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, 2018 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-impl.h"
34
35 /* Computes a^2e (mod B). Uses right-to-left binary algorithm, since
36 typical use will have e small. */
37 static mp_limb_t
38 powsquaredlimb (mp_limb_t a, mp_limb_t e)
39 {
40 mp_limb_t r;
41
42 r = 1;
43 /* if (LIKELY (e != 0)) */
44 do {
45 a *= a;
46 if (e & 1)
47 r *= a;
48 e >>= 1;
49 } while (e != 0);
50
51 return r;
52 }
53
54 /* Compute r such that r^k * y = 1 (mod B^n).
55
56 Iterates
57 r' <-- k^{-1} ((k+1) r - r^{k+1} y) (mod 2^b)
58 using Hensel lifting, each time doubling the number of known bits in r.
59
60 Works just for odd k. Else the Hensel lifting degenerates.
61
62 FIXME:
63
64 (1) Make it work for k == GMP_LIMB_MAX (k+1 below overflows).
65
66 (2) Rewrite iteration as
67 r' <-- r - k^{-1} r (r^k y - 1)
68 and take advantage of the zero low part of r^k y - 1.
69
70 (3) Use wrap-around trick.
71
72 (4) Use a small table to get starting value.
73
74 Scratch need: bn + (((bn + 1) >> 1) + 1) + scratch for mpn_powlo
75 Currently mpn_powlo requires 3*bn
76 so that 5*bn is surely enough, where bn = ceil (bnb / GMP_NUMB_BITS).
77 */
78
79 void
80 mpn_brootinv (mp_ptr rp, mp_srcptr yp, mp_size_t bn, mp_limb_t k, mp_ptr tp)
81 {
82 mp_ptr tp2, tp3;
83 mp_limb_t kinv, k2, r0, y0;
84 mp_size_t order[GMP_LIMB_BITS + 1];
85 int d;
86
87 ASSERT (bn > 0);
88 ASSERT ((k & 1) != 0);
89
90 tp2 = tp + bn;
91 tp3 = tp + bn + ((bn + 3) >> 1);
92 k2 = (k >> 1) + 1; /* (k + 1) / 2 , but avoid k+1 overflow */
93
94 binvert_limb (kinv, k);
95
96 /* 4-bit initial approximation:
97
98 y%16 | 1 3 5 7 9 11 13 15,
99 k%4 +-------------------------+k2%2
100 1 | 1 11 13 7 9 3 5 15 | 1
101 3 | 1 3 5 7 9 11 13 15 | 0
102
103 */
104 y0 = yp[0];
105
106 r0 = y0 ^ (((y0 << 1) ^ (y0 << 2)) & (k2 << 3) & 8); /* 4 bits */
107 r0 = kinv * (k2 * r0 * 2 - y0 * powsquaredlimb(r0, k2 & 0x3f)); /* 8 bits */
108 r0 = kinv * (k2 * r0 * 2 - y0 * powsquaredlimb(r0, k2 & 0x3fff)); /* 16 bits */
109 #if GMP_NUMB_BITS > 16
110 {
111 unsigned prec = 16;
112 do
113 {
114 r0 = kinv * (k2 * r0 * 2 - y0 * powsquaredlimb(r0, k2));
115 prec *= 2;
116 }
117 while (prec < GMP_NUMB_BITS);
118 }
119 #endif
120
121 rp[0] = r0;
122 if (bn == 1)
123 return;
124
125 d = 0;
126 for (; bn != 2; bn = (bn + 1) >> 1)
127 order[d++] = bn;
128
129 order[d] = 2;
130 bn = 1;
131
132 do
133 {
134 mpn_sqr (tp, rp, bn); /* Result may overlap tp2 */
135 tp2[bn] = mpn_mul_1 (tp2, rp, bn, k2 << 1);
136
137 bn = order[d];
138
139 mpn_powlo (rp, tp, &k2, 1, bn, tp3);
140 mpn_mullo_n (tp, yp, rp, bn);
141
142 /* mpn_sub (tp, tp2, ((bn + 1) >> 1) + 1, tp, bn); */
143 /* The function above is not handled, ((bn + 1) >> 1) + 1 <= bn*/
144 {
145 mp_size_t pbn = (bn + 3) >> 1; /* Size of tp2 */
146 int borrow;
147 borrow = mpn_sub_n (tp, tp2, tp, pbn) != 0;
148 if (bn > pbn) /* 3 < bn */
149 {
150 if (borrow)
151 mpn_com (tp + pbn, tp + pbn, bn - pbn);
152 else
153 mpn_neg (tp + pbn, tp + pbn, bn - pbn);
154 }
155 }
156 mpn_pi1_bdiv_q_1 (rp, tp, bn, k, kinv, 0);
157 }
158 while (--d >= 0);
159 }