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| 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=n
mbrthry&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 e
nd; | |
| 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 } | |
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Delete bundled copy of OpenSSL and replace with README. (Closed)
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