bn_sqrt.c revision 296341
1/* crypto/bn/bn_sqrt.c */ 2/* 3 * Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de> and Bodo 4 * Moeller for the OpenSSL project. 5 */ 6/* ==================================================================== 7 * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved. 8 * 9 * Redistribution and use in source and binary forms, with or without 10 * modification, are permitted provided that the following conditions 11 * are met: 12 * 13 * 1. Redistributions of source code must retain the above copyright 14 * notice, this list of conditions and the following disclaimer. 15 * 16 * 2. Redistributions in binary form must reproduce the above copyright 17 * notice, this list of conditions and the following disclaimer in 18 * the documentation and/or other materials provided with the 19 * distribution. 20 * 21 * 3. All advertising materials mentioning features or use of this 22 * software must display the following acknowledgment: 23 * "This product includes software developed by the OpenSSL Project 24 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 25 * 26 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 27 * endorse or promote products derived from this software without 28 * prior written permission. For written permission, please contact 29 * openssl-core@openssl.org. 30 * 31 * 5. Products derived from this software may not be called "OpenSSL" 32 * nor may "OpenSSL" appear in their names without prior written 33 * permission of the OpenSSL Project. 34 * 35 * 6. Redistributions of any form whatsoever must retain the following 36 * acknowledgment: 37 * "This product includes software developed by the OpenSSL Project 38 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 39 * 40 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 41 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 42 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 43 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 44 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 45 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 46 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 47 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 48 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 49 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 50 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 51 * OF THE POSSIBILITY OF SUCH DAMAGE. 52 * ==================================================================== 53 * 54 * This product includes cryptographic software written by Eric Young 55 * (eay@cryptsoft.com). This product includes software written by Tim 56 * Hudson (tjh@cryptsoft.com). 57 * 58 */ 59 60#include "cryptlib.h" 61#include "bn_lcl.h" 62 63BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 64/* 65 * Returns 'ret' such that ret^2 == a (mod p), using the Tonelli/Shanks 66 * algorithm (cf. Henri Cohen, "A Course in Algebraic Computational Number 67 * Theory", algorithm 1.5.1). 'p' must be prime! 