*N, x, N);
171 }
172 }
173
174 const size_t N2 = N / 2;
175
176 const word* x0 = x;
177 const word* x1 = x + N2;
178 word* z0 = z;
179 word* z1 = z + N;
180
181 word* ws0 = workspace;
182 word* ws1 = workspace + N;
183
184 clear_mem(workspace, 2*N);
185
186 // See comment in karatsuba_mul
187 bigint_sub_abs(z0, x0, x1, N2, workspace);
188 karatsuba_sqr(ws0, z0, N2, ws1);
189
190 karatsuba_sqr(z0, x0, N2, ws1);
191 karatsuba_sqr(z1, x1, N2, ws1);
192
193 const word ws_carry = bigint_add3_nc(ws1, z0, N, z1, N);
194 word z_carry = bigint_add2_nc(z + N2, N, ws1, N);
195
196 z_carry += bigint_add2_nc(z + N + N2, N2, &ws_carry, 1);
197 bigint_add2_nc(z + N + N2, N2, &z_carry, 1);
198
199 /*
200 * This is only actually required if cmp (result of bigint_sub_abs) is != 0,
201 * however if cmp==0 then ws0[0:N] == 0 and avoiding the jump hides a
202 * timing channel.
203 */
204 bigint_sub2(z + N2, 2*N-N2, ws0, N);
205 }
206
207 /*
208 * Pick a good size for the Karatsuba multiply
209 */
210 size_t karatsuba_size(size_t z_size,
211 size_t x_size, size_t x_sw,
212 size_t y_size, size_t y_sw)
213 {
214 if(x_sw > x_size || x_sw > y_size || y_sw > x_size || y_sw > y_size)
215 return 0;
216
217 if(((x_size == x_sw) && (x_size % 2)) ||
218 ((y_size == y_sw) && (y_size % 2)))
219 return 0;
220
221 const size_t start = (x_sw > y_sw) ? x_sw : y_sw;
222 const size_t end = (x_size < y_size) ? x_size : y_size;
223
224 if(start == end)
225 {
226 if(start % 2)
227 return 0;
228 return start;
229 }
230
231 for(size_t j = start; j <= end; ++j)
232 {
233 if(j % 2)
234 continue;
235
236 if(2*j > z_size)
237 return 0;
238
239 if(x_sw <= j && j <= x_size && y_sw <= j && j <= y_size)
240 {
241 if(j % 4 == 2 &&
242 (j+2) <= x_size && (j+2) <= y_size && 2*(j+2) <= z_size)
243 return j+2;
244 return j;
245 }
246 }
247
248 return 0;
249 }
250
251 /*
252 * Pick a good size for the Karatsuba squaring
253 */
254 size_t karatsuba_size(size_t z_size, size_t x_size, size_t x_sw)
255 {
256 if(x_sw == x_size)
257 {
258 if(x_sw % 2)
259 return 0;
260 return x_sw;
261 }
262
263 for(size_t j = x_sw; j <= x_size; ++j)
264 {
265 if(j % 2)
266 continue;
267
268 if(2*j > z_size)
269 return 0;
270
271 if(j % 4 == 2 && (j+2) <= x_size && 2*(j+2) <= z_size)
272 return j+2;
273 return j;
274 }
275
276 return 0;
277 }
278
279 template<size_t SZ>
280 inline bool sized_for_comba_mul(size_t x_sw, size_t x_size,
281 size_t y_sw, size_t y_size,
282 size_t z_size)
283 {
284 return (x_sw <= SZ && x_size >= SZ &&
285 y_sw <= SZ && y_size >= SZ &&
286 z_size >= 2*SZ);
287 }
288
289 template<size_t SZ>
290 inline bool sized_for_comba_sqr(size_t x_sw, size_t x_size,
291 size_t z_size)
292 {
293 return (x_sw <= SZ && x_size >= SZ && z_size >= 2*SZ);
294 }
295
296 }
297
298 void bigint_mul(word z[], size_t z_size,
299 const word x[], size_t x_size, size_t x_sw,
