1 /* Written by Lenka Fibikova <fibikova (at) exp-math.uni-essen.de> 2 * and Bodo Moeller for the OpenSSL project. */ 3 /* ==================================================================== 4 * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved. 5 * 6 * Redistribution and use in source and binary forms, with or without 7 * modification, are permitted provided that the following conditions 8 * are met: 9 * 10 * 1. Redistributions of source code must retain the above copyright 11 * notice, this list of conditions and the following disclaimer. 12 * 13 * 2. Redistributions in binary form must reproduce the above copyright 14 * notice, this list of conditions and the following disclaimer in 15 * the documentation and/or other materials provided with the 16 * distribution. 17 * 18 * 3. All advertising materials mentioning features or use of this 19 * software must display the following acknowledgment: 20 * "This product includes software developed by the OpenSSL Project 21 * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" 22 * 23 * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to 24 * endorse or promote products derived from this software without 25 * prior written permission. For written permission, please contact 26 * openssl-core (at) openssl.org. 27 * 28 * 5. Products derived from this software may not be called "OpenSSL" 29 * nor may "OpenSSL" appear in their names without prior written 30 * permission of the OpenSSL Project. 31 * 32 * 6. Redistributions of any form whatsoever must retain the following 33 * acknowledgment: 34 * "This product includes software developed by the OpenSSL Project 35 * for use in the OpenSSL Toolkit (http://www.openssl.org/)" 36 * 37 * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY 38 * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 39 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR 40 * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR 41 * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, 42 * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT 43 * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; 44 * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 45 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, 46 * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) 47 * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED 48 * OF THE POSSIBILITY OF SUCH DAMAGE. 49 * ==================================================================== 50 * 51 * This product includes cryptographic software written by Eric Young 52 * (eay (at) cryptsoft.com). This product includes software written by Tim 53 * Hudson (tjh (at) cryptsoft.com). */ 54 55 #include <openssl/bn.h> 56 57 #include <openssl/err.h> 58 59 #include "internal.h" 60 61 62 BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) { 63 // Compute a square root of |a| mod |p| using the Tonelli/Shanks algorithm 64 // (cf. Henri Cohen, "A Course in Algebraic Computational Number Theory", 65 // algorithm 1.5.1). |p| is assumed to be a prime. 