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
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.01//EN" "http://www.w3.org/TR/html4/strict.dtd">
<html>
<head>
<meta http-equiv="content-type" content="text/html; charset=UTF-8">
<title>~/ntl-11.4.2/doc/ZZ_pXFactoring.cpp.html</title>
<meta name="Generator" content="Vim/8.0">
<meta name="plugin-version" content="vim7.4_v2">
<meta name="syntax" content="cpp">
<meta name="settings" content="use_css,pre_wrap,no_foldcolumn,expand_tabs,prevent_copy=">
<meta name="colorscheme" content="macvim">
<style type="text/css">
<!--
pre { white-space: pre-wrap; font-family: monospace; color: #000000; background-color: #ffffff; }
body { font-family: monospace; color: #000000; background-color: #ffffff; }
* { font-size: 1em; }
.String { color: #4a708b; }
.PreProc { color: #1874cd; }
.Constant { color: #ff8c00; }
.Comment { color: #0000ee; font-style: italic; }
.Type { color: #008b00; font-weight: bold; }
-->
</style>
<script type='text/javascript'>
<!--
-->
</script>
</head>
<body>
<pre id='vimCodeElement'>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment">MODULE: ZZ_pXFactoring</span>
<span class="Comment">SUMMARY:</span>
<span class="Comment">Routines are provided for factorization of polynomials over ZZ_p, as</span>
<span class="Comment">well as routines for related problems such as testing irreducibility</span>
<span class="Comment">and constructing irreducible polynomials of given degree.</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="PreProc">#include </span><span class="String"><NTL/ZZ_pX.h></span>
<span class="PreProc">#include </span><span class="String"><NTL/pair_ZZ_pX_long.h></span>
<span class="Type">void</span> SquareFreeDecomp(vec_pair_ZZ_pX_long& u, <span class="Type">const</span> ZZ_pX& f);
vec_pair_ZZ_pX_long SquareFreeDecomp(<span class="Type">const</span> ZZ_pX& f);
<span class="Comment">// Performs square-free decomposition. f must be monic. If f =</span>
<span class="Comment">// prod_i g_i^i, then u is set to a lest of pairs (g_i, i). The list</span>
<span class="Comment">// is is increasing order of i, with trivial terms (i.e., g_i = 1)</span>
<span class="Comment">// deleted.</span>
<span class="Type">void</span> FindRoots(vec_ZZ_p& x, <span class="Type">const</span> ZZ_pX& f);
vec_ZZ_p FindRoots(<span class="Type">const</span> ZZ_pX& f);
<span class="Comment">// f is monic, and has deg(f) distinct roots. returns the list of</span>
<span class="Comment">// roots</span>
<span class="Type">void</span> FindRoot(ZZ_p& root, <span class="Type">const</span> ZZ_pX& f);
ZZ_p FindRoot(<span class="Type">const</span> ZZ_pX& f);
<span class="Comment">// finds a single root of f. assumes that f is monic and splits into</span>
<span class="Comment">// distinct linear factors</span>
<span class="Type">void</span> SFBerlekamp(vec_ZZ_pX& factors, <span class="Type">const</span> ZZ_pX& f, <span class="Type">long</span> verbose=<span class="Constant">0</span>);
vec_ZZ_pX SFBerlekamp(<span class="Type">const</span> ZZ_pX& f, <span class="Type">long</span> verbose=<span class="Constant">0</span>);
<span class="Comment">// Assumes f is square-free and monic. returns list of factors of f.</span>
<span class="Comment">// Uses "Berlekamp" approach, as described in detail in [Shoup,</span>
<span class="Comment">// J. Symbolic Comp. 20:363-397, 1995].</span>
<span class="Type">void</span> berlekamp(vec_pair_ZZ_pX_long& factors, <span class="Type">const</span> ZZ_pX& f,
<span class="Type">long</span> verbose=<span class="Constant">0</span>);
