1/* 2 * Copyright 2000-2022 The OpenSSL Project Authors. All Rights Reserved. 3 * 4 * Licensed under the Apache License 2.0 (the "License"). You may not use 5 * this file except in compliance with the License. You can obtain a copy 6 * in the file LICENSE in the source distribution or at 7 * https://www.openssl.org/source/license.html 8 */ 9 10#include "internal/cryptlib.h" 11#include "bn_local.h" 12 13BIGNUM *BN_mod_sqrt(BIGNUM *in, const BIGNUM *a, const BIGNUM *p, BN_CTX *ctx) 14/* 15 * Returns 'ret' such that ret^2 == a (mod p), using the Tonelli/Shanks 16 * algorithm (cf. Henri Cohen, "A Course in Algebraic Computational Number 17 * Theory", algorithm 1.5.1). 'p' must be prime, otherwise an error or 18 * an incorrect "result" will be returned. 19 */ 20{ 21 BIGNUM *ret = in; 22 int err = 1; 23 int r; 24 BIGNUM *A, *b, *q, *t, *x, *y; 25 int e, i, j; 26 int used_ctx = 0; 27 28 if (!BN_is_odd(p) || BN_abs_is_word(p, 1)) { 29 if (BN_abs_is_word(p, 2)) { 30 if (ret == NULL) 31 ret = BN_new(); 32 if (ret == NULL) 33 goto end; 34 if (!BN_set_word(ret, BN_is_bit_set(a, 0))) { 35 if (ret != in) 36 BN_free(ret); 37 return NULL; 38 } 39 bn_check_top(ret); 40 return ret; 41 } 42 43 ERR_raise(ERR_LIB_BN, BN_R_P_IS_NOT_PRIME); 44 return NULL; 45 } 46 47 if (BN_is_zero(a) || BN_is_one(a)) { 48 if (ret == NULL) 49 ret = BN_new(); 50 if (ret == NULL) 51 goto end; 52 if (!BN_set_word(ret, BN_is_one(a))) { 53 if (ret != in) 54 BN_free(ret); 55 return NULL; 56 } 57 bn_check_top(ret); 58 return ret; 59 } 60 61 BN_CTX_start(ctx); 62 used_ctx = 1; 63 A = BN_CTX_get(ctx); 64 b = BN_CTX_get(ctx); 65 q = BN_CTX_get(ctx); 66 t = BN_CTX_get(ctx); 67 x = BN_CTX_get(ctx); 68 y = BN_CTX_get(ctx); 69 if (y == NULL) 70 goto end; 71 72 if (ret == NULL) 73 ret = BN_new(); 74 if (ret == NULL) 75 goto end; 76 77 /* A = a mod p */ 78 if (!BN_nnmod(A, a, p, ctx)) 79 goto end; 80 81 /* now write |p| - 1 as 2^e*q where q is odd */ 82 e = 1; 83 while (!BN_is_bit_set(p, e)) 84 e++; 85 /* we'll set q later (if needed) */ 86 87 if (e == 1) { 88 /*- 89 * The easy case: (|p|-1)/2 is odd, so 2 has an inverse 90 * modulo (|p|-1)/2, and square roots can be computed 91 * directly by modular exponentiation. 92 * We have 93 * 2 * (|p|+1)/4 == 1 (mod (|p|-1)/2), 94 * so we can use exponent (|p|+1)/4, i.e. (|p|-3)/4 + 1. 95 */ 96 if (!BN_rshift(q, p, 2)) 97 goto end; 98 q->neg = 0; 99 if (!BN_add_word(q, 1)) 100 goto end; 101 if (!BN_mod_exp(ret, A, q, p, ctx)) 102 goto end; 103 err = 0; 104 goto vrfy; 105 } 106 107 if (e == 2) { 108 /*- 109 * |p| == 5 (mod 8) 110 * 111 * In this case 2 is always a non-square since 112 * Legendre(2,p) = (-1)^((p^2-1)/8) for any odd prime. 113 * So if a really is a square, then 2*a is a non-square. 