1 /* 2 * Copyright 1995-2019 The OpenSSL Project Authors. All Rights Reserved. 3 * 4 * Licensed under the OpenSSL license (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 <assert.h> 11 #include <openssl/bn.h> 12 #include "internal/cryptlib.h" 13 #include "bn_local.h" 14 15 /* The old slow way */ 16 #if 0 17 int BN_div(BIGNUM *dv, BIGNUM *rem, const BIGNUM *m, const BIGNUM *d, 18 BN_CTX *ctx) 19 { 20 int i, nm, nd; 21 int ret = 0; 22 BIGNUM *D; 23 24 bn_check_top(m); 25 bn_check_top(d); 26 if (BN_is_zero(d)) { 27 BNerr(BN_F_BN_DIV, BN_R_DIV_BY_ZERO); 28 return 0; 29 } 30 31 if (BN_ucmp(m, d) < 0) { 32 if (rem != NULL) { 33 if (BN_copy(rem, m) == NULL) 34 return 0; 35 } 36 if (dv != NULL) 37 BN_zero(dv); 38 return 1; 39 } 40 41 BN_CTX_start(ctx); 42 D = BN_CTX_get(ctx); 43 if (dv == NULL) 44 dv = BN_CTX_get(ctx); 45 if (rem == NULL) 46 rem = BN_CTX_get(ctx); 47 if (D == NULL || dv == NULL || rem == NULL) 48 goto end; 49 50 nd = BN_num_bits(d); 51 nm = BN_num_bits(m); 52 if (BN_copy(D, d) == NULL) 53 goto end; 54 if (BN_copy(rem, m) == NULL) 55 goto end; 56 57 /* 58 * The next 2 are needed so we can do a dv->d[0]|=1 later since 59 * BN_lshift1 will only work once there is a value :-) 60 */ 61 BN_zero(dv); 62 if (bn_wexpand(dv, 1) == NULL) 63 goto end; 64 dv->top = 1; 65 66 if (!BN_lshift(D, D, nm - nd)) 67 goto end; 68 for (i = nm - nd; i >= 0; i--) { 69 if (!BN_lshift1(dv, dv)) 70 goto end; 71 if (BN_ucmp(rem, D) >= 0) { 72 dv->d[0] |= 1; 73 if (!BN_usub(rem, rem, D)) 74 goto end; 75 } 76 /* CAN IMPROVE (and have now :=) */ 77 if (!BN_rshift1(D, D)) 78 goto end; 79 } 80 rem->neg = BN_is_zero(rem) ? 0 : m->neg; 81 dv->neg = m->neg ^ d->neg; 82 ret = 1; 83 end: 84 BN_CTX_end(ctx); 85 return ret; 86 } 87 88 #else 89 90 # if defined(BN_DIV3W) 91 BN_ULONG bn_div_3_words(const BN_ULONG *m, BN_ULONG d1, BN_ULONG d0); 92 # elif 0 93 /* 94 * This is #if-ed away, because it's a reference for assembly implementations, 95 * where it can and should be made constant-time. But if you want to test it, 96 * just replace 0 with 1. 97 */ 98 # if BN_BITS2 == 64 && defined(__SIZEOF_INT128__) && __SIZEOF_INT128__==16 99 # undef BN_ULLONG 100 # define BN_ULLONG __uint128_t 101 # define BN_LLONG 102 # endif 103 104 # ifdef BN_LLONG 105 # define BN_DIV3W 106 /* 107 * Interface is somewhat quirky, |m| is pointer to most significant limb, 108 * and less significant limb is referred at |m[-1]|. This means that caller 109 * is responsible for ensuring that |m[-1]| is valid. Second condition that 110 * has to be met is that |d0|'s most significant bit has to be set. Or in 111 * other words divisor has to be "bit-aligned to the left." bn_div_fixed_top 112 * does all this. The subroutine considers four limbs, two of which are 113 * "overlapping," hence the name... 114 */ 115 static BN_ULONG bn_div_3_words(const BN_ULONG *m, BN_ULONG d1, BN_ULONG d0) 116 { 117 BN_ULLONG R = ((BN_ULLONG)m[0] << BN_BITS2) | m[-1]; 118 BN_ULLONG D = ((BN_ULLONG)d0 << BN_BITS2) | d1; 119 BN_ULONG Q = 0, mask; 120 int i; 121 122 for (i = 0; i < BN_BITS2; i++) { 123 Q <<= 1; 124 if (R >= D) { 125 Q |= 1; 126 R -= D; 127 } 128 D >>= 1; 129 } 130 131 mask = 0 - (Q >> (BN_BITS2 - 1)); /* does it overflow? */ 132 133 Q <<= 1; 134 Q |= (R >= D); 135 136 return (Q | mask) & BN_MASK2; 137 } 138 # endif 139 # endif 140 141 static int bn_left_align(BIGNUM *num) 142 { 143 BN_ULONG *d = num->d, n, m, rmask; 144 int top = num->top; 145 int rshift = BN_num_bits_word(d[top - 1]), lshift, i; 146 147 lshift = BN_BITS2 - rshift; 148 rshift %= BN_BITS2; /* say no to undefined behaviour */ 149 rmask = (BN_ULONG)0 - rshift; /* rmask = 0 - (rshift != 0) */ 150 rmask |= rmask >> 8; 151 152 for (i = 0, m = 0; i < top; i++) { 153 n = d[i]; 154 d[i] = ((n << lshift) | m) & BN_MASK2; 155 m = (n >> rshift) & rmask; 156 } 157 158 return lshift; 159 } 160 161 # if !defined(OPENSSL_NO_ASM) && !defined(OPENSSL_NO_INLINE_ASM) \ 162 && !defined(PEDANTIC) && !defined(BN_DIV3W) 163 # if defined(__GNUC__) && __GNUC__>=2 164 # if defined(__i386) || defined (__i386__) 165 /*- 166 * There were two reasons for implementing this template: 167 * - GNU C generates a call to a function (__udivdi3 to be exact) 168 * in reply to ((((BN_ULLONG)n0)<<BN_BITS2)|n1)/d0 (I fail to 169 * understand why...); 170 * - divl doesn't only calculate quotient, but also leaves 171 * remainder in %edx which we can definitely use here:-) 172 */ 173 # undef bn_div_words 174 # define bn_div_words(n0,n1,d0) \ 175 ({ asm volatile ( \ 176 "divl %4" \ 177 : "=a"(q), "=d"(rem) \ 178 : "a"(n1), "d"(n0), "r"(d0) \ 179 : "cc"); \ 180 q; \ 181 }) 182 # define REMAINDER_IS_ALREADY_CALCULATED 183 # elif defined(__x86_64) && defined(SIXTY_FOUR_BIT_LONG) 184 /* 185 * Same story here, but it's 128-bit by 64-bit division. Wow! 186 */ 187 # undef bn_div_words 188 # define bn_div_words(n0,n1,d0) \ 189 ({ asm volatile ( \ 190 "divq %4" \ 191 : "=a"(q), "=d"(rem) \ 192 : "a"(n1), "d"(n0), "r"(d0) \ 193 : "cc"); \ 194 q; \ 195 }) 196 # define REMAINDER_IS_ALREADY_CALCULATED 197 # endif /* __<cpu> */ 198 # endif /* __GNUC__ */ 199 # endif /* OPENSSL_NO_ASM */ 200 201 /*- 202 * BN_div computes dv := num / divisor, rounding towards 203 * zero, and sets up rm such that dv*divisor + rm = num holds. 204 * Thus: 205 * dv->neg == num->neg ^ divisor->neg (unless the result is zero) 206 * rm->neg == num->neg (unless the remainder is zero) 207 * If 'dv' or 'rm' is NULL, the respective value is not returned. 208 */ 209 int BN_div(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, const BIGNUM *divisor, 210 BN_CTX *ctx) 211 { 212 int ret; 213 214 if (BN_is_zero(divisor)) { 215 BNerr(BN_F_BN_DIV, BN_R_DIV_BY_ZERO); 216 return 0; 217 } 218 219 /* 220 * Invalid zero-padding would have particularly bad consequences so don't 221 * just rely on bn_check_top() here (bn_check_top() works only for 222 * BN_DEBUG builds) 223 */ 224 if (divisor->d[divisor->top - 1] == 0) { 225 BNerr(BN_F_BN_DIV, BN_R_NOT_INITIALIZED); 226 return 0; 227 } 228 229 ret = bn_div_fixed_top(dv, rm, num, divisor, ctx); 230 231 if (ret) { 232 if (dv != NULL) 233 bn_correct_top(dv); 234 if (rm != NULL) 235 bn_correct_top(rm); 236 } 237 238 return ret; 239 } 240 241 /* 242 * It's argued that *length* of *significant* part of divisor is public. 