68 */ 69{ 70 BIGNUM *ret = in; 71 int err = 1; 72 int r; 73 BIGNUM *A, *b, *q, *t, *x, *y; 74 int e, i, j; 75 76 if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) { 77 if (BN_abs_is_word(p, 2)) { 78 if (ret == NULL) 79 ret = BN_new(); 80 if (ret == NULL) 81 goto end; 82 if (!BN_set_word(ret, BN_is_bit_set(a, 0))) { 83 if (ret != in) 84 BN_free(ret); 85 return NULL; 86 } 87 bn_check_top(ret); 88 return ret; 89 } 90 91 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); 92 return (NULL); 93 } 94 95 if (BN_is_zero(a) || BN_is_one(a)) { 96 if (ret == NULL) 97 ret = BN_new(); 98 if (ret == NULL) 99 goto end; 100 if (!BN_set_word(ret, BN_is_one(a))) { 101 if (ret != in) 102 BN_free(ret); 103 return NULL; 104 } 105 bn_check_top(ret); 106 return ret; 107 } 108 109 BN_CTX_start(ctx); 110 A = BN_CTX_get(ctx); 111 b = BN_CTX_get(ctx); 112 q = BN_CTX_get(ctx); 113 t = BN_CTX_get(ctx); 114 x = BN_CTX_get(ctx); 115 y = BN_CTX_get(ctx); 116 if (y == NULL) 117 goto end; 118 119 if (ret == NULL) 120 ret = BN_new(); 121 if (ret == NULL) 122 goto end; 123 124 /* A = a mod p */ 125 if (!BN_nnmod(A, a, p, ctx)) 126 goto end; 127 128 /* now write |p| - 1 as 2^e*q where q is odd */ 129 e = 1; 130 while (!BN_is_bit_set(p, e)) 131 e++; 132 /* we'll set q later (if needed) */ 133 134 if (e == 1) { 135 /*- 136 * The easy case: (|p|-1)/2 is odd, so 2 has an inverse 137 * modulo (|p|-1)/2, and square roots can be computed 138 * directly by modular exponentiation. 139 * We have 140 * 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2), 141 * so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1. 142 */ 143 if (!BN_rshift(q, p, 2)) 144 goto end; 145 q->neg = 0; 146 if (!BN_add_word(q, 1)) 147 goto end; 148 if (!BN_mod_exp(ret, A, q, p, ctx)) 149 goto end; 150 err = 0; 151 goto vrfy; 152 } 153 154 if (e == 2) { 155 /*- 156 * |p| == 5 (mod 8) 157 * 158 * In this case 2 is always a non-square since 159 * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime. 160 * So if a really is a square, then 2*a is a non-square. 161 * Thus for 162 * b := (2*a)^((|p|-5)/8), 163 * i := (2*a)*b^2 164 * we have 165 * i^2 = (2*a)^((1 + (|p|-5)/4)*2) 166 * = (2*a)^((p-1)/2) 167 * = -1; 168 * so if we set 169 * x := a*b*(i-1), 170 * then 171 * x^2 = a^2 * b^2 * (i^2 - 2*i + 1) 172 * = a^2 * b^2 * (-2*i) 173 * = a*(-i)*(2*a*b^2) 174 * = a*(-i)*i 175 * = a. 176 * 177 * (This is due to A.O.L. Atkin, 178 * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>, 179 * November 1992.) 180 */ 181 182 /* t := 2*a */ 183 if (!BN_mod_lshift1_quick(t, A, p)) 184 goto end; 185 186 /* b := (2*a)^((|p|-5)/8) */ 187 if (!BN_rshift(q, p, 3)) 188 goto end; 189 q->neg = 0; 190 if (!BN_mod_exp(b, t, q, p, ctx)) 191 goto end; 192 193 /* y := b^2 */ 194 if (!BN_mod_sqr(y, b, p, ctx)) 195 goto end; 196 197 /* t := (2*a)*b^2 - 1 */ 198 if (!BN_mod_mul(t, t, y, p, ctx)) 199 goto end; 200 if (!BN_sub_word(t, 1)) 201 goto end; 202 203 /* x = a*b*t */ 204 if (!BN_mod_mul(x, A, b, p, ctx)) 205 goto end; 206 if (!BN_mod_mul(x, x, t, p, ctx)) 207 goto end; 208 209 if (!BN_copy(ret, x)) 210 goto end; 211 err = 0; 212 goto vrfy; 213 } 214 215 /* 216 * e > 2, so we really have to use the Tonelli/Shanks algorithm. First, 217 * find some y that is not a square. 218 */ 219 if (!BN_copy(q, p)) 220 goto end; /* use 'q' as temp */ 221 q->neg = 0; 222 i = 2; 223 do { 224 /* 225 * For efficiency, try small numbers first; if this fails, try random 226 * numbers. 227 */ 228 if (i < 22) { 229 if (!BN_set_word(y, i)) 230 goto end; 231 } else { 232 if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) 233 goto end; 234 if (BN_ucmp(y, p) >= 0) { 235 if (!(p->neg ? BN_add : BN_sub) (y, y, p)) 236 goto end; 237 } 238 /* now 0 <= y < |p| */ 239 if (BN_is_zero(y)) 240 if (!BN_set_word(y, i)) 241 goto end; 242 } 243 244 r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */ 245 if (r < -1) 246 goto end; 247 if (r == 0) { 248 /* m divides p */ 249 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); 250 goto end; 251 } 252 } 253 while (r == 1 && ++i < 82); 254 255 if (r != -1) { 256 /* 257 * Many rounds and still no non-square -- this is more likely a bug 258 * than just bad luck. Even if p is not prime, we should have found 259 * some y such that r == -1. 