114 * Thus for 115 * b := (2*a)^((|p|-5)/8), 116 * i := (2*a)*b^2 117 * we have 118 * i^2 = (2*a)^((1 + (|p|-5)/4)*2) 119 * = (2*a)^((p-1)/2) 120 * = -1; 121 * so if we set 122 * x := a*b*(i-1), 123 * then 124 * x^2 = a^2 * b^2 * (i^2 - 2*i + 1) 125 * = a^2 * b^2 * (-2*i) 126 * = a*(-i)*(2*a*b^2) 127 * = a*(-i)*i 128 * = a. 129 * 130 * (This is due to A.O.L. Atkin, 131 * Subject: Square Roots and Cognate Matters modulo p=8n+5. 132 * URL: https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind9211&L=NMBRTHRY&P=4026 133 * November 1992.) 134 */ 135 136 /* t := 2*a */ 137 if (!BN_mod_lshift1_quick(t, A, p)) 138 goto end; 139 140 /* b := (2*a)^((|p|-5)/8) */ 141 if (!BN_rshift(q, p, 3)) 142 goto end; 143 q->neg = 0; 144 if (!BN_mod_exp(b, t, q, p, ctx)) 145 goto end; 146 147 /* y := b^2 */ 148 if (!BN_mod_sqr(y, b, p, ctx)) 149 goto end; 150 151 /* t := (2*a)*b^2 - 1 */ 152 if (!BN_mod_mul(t, t, y, p, ctx)) 153 goto end; 154 if (!BN_sub_word(t, 1)) 155 goto end; 156 157 /* x = a*b*t */ 158 if (!BN_mod_mul(x, A, b, p, ctx)) 159 goto end; 160 if (!BN_mod_mul(x, x, t, p, ctx)) 161 goto end; 162 163 if (!BN_copy(ret, x)) 164 goto end; 165 err = 0; 166 goto vrfy; 167 } 168 169 /* 170 * e > 2, so we really have to use the Tonelli/Shanks algorithm. First, 171 * find some y that is not a square. 172 */ 173 if (!BN_copy(q, p)) 174 goto end; /* use 'q' as temp */ 175 q->neg = 0; 176 i = 2; 177 do { 178 /* 179 * For efficiency, try small numbers first; if this fails, try random 180 * numbers. 181 */ 182 if (i < 22) { 183 if (!BN_set_word(y, i)) 184 goto end; 185 } else { 186 if (!BN_priv_rand_ex(y, BN_num_bits(p), 0, 0, 0, ctx)) 187 goto end; 188 if (BN_ucmp(y, p) >= 0) { 189 if (!(p->neg ? BN_add : BN_sub) (y, y, p)) 190 goto end; 191 } 192 /* now 0 <= y < |p| */ 193 if (BN_is_zero(y)) 194 if (!BN_set_word(y, i)) 195 goto end; 196 } 197 198 r = BN_kronecker(y, q, ctx); /* here 'q' is |p| */ 199 if (r < -1) 200 goto end; 201 if (r == 0) { 202 /* m divides p */ 203 ERR_raise(ERR_LIB_BN, BN_R_P_IS_NOT_PRIME); 204 goto end; 205 } 206 } 207 while (r == 1 && ++i < 82); 208 209 if (r != -1) { 210 /* 211 * Many rounds and still no non-square -- this is more likely a bug 212 * than just bad luck. Even if p is not prime, we should have found 213 * some y such that r == -1. 214 */ 215 ERR_raise(ERR_LIB_BN, BN_R_TOO_MANY_ITERATIONS); 216 goto end; 217 } 218 219 /* Here's our actual 'q': */ 220 if (!BN_rshift(q, q, e)) 221 goto end; 222 223 /* 224 * Now that we have some non-square, we can find an element of order 2^e 225 * by computing its q'th power. 226 */ 227 if (!BN_mod_exp(y, y, q, p, ctx)) 228 goto end; 229 if (BN_is_one(y)) { 230 ERR_raise(ERR_LIB_BN, BN_R_P_IS_NOT_PRIME); 231 goto end; 232 } 233 234 /*- 235 * Now we know that (if p is indeed prime) there is an integer 236 * k, 0 <= k < 2^e, such that 237 * 238 * a^q * y^k == 1 (mod p). 