1 /*
2 * Copyright (c) 2008-2024 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 * 2. Redistributions in binary form must reproduce the above copyright
11 * notice, this list of conditions and the following disclaimer in the
12 * documentation and/or other materials provided with the distribution.
13 *
14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND
15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE
18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL
19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY
23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF
24 * SUCH DAMAGE.
25 */
26
27
28 #include <assert.h>
29
30 #include "constants.h"
31 #include "crt.h"
32 #include "numbertheory.h"
33 #include "mpdecimal.h"
34 #include "typearith.h"
35 #include "umodarith.h"
36
37
38 /* Bignum: Chinese Remainder Theorem, extends the maximum transform length. */
39
40
41 /* Multiply P1P2 by v, store result in w. */
42 static inline void
_crt_mulP1P2_3(mpd_uint_t w[3],mpd_uint_t v)43 _crt_mulP1P2_3(mpd_uint_t w[3], mpd_uint_t v)
44 {
45 mpd_uint_t hi1, hi2, lo;
46
47 _mpd_mul_words(&hi1, &lo, LH_P1P2, v);
48 w[0] = lo;
49
50 _mpd_mul_words(&hi2, &lo, UH_P1P2, v);
51 lo = hi1 + lo;
52 if (lo < hi1) hi2++;
53
54 w[1] = lo;
55 w[2] = hi2;
56 }
57
58 /* Add 3 words from v to w. The result is known to fit in w. */
59 static inline void
_crt_add3(mpd_uint_t w[3],mpd_uint_t v[3])60 _crt_add3(mpd_uint_t w[3], mpd_uint_t v[3])
61 {
62 mpd_uint_t carry;
63
64 w[0] = w[0] + v[0];
65 carry = (w[0] < v[0]);
66
67 w[1] = w[1] + v[1];
68 if (w[1] < v[1]) w[2]++;
69
70 w[1] = w[1] + carry;
71 if (w[1] < carry) w[2]++;
72
73 w[2] += v[2];
74 }
75
76 /* Divide 3 words in u by v, store result in w, return remainder. */
77 static inline mpd_uint_t
_crt_div3(mpd_uint_t * w,const mpd_uint_t * u,mpd_uint_t v)78 _crt_div3(mpd_uint_t *w, const mpd_uint_t *u, mpd_uint_t v)
79 {
80 mpd_uint_t r1 = u[2];
81 mpd_uint_t r2;
82
83 if (r1 < v) {
84 w[2] = 0;
85 }
86 else {
87 _mpd_div_word(&w[2], &r1, u[2], v); /* GCOV_NOT_REACHED */
88 }
89
90 _mpd_div_words(&w[1], &r2, r1, u[1], v);
91 _mpd_div_words(&w[0], &r1, r2, u[0], v);
92
93 return r1;
94 }
95
96
97 /*
98 * Chinese Remainder Theorem:
99 * Algorithm from Joerg Arndt, "Matters Computational",
100 * Chapter 37.4.1 [http://www.jjj.de/fxt/]
101 *
102 * See also Knuth, TAOCP, Volume 2, 4.3.2, exercise 7.
103 */
104
105 /*
106 * CRT with carry: x1, x2, x3 contain numbers modulo p1, p2, p3. For each
107 * triple of members of the arrays, find the unique z modulo p1*p2*p3, with
108 * zmax = p1*p2*p3 - 1.
109 *
110 * In each iteration of the loop, split z into result[i] = z % MPD_RADIX
111 * and carry = z / MPD_RADIX. Let N be the size of carry[] and cmax the
112 * maximum carry.
113 *
114 * Limits for the 32-bit build:
115 *
116 * N = 2**96
117 * cmax = 7711435591312380274
118 *
119 * Limits for the 64 bit build:
120 *
121 * N = 2**192
122 * cmax = 627710135393475385904124401220046371710
123 *
124 * The following statements hold for both versions:
125 *
126 * 1) cmax + zmax < N, so the addition does not overflow.
127 *
128 * 2) (cmax + zmax) / MPD_RADIX == cmax.
129 *
130 * 3) If c <= cmax, then c_next = (c + zmax) / MPD_RADIX <= cmax.
131 */
132 void
crt3(mpd_uint_t * x1,mpd_uint_t * x2,mpd_uint_t * x3,mpd_size_t rsize)133 crt3(mpd_uint_t *x1, mpd_uint_t *x2, mpd_uint_t *x3, mpd_size_t rsize)
134 {
135 mpd_uint_t p1 = mpd_moduli[P1];
136 mpd_uint_t umod;
137 #ifdef PPRO
138 double dmod;
139 uint32_t dinvmod[3];
140 #endif
141 mpd_uint_t a1, a2, a3;
142 mpd_uint_t s;
143 mpd_uint_t z[3], t[3];
144 mpd_uint_t carry[3] = {0,0,0};
145 mpd_uint_t hi, lo;
146 mpd_size_t i;
147
148 for (i = 0; i < rsize; i++) {
149
150 a1 = x1[i];
151 a2 = x2[i];
152 a3 = x3[i];
153
154 SETMODULUS(P2);
155 s = ext_submod(a2, a1, umod);
156 s = MULMOD(s, INV_P1_MOD_P2);
157
158 _mpd_mul_words(&hi, &lo, s, p1);
159 lo = lo + a1;
160 if (lo < a1) hi++;
161
162 SETMODULUS(P3);
163 s = dw_submod(a3, hi, lo, umod);
164 s = MULMOD(s, INV_P1P2_MOD_P3);
165
166 z[0] = lo;
167 z[1] = hi;
168 z[2] = 0;
169
170 _crt_mulP1P2_3(t, s);
171 _crt_add3(z, t);
172 _crt_add3(carry, z);
173
174 x1[i] = _crt_div3(carry, carry, MPD_RADIX);
175 }
176
177 assert(carry[0] == 0 && carry[1] == 0 && carry[2] == 0);
178 }
179