260 */ 261 BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS); 262 goto end; 263 } 264 265 /* Here's our actual 'q': */ 266 if (!BN_rshift(q, q, e)) 267 goto end; 268 269 /* 270 * Now that we have some non-square, we can find an element of order 2^e 271 * by computing its q'th power. 272 */ 273 if (!BN_mod_exp(y, y, q, p, ctx)) 274 goto end; 275 if (BN_is_one(y)) { 276 BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); 277 goto end; 278 } 279 280 /*- 281 * Now we know that (if p is indeed prime) there is an integer 282 * k, 0 <= k < 2^e, such that 283 * 284 * a^q * y^k == 1 (mod p). 285 * 286 * As a^q is a square and y is not, k must be even. 287 * q+1 is even, too, so there is an element 288 * 289 * X := a^((q+1)/2) * y^(k/2), 290 * 291 * and it satisfies 292 * 293 * X^2 = a^q * a * y^k 294 * = a, 295 * 296 * so it is the square root that we are looking for. 297 */ 298 299 /* t := (q-1)/2 (note that q is odd) */ 300 if (!BN_rshift1(t, q)) 301 goto end; 302 303 /* x := a^((q-1)/2) */ 304 if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */ 305 if (!BN_nnmod(t, A, p, ctx)) 306 goto end; 307 if (BN_is_zero(t)) { 308 /* special case: a == 0 (mod p) */ 309 BN_zero(ret); 310 err = 0; 311 goto end; 312 } else if (!BN_one(x)) 313 goto end; 314 } else { 315 if (!BN_mod_exp(x, A, t, p, ctx)) 316 goto end; 317 if (BN_is_zero(x)) { 318 /* special case: a == 0 (mod p) */ 319 BN_zero(ret); 320 err = 0; 321 goto end; 322 } 323 } 324 325 /* b := a*x^2 (= a^q) */ 326 if (!BN_mod_sqr(b, x, p, ctx)) 327 goto end; 328 if (!BN_mod_mul(b, b, A, p, ctx)) 329 goto end; 330 331 /* x := a*x (= a^((q+1)/2)) */ 332 if (!BN_mod_mul(x, x, A, p, ctx)) 333 goto end; 334 335 while (1) { 336 /*- 337 * Now b is a^q * y^k for some even k (0 <= k < 2^E 338 * where E refers to the original value of e, which we 339 * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2). 340 * 341 * We have a*b = x^2, 342 * y^2^(e-1) = -1, 343 * b^2^(e-1) = 1. 344 */ 345 346 if (BN_is_one(b)) { 347 if (!BN_copy(ret, x)) 348 goto end; 349 err = 0; 350 goto vrfy; 351 } 352 353 /* find smallest i such that b^(2^i) = 1 */ 354 i = 1; 355 if (!BN_mod_sqr(t, b, p, ctx)) 356 goto end; 357 while (!BN_is_one(t)) { 358 i++; 359 if (i == e) { 360 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); 361 goto end; 362 } 363 if (!BN_mod_mul(t, t, t, p, ctx)) 364 goto end; 365 } 366 367 /* t := y^2^(e - i - 1) */ 368 if (!BN_copy(t, y)) 369 goto end; 370 for (j = e - i - 1; j > 0; j--) { 371 if (!BN_mod_sqr(t, t, p, ctx)) 372 goto end; 373 } 374 if (!BN_mod_mul(y, t, t, p, ctx)) 375 goto end; 376 if (!BN_mod_mul(x, x, t, p, ctx)) 377 goto end; 378 if (!BN_mod_mul(b, b, y, p, ctx)) 379 goto end; 380 e = i; 381 } 382 383 vrfy: 384 if (!err) { 385 /* 386 * verify the result -- the input might have been not a square (test 387 * added in 0.9.8) 388 */ 389 390 if (!BN_mod_sqr(x, ret, p, ctx)) 391 err = 1; 392 393 if (!err && 0 != BN_cmp(x, A)) { 394 BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); 395 err = 1; 396 } 397 } 398 399 end: 400 if (err) { 401 if (ret != NULL && ret != in) { 402 BN_clear_free(ret); 403 } 404 ret = NULL; 405 } 406 BN_CTX_end(ctx); 407 bn_check_top(ret); 408 return ret; 409} 410