239 * 240 * As a^q is a square and y is not, k must be even. 241 * q+1 is even, too, so there is an element 242 * 243 * X := a^((q+1)/2) * y^(k/2), 244 * 245 * and it satisfies 246 * 247 * X^2 = a^q * a * y^k 248 * = a, 249 * 250 * so it is the square root that we are looking for. 251 */ 252 253 /* t := (q-1)/2 (note that q is odd) */ 254 if (!BN_rshift1(t, q)) 255 goto end; 256 257 /* x := a^((q-1)/2) */ 258 if (BN_is_zero(t)) { /* special case: p = 2^e + 1 */ 259 if (!BN_nnmod(t, A, p, ctx)) 260 goto end; 261 if (BN_is_zero(t)) { 262 /* special case: a == 0 (mod p) */ 263 BN_zero(ret); 264 err = 0; 265 goto end; 266 } else if (!BN_one(x)) 267 goto end; 268 } else { 269 if (!BN_mod_exp(x, A, t, p, ctx)) 270 goto end; 271 if (BN_is_zero(x)) { 272 /* special case: a == 0 (mod p) */ 273 BN_zero(ret); 274 err = 0; 275 goto end; 276 } 277 } 278 279 /* b := a*x^2 (= a^q) */ 280 if (!BN_mod_sqr(b, x, p, ctx)) 281 goto end; 282 if (!BN_mod_mul(b, b, A, p, ctx)) 283 goto end; 284 285 /* x := a*x (= a^((q+1)/2)) */ 286 if (!BN_mod_mul(x, x, A, p, ctx)) 287 goto end; 288 289 while (1) { 290 /*- 291 * Now b is a^q * y^k for some even k (0 <= k < 2^E 292 * where E refers to the original value of e, which we 293 * don't keep in a variable), and x is a^((q+1)/2) * y^(k/2). 294 * 295 * We have a*b = x^2, 296 * y^2^(e-1) = -1, 297 * b^2^(e-1) = 1. 298 */ 299 300 if (BN_is_one(b)) { 301 if (!BN_copy(ret, x)) 302 goto end; 303 err = 0; 304 goto vrfy; 305 } 306 307 /* Find the smallest i, 0 < i < e, such that b^(2^i) = 1. */ 308 for (i = 1; i < e; i++) { 309 if (i == 1) { 310 if (!BN_mod_sqr(t, b, p, ctx)) 311 goto end; 312 313 } else { 314 if (!BN_mod_mul(t, t, t, p, ctx)) 315 goto end; 316 } 317 if (BN_is_one(t)) 318 break; 319 } 320 /* If not found, a is not a square or p is not prime. */ 321 if (i >= e) { 322 ERR_raise(ERR_LIB_BN, BN_R_NOT_A_SQUARE); 323 goto end; 324 } 325 326 /* t := y^2^(e - i - 1) */ 327 if (!BN_copy(t, y)) 328 goto end; 329 for (j = e - i - 1; j > 0; j--) { 330 if (!BN_mod_sqr(t, t, p, ctx)) 331 goto end; 332 } 333 if (!BN_mod_mul(y, t, t, p, ctx)) 334 goto end; 335 if (!BN_mod_mul(x, x, t, p, ctx)) 336 goto end; 337 if (!BN_mod_mul(b, b, y, p, ctx)) 338 goto end; 339 e = i; 340 } 341 342 vrfy: 343 if (!err) { 344 /* 345 * verify the result -- the input might have been not a square (test 346 * added in 0.9.8) 347 */ 348 349 if (!BN_mod_sqr(x, ret, p, ctx)) 350 err = 1; 351 352 if (!err && 0 != BN_cmp(x, A)) { 353 ERR_raise(ERR_LIB_BN, BN_R_NOT_A_SQUARE); 354 err = 1; 355 } 356 } 357 358 end: 359 if (err) { 360 if (ret != in) 361 BN_clear_free(ret); 362 ret = NULL; 363 } 364 if (used_ctx) 365 BN_CTX_end(ctx); 366 bn_check_top(ret); 367 return ret; 368} 369