66 67 BIGNUM *ret = in; 68 int err = 1; 69 int r; 70 BIGNUM *A, *b, *q, *t, *x, *y; 71 int e, i, j; 72 73 if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) { 74 if (BN_abs_is_word(p, 2)) { 75 if (ret == NULL) { 76 ret = BN_new(); 77 } 78 if (ret == NULL) { 79 goto end; 80 } 81 if (!BN_set_word(ret, BN_is_bit_set(a, 0))) { 82 if (ret != in) { 83 BN_free(ret); 84 } 85 return NULL; 86 } 87 return ret; 88 } 89 90 OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME); 91 return (NULL); 92 } 93 94 if (BN_is_zero(a) || BN_is_one(a)) { 95 if (ret == NULL) { 96 ret = BN_new(); 97 } 98 if (ret == NULL) { 99 goto end; 100 } 101 if (!BN_set_word(ret, BN_is_one(a))) { 102 if (ret != in) { 103 BN_free(ret); 104 } 105 return NULL; 106 } 107 return ret; 108 } 109 110 BN_CTX_start(ctx); 111 A = BN_CTX_get(ctx); 112 b = BN_CTX_get(ctx); 113 q = BN_CTX_get(ctx); 114 t = BN_CTX_get(ctx); 115 x = BN_CTX_get(ctx); 116 y = BN_CTX_get(ctx); 117 if (y == NULL) { 118 goto end; 119 } 120 121 if (ret == NULL) { 122 ret = BN_new(); 123 } 124 if (ret == NULL) { 125 goto end; 126 } 127 128 // A = a mod p 129 if (!BN_nnmod(A, a, p, ctx)) { 130 goto end; 131 } 132 133 // now write |p| - 1 as 2^e*q where q is odd 134 e = 1; 135 while (!BN_is_bit_set(p, e)) { 136 e++; 137 } 138 // we'll set q later (if needed) 139 140 if (e == 1) { 141 // The easy case: (|p|-1)/2 is odd, so 2 has an inverse 142 // modulo (|p|-1)/2, and square roots can be computed 143 // directly by modular exponentiation. 144 // We have 145 // 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2), 146 // so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1. 147 if (!BN_rshift(q, p, 2)) { 148 goto end; 149 } 150 q->neg = 0; 151 if (!BN_add_word(q, 1) || 152 !BN_mod_exp_mont(ret, A, q, p, ctx, NULL)) { 153 goto end; 154 } 155 err = 0; 156 goto vrfy; 157 } 158 159 if (e == 2) { 160 // |p| == 5 (mod 8) 161 // 162 // In this case 2 is always a non-square since 163 // Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime. 164 // So if a really is a square, then 2*a is a non-square. 165 // Thus for 166 // b := (2*a)^((|p|-5)/8), 167 // i := (2*a)*b^2 168 // we have 169 // i^2 = (2*a)^((1 + (|p|-5)/4)*2) 170 // = (2*a)^((p-1)/2) 171 // = -1; 172 // so if we set 173 // x := a*b*(i-1), 174 // then 175 // x^2 = a^2 * b^2 * (i^2 - 2*i + 1) 176 // = a^2 * b^2 * (-2*i) 177 // = a*(-i)*(2*a*b^2) 178 // = a*(-i)*i 179 // = a. 180 // 181 // (This is due to A.O.L. Atkin, 182 // <URL: 183 //http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>, 184 // November 1992.) 185 186 // t := 2*a 187 if (!BN_mod_lshift1_quick(t, A, p)) { 188 goto end; 189 } 190 191 // b := (2*a)^((|p|-5)/8) 192 if (!BN_rshift(q, p, 3)) { 193 goto end; 194 } 195 q->neg = 0; 196 if (!BN_mod_exp_mont(b, t, q, p, ctx, NULL)) { 197 goto end; 198 } 199 200 // y := b^2 201 if (!BN_mod_sqr(y, b, p, ctx)) { 202 goto end; 203 } 204 205 // t := (2*a)*b^2 - 1 206 if (!BN_mod_mul(t, t, y, p, ctx) || 207 !BN_sub_word(t, 1)) { 208 goto end; 209 } 210 211 // x = a*b*t 212 if (!BN_mod_mul(x, A, b, p, ctx) || 213 !BN_mod_mul(x, x, t, p, ctx)) { 214 goto end; 215 } 216 217 if (!BN_copy(ret, x)) { 218 goto end; 219 } 220 err = 0; 221 goto vrfy; 222 } 223 224 // e > 2, so we really have to use the Tonelli/Shanks algorithm. 