Alexandre Savard | 1b09e31 | 2012-08-07 20:33:29 -0400 | [diff] [blame] | 1 | /* crypto/bn/bn_sqrt.c */ |
| 2 | /* Written by Lenka Fibikova <fibikova@exp-math.uni-essen.de> |
| 3 | * and Bodo Moeller for the OpenSSL project. */ |
| 4 | /* ==================================================================== |
| 5 | * Copyright (c) 1998-2000 The OpenSSL Project. All rights reserved. |
| 6 | * |
| 7 | * Redistribution and use in source and binary forms, with or without |
| 8 | * modification, are permitted provided that the following conditions |
| 9 | * are met: |
| 10 | * |
| 11 | * 1. Redistributions of source code must retain the above copyright |
| 12 | * notice, this list of conditions and the following disclaimer. |
| 13 | * |
| 14 | * 2. Redistributions in binary form must reproduce the above copyright |
| 15 | * notice, this list of conditions and the following disclaimer in |
| 16 | * the documentation and/or other materials provided with the |
| 17 | * distribution. |
| 18 | * |
| 19 | * 3. All advertising materials mentioning features or use of this |
| 20 | * software must display the following acknowledgment: |
| 21 | * "This product includes software developed by the OpenSSL Project |
| 22 | * for use in the OpenSSL Toolkit. (http://www.openssl.org/)" |
| 23 | * |
| 24 | * 4. The names "OpenSSL Toolkit" and "OpenSSL Project" must not be used to |
| 25 | * endorse or promote products derived from this software without |
| 26 | * prior written permission. For written permission, please contact |
| 27 | * openssl-core@openssl.org. |
| 28 | * |
| 29 | * 5. Products derived from this software may not be called "OpenSSL" |
| 30 | * nor may "OpenSSL" appear in their names without prior written |
| 31 | * permission of the OpenSSL Project. |
| 32 | * |
| 33 | * 6. Redistributions of any form whatsoever must retain the following |
| 34 | * acknowledgment: |
| 35 | * "This product includes software developed by the OpenSSL Project |
| 36 | * for use in the OpenSSL Toolkit (http://www.openssl.org/)" |
| 37 | * |
| 38 | * THIS SOFTWARE IS PROVIDED BY THE OpenSSL PROJECT ``AS IS'' AND ANY |
| 39 | * EXPRESSED OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE |
| 40 | * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR |
| 41 | * PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE OpenSSL PROJECT OR |
| 42 | * ITS CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, |
| 43 | * SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT |
| 44 | * NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; |
| 45 | * LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) |
| 46 | * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, |
| 47 | * STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) |
| 48 | * ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED |
| 49 | * OF THE POSSIBILITY OF SUCH DAMAGE. |
| 50 | * ==================================================================== |
| 51 | * |
| 52 | * This product includes cryptographic software written by Eric Young |
| 53 | * (eay@cryptsoft.com). This product includes software written by Tim |
| 54 | * Hudson (tjh@cryptsoft.com). |
| 55 | * |