243 * Even if it's private modulus that is. Again, *length* is assumed 244 * public, but not *value*. Former is likely to be pre-defined by 245 * algorithm with bit granularity, though below subroutine is invariant 246 * of limb length. Thanks to this assumption we can require that |divisor| 247 * may not be zero-padded, yet claim this subroutine "constant-time"(*). 248 * This is because zero-padded dividend, |num|, is tolerated, so that 249 * caller can pass dividend of public length(*), but with smaller amount 250 * of significant limbs. This naturally means that quotient, |dv|, would 251 * contain correspongly less significant limbs as well, and will be zero- 252 * padded accordingly. Returned remainder, |rm|, will have same bit length 253 * as divisor, also zero-padded if needed. These actually leave sign bits 254 * in ambiguous state. In sense that we try to avoid negative zeros, while 255 * zero-padded zeros would retain sign. 256 * 257 * (*) "Constant-time-ness" has two pre-conditions: 258 * 259 * - availability of constant-time bn_div_3_words; 260 * - dividend is at least as "wide" as divisor, limb-wise, zero-padded 261 * if so required, which shouldn't be a privacy problem, because 262 * divisor's length is considered public; 263 */ 264 int bn_div_fixed_top(BIGNUM *dv, BIGNUM *rm, const BIGNUM *num, 265 const BIGNUM *divisor, BN_CTX *ctx) 266 { 267 int norm_shift, i, j, loop; 268 BIGNUM *tmp, *snum, *sdiv, *res; 269 BN_ULONG *resp, *wnum, *wnumtop; 270 BN_ULONG d0, d1; 271 int num_n, div_n; 272 273 assert(divisor->top > 0 && divisor->d[divisor->top - 1] != 0); 274 275 bn_check_top(num); 276 bn_check_top(divisor); 277 bn_check_top(dv); 278 bn_check_top(rm); 279 280 BN_CTX_start(ctx); 281 res = (dv == NULL) ? BN_CTX_get(ctx) : dv; 282 tmp = BN_CTX_get(ctx); 283 snum = BN_CTX_get(ctx); 284 sdiv = BN_CTX_get(ctx); 285 if (sdiv == NULL) 286 goto err; 287 288 /* First we normalise the numbers */ 289 if (!BN_copy(sdiv, divisor)) 290 goto err; 291 norm_shift = bn_left_align(sdiv); 292 sdiv->neg = 0; 293 /* 294 * Note that bn_lshift_fixed_top's output is always one limb longer 295 * than input, even when norm_shift is zero. This means that amount of 296 * inner loop iterations is invariant of dividend value, and that one 297 * doesn't need to compare dividend and divisor if they were originally 298 * of the same bit length. 299 */ 300 if (!(bn_lshift_fixed_top(snum, num, norm_shift))) 301 goto err; 302 303 div_n = sdiv->top; 304 num_n = snum->top; 305 306 if (num_n <= div_n) { 307 /* caller didn't pad dividend -> no constant-time guarantee... */ 308 if (bn_wexpand(snum, div_n + 1) == NULL) 309 goto err; 310 memset(&(snum->d[num_n]), 0, (div_n - num_n + 1) * sizeof(BN_ULONG)); 311 snum->top = num_n = div_n + 1; 312 } 313 314 loop = num_n - div_n; 315 /* 316 * Lets setup a 'window' into snum This is the part that corresponds to 317 * the current 'area' being divided 318 */ 319 wnum = &(snum->d[loop]); 320 wnumtop = &(snum->d[num_n - 1]); 321 322 /* Get the top 2 words of sdiv */ 323 d0 = sdiv->d[div_n - 1]; 324 d1 = (div_n == 1) ? 