225 // First, find some y that is not a square. 226 if (!BN_copy(q, p)) { 227 goto end; // use 'q' as temp 228 } 229 q->neg = 0; 230 i = 2; 231 do { 232 // For efficiency, try small numbers first; 233 // if this fails, try random numbers. 234 if (i < 22) { 235 if (!BN_set_word(y, i)) { 236 goto end; 237 } 238 } else { 239 if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) { 240 goto end; 241 } 242 if (BN_ucmp(y, p) >= 0) { 243 if (!(p->neg ? BN_add : BN_sub)(y, y, p)) { 244 goto end; 245 } 246 } 247 // now 0 <= y < |p| 248 if (BN_is_zero(y)) { 249 if (!BN_set_word(y, i)) { 250 goto end; 251 } 252 } 253 } 254 255 r = bn_jacobi(y, q, ctx); // here 'q' is |p| 256 if (r < -1) { 257 goto end; 258 } 259 if (r == 0) { 260 // m divides p 261 OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME); 262 goto end; 263 } 264 } while (r == 1 && ++i < 82); 265 266 if (r != -1) { 267 // Many rounds and still no non-square -- this is more likely 268 // a bug than just bad luck. 269 // Even if p is not prime, we should have found some y 270 // such that r == -1. 271 OPENSSL_PUT_ERROR(BN, BN_R_TOO_MANY_ITERATIONS); 272 goto end; 273 } 274 275 // Here's our actual 'q': 276 if (!BN_rshift(q, q, e)) { 277 goto end; 278 } 279 280 // Now that we have some non-square, we can find an element 281 // of order 2^e by computing its q'th power. 282 if (!BN_mod_exp_mont(y, y, q, p, ctx, NULL)) { 283 goto end; 284 } 285 if (BN_is_one(y)) { 286 OPENSSL_PUT_ERROR(BN, BN_R_P_IS_NOT_PRIME); 287 goto end; 288 } 289 290 // Now we know that (if p is indeed prime) there is an integer 291 // k, 0 <= k < 2^e, such that 292 // 293 // a^q * y^k == 1 (mod p). 294 // 295 // As a^q is a square and y is not, k must be even. 296 // q+1 is even, too, so there is an element 297 // 298 // X := a^((q+1)/2) * y^(k/2), 299 // 300 // and it satisfies 301 // 302 // X^2 = a^q * a * y^k 303 // = a, 304 // 305 // so it is the square root that we are looking for. 306 307 // t := (q-1)/2 (note that q is odd) 308 if (!BN_rshift1(t, q)) { 309 goto end; 310 } 311 312 // x := a^((q-1)/2) 313 if (BN_is_zero(t)) // special case: p = 2^e + 1 314 { 315 if (!BN_nnmod(t, A, p, ctx)) { 316 goto end; 317 } 318 if (BN_is_zero(t)) { 319 // special case: a == 0 (mod p) 320 BN_zero(ret); 321 err = 0; 322 goto end; 323 } else if (!BN_one(x)) { 324 goto end; 325 } 326 } else { 327 if (!BN_mod_exp_mont(x, A, t, p, ctx, NULL)) { 328 goto end; 329 } 330 if (BN_is_zero(x)) { 331 // special case: a == 0 (mod p) 332 BN_zero(ret); 333 err = 0; 334 goto end; 335 } 336 } 337 338 // b := a*x^2 (= a^q) 339 if (!BN_mod_sqr(b, x, p, ctx) || 340 !BN_mod_mul(b, b, A, p, ctx)) { 341 goto end; 342 } 343 344 // x := a*x (= a^((q+1)/2)) 345 if (!BN_mod_mul(x, x, A, p, ctx)) { 346 goto end; 347 } 348 349 while (1) { 350 // Now b is a^q * y^k for some even k (0 <= k < 2^E 351 // where E refers to the original value of e, which we 352 // don't keep in a variable), and x is a^((q+1)/2) * y^(k/2). 353 // 354 // We have a*b = x^2, 355 // y^2^(e-1) = -1, 356 // b^2^(e-1) = 1. 