| 56 | */ |
| 57 | |
| 58 | #include "cryptlib.h" |
| 59 | #include "bn_lcl.h" |
| 60 | |
| 61 | |
| 62 | BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) |
| 63 | /* Returns 'ret' such that |
| 64 | * ret^2 == a (mod p), |
| 65 | * using the Tonelli/Shanks algorithm (cf. Henri Cohen, "A Course |
| 66 | * in Algebraic Computational Number Theory", algorithm 1.5.1). |
| 67 | * '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 | { |
| 78 | if (BN_abs_is_word(p, 2)) |
| 79 | { |
| 80 | if (ret == NULL) |
| 81 | ret = BN_new(); |
| 82 | if (ret == NULL) |
| 83 | goto end; |
| 84 | if (!BN_set_word(ret, BN_is_bit_set(a, 0))) |
| 85 | { |
| 86 | if (ret != in) |
| 87 | BN_free(ret); |
| 88 | return NULL; |
| 89 | } |
| 90 | bn_check_top(ret); |
| 91 | return ret; |
| 92 | } |
| 93 | |
| 94 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); |
| 95 | return(NULL); |
| 96 | } |
| 97 | |
| 98 | if (BN_is_zero(a) || BN_is_one(a)) |
| 99 | { |
| 100 | if (ret == NULL) |
| 101 | ret = BN_new(); |
| 102 | if (ret == NULL) |
| 103 | goto end; |
| 104 | if (!BN_set_word(ret, BN_is_one(a))) |
| 105 | { |
| 106 | if (ret != in) |
| 107 | BN_free(ret); |
| 108 | return NULL; |
| 109 | } |
| 110 | bn_check_top(ret); |
| 111 | return ret; |
| 112 | } |
| 113 | |
| 114 | BN_CTX_start(ctx); |
| 115 | A = BN_CTX_get(ctx); |
| 116 | b = BN_CTX_get(ctx); |
| 117 | q = BN_CTX_get(ctx); |
| 118 | t = BN_CTX_get(ctx); |
| 119 | x = BN_CTX_get(ctx); |
| 120 | y = BN_CTX_get(ctx); |
| 121 | if (y == NULL) goto end; |
| 122 | |
| 123 | if (ret == NULL) |
| 124 | ret = BN_new(); |
| 125 | if (ret == NULL) goto end; |
| 126 | |
| 127 | /* A = a mod p */ |
| 128 | if (!BN_nnmod(A, a, p, ctx)) goto end; |
| 129 | |
| 130 | /* now write |p| - 1 as 2^e*q where q is odd */ |
| 131 | e = 1; |
| 132 | while (!BN_is_bit_set(p, e)) |
| 133 | e++; |
| 134 | /* we'll set q later (if needed) */ |
| 135 | |
| 136 | if (e == 1) |
| 137 | { |
| 138 | /* The easy case: (|p|-1)/2 is odd, so 2 has an inverse |
| 139 | * modulo (|p|-1)/2, and square roots can be computed |
| 140 | * directly by modular exponentiation. |
| 141 | * We have |
| 142 | * 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2), |
| 143 | * so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1. |
| 144 | */ |
| 145 | if (!BN_rshift(q, p, 2)) goto end; |
| 146 | q->neg = 0; |
| 147 | if (!BN_add_word(q, 1)) goto end; |
| 148 | if (!BN_mod_exp(ret, A, q, p, ctx)) goto end; |
| 149 | err = 0; |
| 150 | goto vrfy; |
| 151 | } |
| 152 | |
| 153 | if (e == 2) |
| 154 | { |
| 155 | /* |p| == 5 (mod 8) |
| 156 | * |
| 157 | * In this case 2 is always a non-square since |
| 158 | * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime. |
| 159 | * So if a really is a square, then 2*a is a non-square. |
| 160 | * Thus for |
| 161 | * b := (2*a)^((|p|-5)/8), |
| 162 | * i := (2*a)*b^2 |
| 163 | * we have |
| 164 | * i^2 = (2*a)^((1 + (|p|-5)/4)*2) |
| 165 | * = (2*a)^((p-1)/2) |
| 166 | * = -1; |
| 167 | * so if we set |
| 168 | * x := a*b*(i-1), |
| 169 | * then |
| 170 | * x^2 = a^2 * b^2 * (i^2 - 2*i + 1) |
| 171 | * = a^2 * b^2 * (-2*i) |
| 172 | * = a*(-i)*(2*a*b^2) |
