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
/*
Copyright (C) 2020 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.h"
#include "fmpz_vec.h"
#include "fmpz_poly.h"
ulong
_fmpz_poly_is_cyclotomic(const fmpz * poly, slong len)
{
ulong * phi;
ulong i, d, p, q, N1, N2;
ulong res;
double U;
fmpz_poly_t tmp;
d = len - 1;
if ((slong) d < 1)
return 0;
if (d == 1)
{
if (fmpz_is_one(poly + 1))
{
if (fmpz_is_one(poly))
return 2;
if (fmpz_equal_si(poly, -1))
return 1;
}
return 0;
}
if (d % 2 != 0)
return 0;
if (!fmpz_is_one(poly))
return 0;
/* Rule out non-palindromes */
for (i = 0; i < d / 2; i++)
{
if (!fmpz_equal(poly + i, poly + d - i))
return 0;
}
/* Compute inverse image of the totient function to find candidate
indices of the cyclotomic polynomial */
/* Determine lower and upper bounds [N1, N2) */
U = d;
for (p = 2; p <= d; p++)
if (d % (p - 1) == 0 && n_is_prime(p))
U = (U * p) / (p - 1);
N1 = d + 1;
N2 = U + 3; /* +3 as safety for possible floating-point rounding */
res = 0;
phi = flint_malloc(N2 * sizeof(ulong));
fmpz_poly_init(tmp);
for (i = 0; i < N2; i++)
phi[i] = i;
for (p = 2; p < N2; p++)
{
if (phi[p] == p)
{
phi[p] = p - 1;
for (q = 2 * p; q < N2; q += p)
phi[q] = (phi[q] / p) * (p - 1);
}
}
for (i = N1; i < N2 && !res; i++)
{
if (phi[i] == d)
{
/* todo: we could avoid O(len) overhead by computing the
factorisation of phi and calling _fmpz_poly_cyclotomic,
checking only the deflated polynomial */
fmpz_poly_cyclotomic(tmp, i);
if (tmp->length == len && _fmpz_vec_equal(poly, tmp->coeffs, len))
res = i;
}
}
flint_free(phi);
fmpz_poly_clear(tmp);
return res;
}
ulong
fmpz_poly_is_cyclotomic(const fmpz_poly_t poly)
{
return _fmpz_poly_is_cyclotomic(poly->coeffs, poly->length);
}