1 /*
2 * Copyright (c) 2008-2020 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
31 #include <assert.h>
32
33 #include "constants.h"
34 #include "crt.h"
35 #include "numbertheory.h"
36 #include "typearith.h"
37 #include "umodarith.h"
38
39
40 /* Bignum: Chinese Remainder Theorem, extends the maximum transform length. */
41
42
43 /* Multiply P1P2 by v, store result in w. */
44 static inline void
45 _crt_mulP1P2_3(mpd_uint_t w[3], mpd_uint_t v)
46 {
47 mpd_uint_t hi1, hi2, lo;
48
49 _mpd_mul_words(&hi1, &lo, LH_P1P2, v);
50 w[0] = lo;
51
52 _mpd_mul_words(&hi2, &lo, UH_P1P2, v);
53 lo = hi1 + lo;
54 if (lo < hi1) hi2++;
55
56 w[1] = lo;
57 w[2] = hi2;
58 }
59
60 /* Add 3 words from v to w. The result is known to fit in w. */
61 static inline void
62 _crt_add3(mpd_uint_t w[3], mpd_uint_t v[3])
63 {
64 mpd_uint_t carry;
65
66 w[0] = w[0] + v[0];
67 carry = (w[0] < v[0]);
68
69 w[1] = w[1] + v[1];
70 if (w[1] < v[1]) w[2]++;
71
72 w[1] = w[1] + carry;
73 if (w[1] < carry) w[2]++;
74
75 w[2] += v[2];
76 }
77
78 /* Divide 3 words in u by v, store result in w, return remainder. */
79 static inline mpd_uint_t
80 _crt_div3(mpd_uint_t *w, const mpd_uint_t *u, mpd_uint_t v)
81 {
82 mpd_uint_t r1 = u[2];
83 mpd_uint_t r2;
84
85 if (r1 < v) {
86 w[2] = 0;
87 }
88 else {
89 _mpd_div_word(&w[2], &r1, u[2], v); /* GCOV_NOT_REACHED */
90 }
91
92 _mpd_div_words(&w[1], &r2, r1, u[1], v);
93 _mpd_div_words(&w[0], &r1, r2, u[0], v);
94
95 return r1;
96 }
97
98
99 /*
100 * Chinese Remainder Theorem:
101 * Algorithm from Joerg Arndt, "Matters Computational",
102 * Chapter 37.4.1 [http://www.jjj.de/fxt/]
103 *
104 * See also Knuth, TAOCP, Volume 2, 4.3.2, exercise 7.
105 */
106
107 /*
108 * CRT with carry: x1, x2, x3 contain numbers modulo p1, p2, p3. For each
109 * triple of members of the arrays, find the unique z modulo p1*p2*p3, with
110 * zmax = p1*p2*p3 - 1.
111 *
112 * In each iteration of the loop, split z into result[i] = z % MPD_RADIX
113 * and carry = z / MPD_RADIX. Let N be the size of carry[] and cmax the
114 * maximum carry.
115 *
116 * Limits for the 32-bit build:
117 *
118 * N = 2**96
119 * cmax = 7711435591312380274
120 *
121 * Limits for the 64 bit build:
122 *
123 * N = 2**192
124 * cmax = 627710135393475385904124401220046371710
125 *
126 * The following statements hold for both versions:
127 *
128 * 1) cmax + zmax < N, so the addition does not overflow.
129 *
130 * 2) (cmax + zmax) / MPD_RADIX == cmax.
131 *
132 * 3) If c <= cmax, then c_next = (c + zmax) / MPD_RADIX <= cmax.
133 */
134 void
135 crt3(mpd_uint_t *x1, mpd_uint_t *x2, mpd_uint_t *x3, mpd_size_t rsize)
136 {
137 mpd_uint_t p1 = mpd_moduli[P1];
138 mpd_uint_t umod;
139 #ifdef PPRO
140 double dmod;
141 uint32_t dinvmod[3];
142 #endif
143 mpd_uint_t a1, a2, a3;
144 mpd_uint_t s;
145 mpd_uint_t z[3], t[3];
146 mpd_uint_t carry[3] = {0,0,0};
147 mpd_uint_t hi, lo;
148 mpd_size_t i;
149
150 for (i = 0; i < rsize; i++) {
151
152 a1 = x1[i];
153 a2 = x2[i];
154 a3 = x3[i];
155
156 SETMODULUS(P2);
157 s = ext_submod(a2, a1, umod);
158 s = MULMOD(s, INV_P1_MOD_P2);
159
160 _mpd_mul_words(&hi, &lo, s, p1);
161 lo = lo + a1;
162 if (lo < a1) hi++;
163
164 SETMODULUS(P3);
165 s = dw_submod(a3, hi, lo, umod);
166 s = MULMOD(s, INV_P1P2_MOD_P3);
167
168 z[0] = lo;
169 z[1] = hi;
170 z[2] = 0;
171
172 _crt_mulP1P2_3(t, s);
173 _crt_add3(z, t);
174 _crt_add3(carry, z);
175
176 x1[i] = _crt_div3(carry, carry, MPD_RADIX);
177 }
178
179 assert(carry[0] == 0 && carry[1] == 0 && carry[2] == 0);
180 }