357 358 if (BN_is_one(b)) { 359 if (!BN_copy(ret, x)) { 360 goto end; 361 } 362 err = 0; 363 goto vrfy; 364 } 365 366 367 // find smallest i such that b^(2^i) = 1 368 i = 1; 369 if (!BN_mod_sqr(t, b, p, ctx)) { 370 goto end; 371 } 372 while (!BN_is_one(t)) { 373 i++; 374 if (i == e) { 375 OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE); 376 goto end; 377 } 378 if (!BN_mod_mul(t, t, t, p, ctx)) { 379 goto end; 380 } 381 } 382 383 384 // t := y^2^(e - i - 1) 385 if (!BN_copy(t, y)) { 386 goto end; 387 } 388 for (j = e - i - 1; j > 0; j--) { 389 if (!BN_mod_sqr(t, t, p, ctx)) { 390 goto end; 391 } 392 } 393 if (!BN_mod_mul(y, t, t, p, ctx) || 394 !BN_mod_mul(x, x, t, p, ctx) || 395 !BN_mod_mul(b, b, y, p, ctx)) { 396 goto end; 397 } 398 e = i; 399 } 400 401 vrfy: 402 if (!err) { 403 // verify the result -- the input might have been not a square 404 // (test added in 0.9.8) 405 406 if (!BN_mod_sqr(x, ret, p, ctx)) { 407 err = 1; 408 } 409 410 if (!err && 0 != BN_cmp(x, A)) { 411 OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE); 412 err = 1; 413 } 414 } 415 416 end: 417 if (err) { 418 if (ret != in) { 419 BN_clear_free(ret); 420 } 421 ret = NULL; 422 } 423 BN_CTX_end(ctx); 424 return ret; 425 } 426 427 int BN_sqrt(BIGNUM *out_sqrt, const BIGNUM *in, BN_CTX *ctx) { 428 BIGNUM *estimate, *tmp, *delta, *last_delta, *tmp2; 429 int ok = 0, last_delta_valid = 0; 430 431 if (in->neg) { 432 OPENSSL_PUT_ERROR(BN, BN_R_NEGATIVE_NUMBER); 433 return 0; 434 } 435 if (BN_is_zero(in)) { 436 BN_zero(out_sqrt); 437 return 1; 438 } 439 440 BN_CTX_start(ctx); 441 if (out_sqrt == in) { 442 estimate = BN_CTX_get(ctx); 443 } else { 444 estimate = out_sqrt; 445 } 446 tmp = BN_CTX_get(ctx); 447 last_delta = BN_CTX_get(ctx); 448 delta = BN_CTX_get(ctx); 449 if (estimate == NULL || tmp == NULL || last_delta == NULL || delta == NULL) { 450 OPENSSL_PUT_ERROR(BN, ERR_R_MALLOC_FAILURE); 451 goto err; 452 } 453 454 // We estimate that the square root of an n-bit number is 2^{n/2}. 455 if (!BN_lshift(estimate, BN_value_one(), BN_num_bits(in)/2)) { 456 goto err; 457 } 458 459 // This is Newton's method for finding a root of the equation |estimate|^2 - 460 // |in| = 0. 461 for (;;) { 462 // |estimate| = 1/2 * (|estimate| + |in|/|estimate|) 463 if (!BN_div(tmp, NULL, in, estimate, ctx) || 464 !BN_add(tmp, tmp, estimate) || 465 !BN_rshift1(estimate, tmp) || 466 // |tmp| = |estimate|^2 467 !BN_sqr(tmp, estimate, ctx) || 468 // |delta| = |in| - |tmp| 469 !BN_sub(delta, in, tmp)) { 470 OPENSSL_PUT_ERROR(BN, ERR_R_BN_LIB); 471 goto err; 472 } 473 474 delta->neg = 0; 475 // The difference between |in| and |estimate| squared is required to always 476 // decrease. This ensures that the loop always terminates, but I don't have 477 // a proof that it always finds the square root for a given square. 478 if (last_delta_valid && BN_cmp(delta, last_delta) >= 0) { 479 break; 480 } 481 482 last_delta_valid = 1; 483 484 tmp2 = last_delta; 485 last_delta = delta; 486 delta = tmp2; 487 } 488 489 if (BN_cmp(tmp, in) != 0) { 490 OPENSSL_PUT_ERROR(BN, BN_R_NOT_A_SQUARE); 491 goto err; 492 } 493 494 ok = 1; 495 496 err: 497 if (ok && out_sqrt == in && !BN_copy(out_sqrt, estimate)) { 498 ok = 0; 499 } 500 BN_CTX_end(ctx); 501 return ok; 502 } 503