| 173 | * = a*(-i)*i |
| 174 | * = a. |
| 175 | * |
| 176 | * (This is due to A.O.L. Atkin, |
| 177 | * <URL: http://listserv.nodak.edu/scripts/wa.exe?A2=ind9211&L=nmbrthry&O=T&P=562>, |
| 178 | * November 1992.) |
| 179 | */ |
| 180 | |
| 181 | /* t := 2*a */ |
| 182 | if (!BN_mod_lshift1_quick(t, A, p)) goto end; |
| 183 | |
| 184 | /* b := (2*a)^((|p|-5)/8) */ |
| 185 | if (!BN_rshift(q, p, 3)) goto end; |
| 186 | q->neg = 0; |
| 187 | if (!BN_mod_exp(b, t, q, p, ctx)) goto end; |
| 188 | |
| 189 | /* y := b^2 */ |
| 190 | if (!BN_mod_sqr(y, b, p, ctx)) goto end; |
| 191 | |
| 192 | /* t := (2*a)*b^2 - 1*/ |
| 193 | if (!BN_mod_mul(t, t, y, p, ctx)) goto end; |
| 194 | if (!BN_sub_word(t, 1)) goto end; |
| 195 | |
| 196 | /* x = a*b*t */ |
| 197 | if (!BN_mod_mul(x, A, b, p, ctx)) goto end; |
| 198 | if (!BN_mod_mul(x, x, t, p, ctx)) goto end; |
| 199 | |
| 200 | if (!BN_copy(ret, x)) goto end; |
| 201 | err = 0; |
| 202 | goto vrfy; |
| 203 | } |
| 204 | |
| 205 | /* e > 2, so we really have to use the Tonelli/Shanks algorithm. |
| 206 | * First, find some y that is not a square. */ |
| 207 | if (!BN_copy(q, p)) goto end; /* use 'q' as temp */ |
| 208 | q->neg = 0; |
| 209 | i = 2; |
| 210 | do |
| 211 | { |
| 212 | /* For efficiency, try small numbers first; |
| 213 | * if this fails, try random numbers. |
| 214 | */ |
| 215 | if (i < 22) |
| 216 | { |
| 217 | if (!BN_set_word(y, i)) goto end; |
| 218 | } |
| 219 | else |
| 220 | { |
| 221 | if (!BN_pseudo_rand(y, BN_num_bits(p), 0, 0)) goto end; |
| 222 | if (BN_ucmp(y, p) >= 0) |
| 223 | { |
| 224 | if (!(p->neg ? BN_add : BN_sub)(y, y, p)) goto end; |
| 225 | } |
| 226 | /* now 0 <= y < |p| */ |
| 227 | if (BN_is_zero(y)) |
| 228 | if (!BN_set_word(y, i)) goto end; |
| 229 | } |
| 230 | |
| 231 | r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */ |
| 232 | if (r < -1) goto end; |
| 233 | if (r == 0) |
| 234 | { |
| 235 | /* m divides p */ |
| 236 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); |
| 237 | goto end; |
| 238 | } |
| 239 | } |
| 240 | while (r == 1 && ++i < 82); |
| 241 | |
| 242 | if (r != -1) |
| 243 | { |
| 244 | /* Many rounds and still no non-square -- this is more likely |
| 245 | * a bug than just bad luck. |
| 246 | * Even if p is not prime, we should have found some y |
| 247 | * such that r == -1. |
| 248 | */ |
| 249 | BNerr(BN_F_BN_MOD_SQRT, BN_R_TOO_MANY_ITERATIONS); |
| 250 | goto end; |
| 251 | } |
| 252 | |
| 253 | /* Here's our actual 'q': */ |
| 254 | if (!BN_rshift(q, q, e)) goto end; |
| 255 | |
| 256 | /* Now that we have some non-square, we can find an element |
| 257 | * of order 2^e by computing its q'th power. */ |
| 258 | if (!BN_mod_exp(y, y, q, p, ctx)) goto end; |
| 259 | if (BN_is_one(y)) |
| 260 | { |
| 261 | BNerr(BN_F_BN_MOD_SQRT, BN_R_P_IS_NOT_PRIME); |
| 262 | goto end; |
| 263 | } |
| 264 | |
| 265 | /* Now we know that (if p is indeed prime) there is an integer |
| 266 | * k, 0 <= k < 2^e, such that |
| 267 | * |
| 268 | * a^q * y^k == 1 (mod p). |
| 269 | * |
| 270 | * As a^q is a square and y is not, k must be even. |
| 271 | * q+1 is even, too, so there is an element |
| 272 | * |
| 273 | * X := a^((q+1)/2) * y^(k/2), |
