1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
/*
Copyright (C) 2012 Sebastian Pancratz
Copyright (C) 2012 Fredrik Johansson
This file is part of FLINT.
FLINT is free software: you can redistribute it and/or modify it under
the terms of the GNU Lesser General Public License (LGPL) as published
by the Free Software Foundation; either version 3 of the License, or
(at your option) any later version. See <https://www.gnu.org/licenses/>.
*/
#include "ulong_extras.h"
#include "fmpz_vec.h"
#include "padic.h"
/*
Computes the sum $1 + x + x^2 / 2$ reduced modulo $p^N$,
where $x = p^v u$.
Supports aliasing between \code{rop} and $u$.
*/
static void _padic_exp_small(fmpz_t rop, const fmpz_t u, slong v, slong n,
const fmpz_t p, const fmpz_t pN)
{
if (n == 1) /* rop = 1 */
{
fmpz_one(rop);
}
else if (n == 2) /* rop = 1 + x */
{
fmpz_t f;
fmpz_init(f);
fmpz_pow_ui(f, p, v);
fmpz_mul(rop, f, u);
fmpz_add_ui(rop, rop, 1);
fmpz_mod(rop, rop, pN);
fmpz_clear(f);
}
else /* n == 3, rop = 1 + x + x^2 / 2 */
{
fmpz_t f;
fmpz_init(f);
fmpz_pow_ui(f, p, v);
fmpz_mul(rop, f, u);
fmpz_mul(f, rop, rop);
if (fmpz_is_odd(f))
fmpz_add(f, f, pN);
fmpz_fdiv_q_2exp(f, f, 1);
fmpz_add(rop, rop, f);
fmpz_add_ui(rop, rop, 1);
fmpz_clear(f);
}
}
void _padic_exp_rectangular(fmpz_t rop, const fmpz_t u, slong v,
const fmpz_t p, slong N)
{
const slong n = _padic_exp_bound(v, N, p);
fmpz_t pN;
fmpz_init(pN);
fmpz_pow_ui(pN, p, N);
if (n <= 3)
{
_padic_exp_small(rop, u, v, n, p, pN);
}
else
{
const slong k = fmpz_fits_si(p) ?
(n - 1 - 1) / (fmpz_get_si(p) - 1) : 0;
slong i, npows, nsums;
fmpz_t c, f, s, t, sum, pNk;
fmpz *pows;
fmpz_init(pNk);
fmpz_pow_ui(pNk, p, N + k);
npows = n_sqrt(n);
nsums = (n + npows - 1) / npows;
fmpz_init(c);
fmpz_init(f);
fmpz_init(s);
fmpz_init(t);
fmpz_init(sum);
/* Compute pows; pows[i] = x^i. */
pows = _fmpz_vec_init(npows + 1);
fmpz_one(pows + 0);
fmpz_pow_ui(f, p, v);
fmpz_mul(pows + 1, f, u);
for (i = 2; i <= npows; i++)
{
fmpz_mul(pows + i, pows + i - 1, pows + 1);
fmpz_mod(pows + i, pows + i, pNk);
}
fmpz_zero(sum);
fmpz_one(f);
for (i = nsums - 1; i >= 0; i--)
{
slong lo = i * npows;
slong hi = FLINT_MIN(n - 1, lo + npows - 1);
fmpz_zero(s);
fmpz_one(c);
for ( ; hi >= lo; hi--)
{
fmpz_addmul(s, pows + hi - lo, c);
if (hi != 0)
fmpz_mul_ui(c, c, hi);
}
fmpz_mul(t, pows + npows, sum);
fmpz_mul(sum, s, f);
fmpz_add(sum, sum, t);
fmpz_mod(sum, sum, pNk);
fmpz_mul(f, f, c);
}
/* Divide by factorial, TODO: Improve */
/* Note exp(x) is a unit so val(sum) == val(f) */
if (fmpz_remove(sum, sum, p))
fmpz_remove(f, f, p);
_padic_inv(f, f, p, N);
fmpz_mul(rop, sum, f);
_fmpz_vec_clear(pows, npows + 1);
fmpz_clear(c);
fmpz_clear(f);
fmpz_clear(s);
fmpz_clear(t);
fmpz_clear(sum);
fmpz_clear(pNk);
}
fmpz_mod(rop, rop, pN);
fmpz_clear(pN);
}
int padic_exp_rectangular(padic_t rop, const padic_t op, const padic_ctx_t ctx)
{
const slong N = padic_prec(rop);
const slong v = padic_val(op);
const fmpz *p = ctx->p;
if (padic_is_zero(op))
{
padic_one(rop);
return 1;
}
if ((fmpz_equal_ui(p, 2) && v <= 1) || (v <= 0))
{
return 0;
}
else
{
if (v < N)
{
_padic_exp_rectangular(padic_unit(rop),
padic_unit(op), padic_val(op), p, N);
padic_val(rop) = 0;
}
else
{
padic_one(rop);
}
return 1;
}
}