1 /* mpn_brootinv, compute r such that r^k * y = 1 (mod 2^b).
2
3 Contributed to the GNU project by Martin Boij (as part of perfpow.c).
4
5 Copyright 2009, 2010, 2012, 2013 Free Software Foundation, Inc.
6
7 This file is part of the GNU MP Library.
8
9 The GNU MP Library is free software; you can redistribute it and/or modify
10 it under the terms of either:
11
12 * the GNU Lesser General Public License as published by the Free
13 Software Foundation; either version 3 of the License, or (at your
14 option) any later version.
15
16 or
17
18 * the GNU General Public License as published by the Free Software
19 Foundation; either version 2 of the License, or (at your option) any
20 later version.
21
22 or both in parallel, as here.
23
24 The GNU MP Library is distributed in the hope that it will be useful, but
25 WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
26 or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
27 for more details.
28
29 You should have received copies of the GNU General Public License and the
30 GNU Lesser General Public License along with the GNU MP Library. If not,
31 see https://www.gnu.org/licenses/. */
32
33 #include "gmp.h"
34 #include "gmp-impl.h"
35
36 /* Computes a^e (mod B). Uses right-to-left binary algorithm, since
37 typical use will have e small. */
38 static mp_limb_t
powlimb(mp_limb_t a,mp_limb_t e)39 powlimb (mp_limb_t a, mp_limb_t e)
40 {
41 mp_limb_t r;
42
43 for (r = 1; e > 0; e >>= 1, a *= a)
44 if (e & 1)
45 r *= a;
46
47 return r;
48 }
49
50 /* Compute r such that r^k * y = 1 (mod B^n).
51
52 Iterates
53 r' <-- k^{-1} ((k+1) r - r^{k+1} y) (mod 2^b)
54 using Hensel lifting, each time doubling the number of known bits in r.
55
56 Works just for odd k. Else the Hensel lifting degenerates.
57
58 FIXME:
59
60 (1) Make it work for k == GMP_LIMB_MAX (k+1 below overflows).
61
62 (2) Rewrite iteration as
63 r' <-- r - k^{-1} r (r^k y - 1)
64 and take advantage of the zero low part of r^k y - 1.
65
66 (3) Use wrap-around trick.
67
68 (4) Use a small table to get starting value.
69
70 Scratch need: 5*bn, where bn = ceil (bnb / GMP_NUMB_BITS).
71 */
72
73 void
mpn_brootinv(mp_ptr rp,mp_srcptr yp,mp_size_t bn,mp_limb_t k,mp_ptr tp)74 mpn_brootinv (mp_ptr rp, mp_srcptr yp, mp_size_t bn, mp_limb_t k, mp_ptr tp)
75 {
76 mp_ptr tp2, tp3;
77 mp_limb_t kinv, k2, r0, y0;
78 mp_size_t order[GMP_LIMB_BITS + 1];
79 int i, d;
80
81 ASSERT (bn > 0);
82 ASSERT ((k & 1) != 0);
83
84 tp2 = tp + bn;
85 tp3 = tp + 2 * bn;
86 k2 = k + 1;
87
88 binvert_limb (kinv, k);
89
90 /* 4-bit initial approximation:
91
92 y%16 | 1 3 5 7 9 11 13 15,
93 k%4 +-------------------------+k2%4
94 1 | 1 11 13 7 9 3 5 15 | 2
95 3 | 1 3 5 7 9 11 13 15 | 0
96
97 */
98 y0 = yp[0];
99
100 r0 = y0 ^ (((y0 << 1) ^ (y0 << 2)) & (k2 << 2) & 8); /* 4 bits */
101 r0 = kinv * (k2 * r0 - y0 * powlimb(r0, k2 & 0x7f)); /* 8 bits */
102 r0 = kinv * (k2 * r0 - y0 * powlimb(r0, k2 & 0x7fff)); /* 16 bits */
103 #if GMP_NUMB_BITS > 16
104 {
105 unsigned prec = 16;
106 do
107 {
108 r0 = kinv * (k2 * r0 - y0 * powlimb(r0, k2));
109 prec *= 2;
110 }
111 while (prec < GMP_NUMB_BITS);
112 }
113 #endif
114
115 rp[0] = r0;
116 if (bn == 1)
117 return;
118
119 /* This initialization doesn't matter for the result (any garbage is
120 cancelled in the iteration), but proper initialization makes
121 valgrind happier. */
122 MPN_ZERO (rp+1, bn-1);
123
124 d = 0;
125 for (; bn > 1; bn = (bn + 1) >> 1)
126 order[d++] = bn;
127
128 for (i = d - 1; i >= 0; i--)
129 {
130 bn = order[i];
131
132 mpn_mul_1 (tp, rp, bn, k2);
133
134 mpn_powlo (tp2, rp, &k2, 1, bn, tp3);
135 mpn_mullo_n (rp, yp, tp2, bn);
136
137 mpn_sub_n (tp2, tp, rp, bn);
138 mpn_pi1_bdiv_q_1 (rp, tp2, bn, k, kinv, 0);
139 }
140 }
141