| 274 | * |
| 275 | * and it satisfies |
| 276 | * |
| 277 | * X^2 = a^q * a * y^k |
| 278 | * = a, |
| 279 | * |
| 280 | * so it is the square root that we are looking for. |
| 281 | */ |
| 282 | |
| 283 | /* t := (q-1)/2 (note that q is odd) */ |
| 284 | if (!BN_rshift1(t, q)) goto end; |
| 285 | |
| 286 | /* x := a^((q-1)/2) */ |
| 287 | if (BN_is_zero(t)) /* special case: p = 2^e + 1 */ |
| 288 | { |
| 289 | if (!BN_nnmod(t, A, p, ctx)) goto end; |
| 290 | if (BN_is_zero(t)) |
| 291 | { |
| 292 | /* special case: a == 0 (mod p) */ |
| 293 | BN_zero(ret); |
| 294 | err = 0; |
| 295 | goto end; |
| 296 | } |
| 297 | else |
| 298 | if (!BN_one(x)) goto end; |
| 299 | } |
| 300 | else |
| 301 | { |
| 302 | if (!BN_mod_exp(x, A, t, p, ctx)) goto end; |
| 303 | if (BN_is_zero(x)) |
| 304 | { |
| 305 | /* special case: a == 0 (mod p) */ |
| 306 | BN_zero(ret); |
| 307 | err = 0; |
| 308 | goto end; |
| 309 | } |
| 310 | } |
| 311 | |
| 312 | /* b := a*x^2 (= a^q) */ |
| 313 | if (!BN_mod_sqr(b, x, p, ctx)) goto end; |
| 314 | if (!BN_mod_mul(b, b, A, p, ctx)) goto end; |
| 315 | |
| 316 | /* x := a*x (= a^((q+1)/2)) */ |
| 317 | if (!BN_mod_mul(x, x, A, p, ctx)) goto end; |
| 318 | |
| 319 | while (1) |
| 320 | { |
| 321 | /* Now b is a^q * y^k for some even k (0 <= k < 2^E |
| 322 | * where E refers to the original value of e, which we |
| 323 | * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2). |
| 324 | * |
| 325 | * We have a*b = x^2, |
| 326 | * y^2^(e-1) = -1, |
| 327 | * b^2^(e-1) = 1. |
| 328 | */ |
| 329 | |
| 330 | if (BN_is_one(b)) |
| 331 | { |
| 332 | if (!BN_copy(ret, x)) goto end; |
| 333 | err = 0; |
| 334 | goto vrfy; |
| 335 | } |
| 336 | |
| 337 | |
| 338 | /* find smallest i such that b^(2^i) = 1 */ |
| 339 | i = 1; |
| 340 | if (!BN_mod_sqr(t, b, p, ctx)) goto end; |
| 341 | while (!BN_is_one(t)) |
| 342 | { |
| 343 | i++; |
| 344 | if (i == e) |
| 345 | { |
| 346 | BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); |
| 347 | goto end; |
| 348 | } |
| 349 | if (!BN_mod_mul(t, t, t, p, ctx)) goto end; |
| 350 | } |
| 351 | |
| 352 | |
| 353 | /* t := y^2^(e - i - 1) */ |
| 354 | if (!BN_copy(t, y)) goto end; |
| 355 | for (j = e - i - 1; j > 0; j--) |
| 356 | { |
| 357 | if (!BN_mod_sqr(t, t, p, ctx)) goto end; |
| 358 | } |
| 359 | if (!BN_mod_mul(y, t, t, p, ctx)) goto end; |
| 360 | if (!BN_mod_mul(x, x, t, p, ctx)) goto end; |
| 361 | if (!BN_mod_mul(b, b, y, p, ctx)) goto end; |
| 362 | e = i; |
| 363 | } |
| 364 | |
| 365 | vrfy: |
| 366 | if (!err) |
| 367 | { |
| 368 | /* verify the result -- the input might have been not a square |
| 369 | * (test added in 0.9.8) */ |
| 370 | |
| 371 | if (!BN_mod_sqr(x, ret, p, ctx)) |
| 372 | err = 1; |
| 373 | |
| 374 | if (!err && 0 != BN_cmp(x, A)) |
| 375 | { |
| 376 | BNerr(BN_F_BN_MOD_SQRT, BN_R_NOT_A_SQUARE); |
| 377 | err = 1; |
| 378 | } |
| 379 | } |
| 380 | |
| 381 | end: |
| 382 | if (err) |
| 383 | { |
| 384 | if (ret != NULL && ret != in) |
| 385 | { |
| 386 | BN_clear_free(ret); |
| 387 | } |
| 388 | ret = NULL; |
| 389 | } |
| 390 | BN_CTX_end(ctx); |
| 391 | bn_check_top(ret); |
| 392 | return ret; |
| 393 | } |