0 : sdiv->d[div_n - 2]; 325 326 /* Setup quotient */ 327 if (!bn_wexpand(res, loop)) 328 goto err; 329 res->neg = (num->neg ^ divisor->neg); 330 res->top = loop; 331 res->flags |= BN_FLG_FIXED_TOP; 332 resp = &(res->d[loop]); 333 334 /* space for temp */ 335 if (!bn_wexpand(tmp, (div_n + 1))) 336 goto err; 337 338 for (i = 0; i < loop; i++, wnumtop--) { 339 BN_ULONG q, l0; 340 /* 341 * the first part of the loop uses the top two words of snum and sdiv 342 * to calculate a BN_ULONG q such that | wnum - sdiv * q | < sdiv 343 */ 344 # if defined(BN_DIV3W) 345 q = bn_div_3_words(wnumtop, d1, d0); 346 # else 347 BN_ULONG n0, n1, rem = 0; 348 349 n0 = wnumtop[0]; 350 n1 = wnumtop[-1]; 351 if (n0 == d0) 352 q = BN_MASK2; 353 else { /* n0 < d0 */ 354 BN_ULONG n2 = (wnumtop == wnum) ? 0 : wnumtop[-2]; 355 # ifdef BN_LLONG 356 BN_ULLONG t2; 357 358 # if defined(BN_LLONG) && defined(BN_DIV2W) && !defined(bn_div_words) 359 q = (BN_ULONG)(((((BN_ULLONG) n0) << BN_BITS2) | n1) / d0); 360 # else 361 q = bn_div_words(n0, n1, d0); 362 # endif 363 364 # ifndef REMAINDER_IS_ALREADY_CALCULATED 365 /* 366 * rem doesn't have to be BN_ULLONG. The least we 367 * know it's less that d0, isn't it? 368 */ 369 rem = (n1 - q * d0) & BN_MASK2; 370 # endif 371 t2 = (BN_ULLONG) d1 *q; 372 373 for (;;) { 374 if (t2 <= ((((BN_ULLONG) rem) << BN_BITS2) | n2)) 375 break; 376 q--; 377 rem += d0; 378 if (rem < d0) 379 break; /* don't let rem overflow */ 380 t2 -= d1; 381 } 382 # else /* !BN_LLONG */ 383 BN_ULONG t2l, t2h; 384 385 q = bn_div_words(n0, n1, d0); 386 # ifndef REMAINDER_IS_ALREADY_CALCULATED 387 rem = (n1 - q * d0) & BN_MASK2; 388 # endif 389 390 # if defined(BN_UMULT_LOHI) 391 BN_UMULT_LOHI(t2l, t2h, d1, q); 392 # elif defined(BN_UMULT_HIGH) 393 t2l = d1 * q; 394 t2h = BN_UMULT_HIGH(d1, q); 395 # else 396 { 397 BN_ULONG ql, qh; 398 t2l = LBITS(d1); 399 t2h = HBITS(d1); 400 ql = LBITS(q); 401 qh = HBITS(q); 402 mul64(t2l, t2h, ql, qh); /* t2=(BN_ULLONG)d1*q; */ 403 } 404 # endif 405 406 for (;;) { 407 if ((t2h < rem) || ((t2h == rem) && (t2l <= n2))) 408 break; 409 q--; 410 rem += d0; 411 if (rem < d0) 412 break; /* don't let rem overflow */ 413 if (t2l < d1) 414 t2h--; 415 t2l -= d1; 416 } 417 # endif /* !BN_LLONG */ 418 } 419 # endif /* !BN_DIV3W */ 420 421 l0 = bn_mul_words(tmp->d, sdiv->d, div_n, q); 422 tmp->d[div_n] = l0; 423 wnum--; 424 /* 425 * ignore top values of the bignums just sub the two BN_ULONG arrays 426 * with bn_sub_words 427 */ 428 l0 = bn_sub_words(wnum, wnum, tmp->d, div_n + 1); 429 q -= l0; 430 /* 431 * Note: As we have considered only the leading two BN_ULONGs in 432 * the calculation of q, sdiv * q might be greater than wnum (but 433 * then (q-1) * sdiv is less or equal than wnum) 434 */ 435 for (l0 = 0 - l0, j = 0; j < div_n; j++) 436 tmp->d[j] = sdiv->d[j] & l0; 437 l0 = bn_add_words(wnum, wnum, tmp->d, div_n); 438 (*wnumtop) += l0; 439 assert((*wnumtop) == 0); 440 441 /* store part of the result */ 442 *--resp = q; 443 } 444 /* snum holds remainder, it's as wide as divisor */ 445 snum->neg = num->neg; 446 snum->top = div_n; 447 snum->flags |= BN_FLG_FIXED_TOP; 448 if (rm != NULL) 449 bn_rshift_fixed_top(rm, snum, norm_shift); 450 BN_CTX_end(ctx); 451 return 1; 452 err: 453 bn_check_top(rm); 454 BN_CTX_end(ctx); 455 return 0; 456 } 457 #endif 458