300 const word y[], size_t y_size, size_t y_sw,
301 word workspace[], size_t ws_size)
302 {
303 clear_mem(z, z_size);
304
305 if(x_sw == 1)
306 {
307 bigint_linmul3(z, y, y_sw, x[0]);
308 }
309 else if(y_sw == 1)
310 {
311 bigint_linmul3(z, x, x_sw, y[0]);
312 }
313 else if(sized_for_comba_mul<4>(x_sw, x_size, y_sw, y_size, z_size))
314 {
315 bigint_comba_mul4(z, x, y);
316 }
317 else if(sized_for_comba_mul<6>(x_sw, x_size, y_sw, y_size, z_size))
318 {
319 bigint_comba_mul6(z, x, y);
320 }
321 else if(sized_for_comba_mul<8>(x_sw, x_size, y_sw, y_size, z_size))
322 {
323 bigint_comba_mul8(z, x, y);
324 }
325 else if(sized_for_comba_mul<9>(x_sw, x_size, y_sw, y_size, z_size))
326 {
327 bigint_comba_mul9(z, x, y);
328 }
329 else if(sized_for_comba_mul<16>(x_sw, x_size, y_sw, y_size, z_size))
330 {
331 bigint_comba_mul16(z, x, y);
332 }
333 else if(sized_for_comba_mul<24>(x_sw, x_size, y_sw, y_size, z_size))
334 {
335 bigint_comba_mul24(z, x, y);
336 }
337 else if(x_sw < KARATSUBA_MULTIPLY_THRESHOLD ||
338 y_sw < KARATSUBA_MULTIPLY_THRESHOLD ||
339 !workspace)
340 {
341 basecase_mul(z, z_size, x, x_sw, y, y_sw);
342 }
343 else
344 {
345 const size_t N = karatsuba_size(z_size, x_size, x_sw, y_size, y_sw);
346
347 if(N && z_size >= 2*N && ws_size >= 2*N)
348 karatsuba_mul(z, x, y, N, workspace);
349 else
350 basecase_mul(z, z_size, x, x_sw, y, y_sw);
351 }
352 }
353
354 /*
355 * Squaring Algorithm Dispatcher
356 */
357 void bigint_sqr(word z[], size_t z_size,
358 const word x[], size_t x_size, size_t x_sw,
359 word workspace[], size_t ws_size)
360 {
361 clear_mem(z, z_size);
362
363 BOTAN_ASSERT(z_size/2 >= x_sw, "Output size is sufficient");
364
365 if(x_sw == 1)
366 {
367 bigint_linmul3(z, x, x_sw, x[0]);
368 }
369 else if(sized_for_comba_sqr<4>(x_sw, x_size, z_size))
370 {
371 bigint_comba_sqr4(z, x);
372 }
373 else if(sized_for_comba_sqr<6>(x_sw, x_size, z_size))
374 {
375 bigint_comba_sqr6(z, x);
376 }
377 else if(sized_for_comba_sqr<8>(x_sw, x_size, z_size))
378 {
379 bigint_comba_sqr8(z, x);
380 }
381 else if(sized_for_comba_sqr<9>(x_sw, x_size, z_size))
382 {
383 bigint_comba_sqr9(z, x);
384 }
385 else if(sized_for_comba_sqr<16>(x_sw, x_size, z_size))
386 {
387 bigint_comba_sqr16(z, x);
388 }
389 else if(sized_for_comba_sqr<24>(x_sw, x_size, z_size))
390 {
391 bigint_comba_sqr24(z, x);
392 }
393 else if(x_size < KARATSUBA_SQUARE_THRESHOLD || !workspace)
394 {
395 basecase_sqr(z, z_size, x, x_sw);
396 }
397 else
398 {
399 const size_t N = karatsuba_size(z_size, x_size, x_sw);
400
401 if(N && z_size >= 2*N && ws_size >= 2*N)
402 karatsuba_sqr(z, x, N, workspace);
403 else
404 basecase_sqr(z, z_size, x, x_sw);
405 }
406 }
407
408 }
Botan::carry
void carry(int64_t &h0, int64_t &h1)
Definition: ed25519_internal.h:32
Botan::clear_mem
void clear_mem(T *ptr, size_t n)
Definition: mem_ops.h:112
Botan::bigint_sub2
