1 /* 2 * Copyright (c) 2008-2016 Stefan Krah. All rights reserved. 3 * 4 * Redistribution and use in source and binary forms, with or without 5 * modification, are permitted provided that the following conditions 6 * are met: 7 * 8 * 1. Redistributions of source code must retain the above copyright 9 * notice, this list of conditions and the following disclaimer. 10 * 11 * 2. Redistributions in binary form must reproduce the above copyright 12 * notice, this list of conditions and the following disclaimer in the 13 * documentation and/or other materials provided with the distribution. 14 * 15 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS "AS IS" AND 16 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 17 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 18 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 19 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 20 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 21 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 22 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 23 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 24 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 25 * SUCH DAMAGE. 26 */ 27 28 29 #include "mpdecimal.h" 30 #include <stdio.h> 31 #include <assert.h> 32 #include "numbertheory.h" 33 #include "umodarith.h" 34 #include "crt.h" 35 36 37 /* Bignum: Chinese Remainder Theorem, extends the maximum transform length. */ 38 39 40 /* Multiply P1P2 by v, store result in w. */ 41 static inline void 42 _crt_mulP1P2_3(mpd_uint_t w[3], mpd_uint_t v) 43 { 44 mpd_uint_t hi1, hi2, lo; 45 46 _mpd_mul_words(&hi1, &lo, LH_P1P2, v); 47 w[0] = lo; 48 49 _mpd_mul_words(&hi2, &lo, UH_P1P2, v); 50 lo = hi1 + lo; 51 if (lo < hi1) hi2++; 52 53 w[1] = lo; 54 w[2] = hi2; 55 } 56 57 /* Add 3 words from v to w. The result is known to fit in w. */ 58 static inline void 59 _crt_add3(mpd_uint_t w[3], mpd_uint_t v[3]) 60 { 61 mpd_uint_t carry; 62 mpd_uint_t s; 63 64 s = w[0] + v[0]; 65 carry = (s < w[0]); 66 w[0] = s; 67 68 s = w[1] + (v[1] + carry); 69 carry = (s < w[1]); 70 w[1] = s; 71 72 w[2] = w[2] + (v[2] + carry); 73 } 74 75 /* Divide 3 words in u by v, store result in w, return remainder. */ 76 static inline mpd_uint_t 77 _crt_div3(mpd_uint_t *w, const mpd_uint_t *u, mpd_uint_t v) 78 { 79 mpd_uint_t r1 = u[2]; 80 mpd_uint_t r2; 81 82 if (r1 < v) { 83 w[2] = 0; 84 } 85 else { 86 _mpd_div_word(&w[2], &r1, u[2], v); /* GCOV_NOT_REACHED */ 87 } 88 89 _mpd_div_words(&w[1], &r2, r1, u[1], v); 90 _mpd_div_words(&w[0], &r1, r2, u[0], v); 91 92 return r1; 93 } 94 95 96 /* 97 * Chinese Remainder Theorem: 98 * Algorithm from Joerg Arndt, "Matters Computational", 99 * Chapter 37.4.1 [http://www.jjj.de/fxt/] 100 * 101 * See also Knuth, TAOCP, Volume 2, 4.3.2, exercise 7. 102 */ 103 104 /* 105 * CRT with carry: x1, x2, x3 contain numbers modulo p1, p2, p3. For each 106 * triple of members of the arrays, find the unique z modulo p1*p2*p3, with 107 * zmax = p1*p2*p3 - 1. 108 * 109 * In each iteration of the loop, split z into result[i] = z % MPD_RADIX 110 * and carry = z / MPD_RADIX. Let N be the size of carry[] and cmax the 111 * maximum carry. 112 * 113 * Limits for the 32-bit build: 114 * 115 * N = 2**96 116 * cmax = 7711435591312380274 117 * 118 * Limits for the 64 bit build: 119 * 120 * N = 2**192 121 * cmax = 627710135393475385904124401220046371710 122 * 123 * The following statements hold for both versions: 124 * 125 * 1) cmax + zmax < N, so the addition does not overflow. 126 * 127 * 2) (cmax + zmax) / MPD_RADIX == cmax. 128 * 129 * 3) If c <= cmax, then c_next = (c + zmax) / MPD_RADIX <= cmax. 130 */ 131 void 132 crt3(mpd_uint_t *x1, mpd_uint_t *x2, mpd_uint_t *x3, mpd_size_t rsize) 133 { 134 mpd_uint_t p1 = mpd_moduli[P1]; 135 mpd_uint_t umod; 136 #ifdef PPRO 137 double dmod; 138 uint32_t dinvmod[3]; 139 #endif 140 mpd_uint_t a1, a2, a3; 141 mpd_uint_t s; 142 mpd_uint_t z[3], t[3]; 143 mpd_uint_t carry[3] = {0,0,0}; 144 mpd_uint_t hi, lo; 145 mpd_size_t i; 146 147 for (i = 0; i < rsize; i++) { 148 149 a1 = x1[i]; 150 a2 = x2[i]; 151 a3 = x3[i]; 152 153 SETMODULUS(P2); 154 s = ext_submod(a2, a1, umod); 155 s = MULMOD(s, INV_P1_MOD_P2); 156 157 _mpd_mul_words(&hi, &lo, s, p1); 158 lo = lo + a1; 159 if (lo < a1) hi++; 160 161 SETMODULUS(P3); 162 s = dw_submod(a3, hi, lo, umod); 163 s = MULMOD(s, INV_P1P2_MOD_P3); 164 165 z[0] = lo; 166 z[1] = hi; 167 z[2] = 0; 168 169 _crt_mulP1P2_3(t, s); 170 _crt_add3(z, t); 171 _crt_add3(carry, z); 172 173 x1[i] = _crt_div3(carry, carry, MPD_RADIX); 174 } 175 176 assert(carry[0] == 0 && carry[1] == 0 && carry[2] == 0); 177 } 178 179 180