word bigint_sub2(word x[], size_t x_size, const word y[], size_t y_size)
Definition: mp_core.h:300
Botan::bigint_add3_nc
word bigint_add3_nc(word z[], const word x[], size_t x_size, const word y[], size_t y_size)
Definition: mp_core.h:250
Botan::word_madd3
word word_madd3(word a, word b, word c, word *d)
Definition: mp_madd.h:92
Botan::bigint_comba_mul4
void bigint_comba_mul4(word z[8], const word x[4], const word y[4])
Definition: mp_comba.cpp:50
Botan::bigint_comba_mul9
void bigint_comba_mul9(word z[18], const word x[9], const word y[9])
Definition: mp_comba.cpp:474
Botan::bigint_sqr
void bigint_sqr(word z[], size_t z_size, const word x[], size_t x_size, size_t x_sw, word workspace[], size_t ws_size)
Definition: mp_karat.cpp:357
Botan::bigint_comba_mul24
void bigint_comba_mul24(word z[48], const word x[24], const word y[24])
Definition: mp_comba.cpp:1535
BOTAN_ASSERT
#define BOTAN_ASSERT(expr, assertion_made)
Definition: assert.h:55
Botan::bigint_linmul3
void bigint_linmul3(word z[], const word x[], size_t x_size, word y)
Definition: mp_core.h:502
Botan::bigint_comba_sqr16
void bigint_comba_sqr16(word z[32], const word x[16])
Definition: mp_comba.cpp:598
Botan::word8_madd3
word word8_madd3(word z[8], const word x[8], word y, word carry)
Definition: mp_asmi.h:415
Botan::bigint_sub_abs
CT::Mask< word > bigint_sub_abs(word z[], const word x[], const word y[], size_t N, word ws[])
Definition: mp_core.h:377
Botan::bigint_comba_sqr24
void bigint_comba_sqr24(word z[48], const word x[24])
Definition: mp_comba.cpp:1132
Botan::bigint_add2_nc
word bigint_add2_nc(word x[], size_t x_size, const word y[], size_t y_size)
Definition: mp_core.h:227
Botan::bigint_cnd_add_or_sub
void bigint_cnd_add_or_sub(CT::Mask< word > mask, word x[], const word y[], size_t size)
Definition: mp_core.h:139
Botan
Definition: alg_id.cpp:13
Botan::bigint_comba_sqr9
void bigint_comba_sqr9(word z[18], const word x[9])
Definition: mp_comba.cpp:386
Botan::bigint_mul
void bigint_mul(word z[], size_t z_size, const word x[], size_t x_size, size_t x_sw, const word y[], size_t y_size, size_t y_sw, word workspace[], size_t ws_size)
Definition: mp_karat.cpp:298
Botan::bigint_comba_mul8
void bigint_comba_mul8(word z[16], const word x[8], const word y[8])
Definition: mp_comba.cpp:283
Botan::bigint_comba_sqr8
void bigint_comba_sqr8(word z[16], const word x[8])
Definition: mp_comba.cpp:208
Botan::bigint_comba_mul16
void bigint_comba_mul16(word z[32], const word x[16], const word y[16])
Definition: mp_comba.cpp:805
Botan::bigint_comba_mul6
void bigint_comba_mul6(word z[12], const word x[6], const word y[6])
Definition: mp_comba.cpp:141
Botan::bigint_comba_sqr4
void bigint_comba_sqr4(word z[8], const word x[4])
Definition: mp_comba.cpp:17
Botan::bigint_comba_sqr6
void bigint_comba_sqr6(word z[12], const word x[6])
Definition: mp_comba.cpp:89