vec_pair_ZZ_pX_long berlekamp(<span class="Type">const</span> ZZ_pX& f, <span class="Type">long</span> verbose=<span class="Constant">0</span>);
<span class="Comment">// returns a list of factors, with multiplicities. f must be monic.</span>
<span class="Comment">// Calls SFBerlekamp.</span>
<span class="Type">void</span> NewDDF(vec_pair_ZZ_pX_long& factors, <span class="Type">const</span> ZZ_pX& f, <span class="Type">const</span> ZZ_pX& h,
<span class="Type">long</span> verbose=<span class="Constant">0</span>);
vec_pair_ZZ_pX_long NewDDF(<span class="Type">const</span> ZZ_pX& f, <span class="Type">const</span> ZZ_pX& h,
<span class="Type">long</span> verbose=<span class="Constant">0</span>);
<span class="Comment">// This computes a distinct-degree factorization. The input must be</span>
<span class="Comment">// monic and square-free. factors is set to a list of pairs (g, d),</span>
<span class="Comment">// where g is the product of all irreducible factors of f of degree d.</span>
<span class="Comment">// Only nontrivial pairs (i.e., g != 1) are included. The polynomial</span>
<span class="Comment">// h is assumed to be equal to X^p mod f. </span>
<span class="Comment">// This routine implements the baby step/giant step algorithm </span>
<span class="Comment">// of [Kaltofen and Shoup, STOC 1995].</span>
<span class="Comment">// further described in [Shoup, J. Symbolic Comp. 20:363-397, 1995].</span>
<span class="Comment">// NOTE: When factoring "large" polynomials,</span>
<span class="Comment">// this routine uses external files to store some intermediate</span>
<span class="Comment">// results, which are removed if the routine terminates normally.</span>
<span class="Comment">// These files are stored in the current directory under names of the</span>
<span class="Comment">// form tmp-*.</span>
<span class="Comment">// The definition of "large" is controlled by the variable</span>
<span class="Type">extern</span> <span class="Type">thread_local</span> <span class="Type">double</span> ZZ_pXFileThresh
<span class="Comment">// which can be set by the user. If the sizes of the tables</span>
<span class="Comment">// exceeds ZZ_pXFileThresh KB, external files are used.</span>
<span class="Comment">// Initial value is NTL_FILE_THRESH (defined in tools.h).</span>
<span class="Type">void</span> EDF(vec_ZZ_pX& factors, <span class="Type">const</span> ZZ_pX& f, <span class="Type">const</span> ZZ_pX& h,
<span class="Type">long</span> d, <span class="Type">long</span> verbose=<span class="Constant">0</span>);
vec_ZZ_pX EDF(<span class="Type">const</span> ZZ_pX& f, <span class="Type">const</span> ZZ_pX& h,
<span class="Type">long</span> d, <span class="Type">long</span> verbose=<span class="Constant">0</span>);
<span class="Comment">// Performs equal-degree factorization. f is monic, square-free, and</span>
<span class="Comment">// all irreducible factors have same degree. h = X^p mod f. d =</span>
<span class="Comment">// degree of irreducible factors of f. This routine implements the</span>
<span class="Comment">// algorithm of [von zur Gathen and Shoup, Computational Complexity</span>
<span class="Comment">// 2:187-224, 1992].</span>
<span class="Type">void</span> RootEDF(vec_ZZ_pX& factors, <span class="Type">const</span> ZZ_pX& f, <span class="Type">long</span> verbose=<span class="Constant">0</span>);
vec_ZZ_pX RootEDF(<span class="Type">const</span> ZZ_pX& f, <span class="Type">long</span> verbose=<span class="Constant">0</span>);
<span class="Comment">// EDF for d==1</span>
<span class="Type">void</span> SFCanZass(vec_ZZ_pX& factors, <span class="Type">const</span> ZZ_pX& f, <span class="Type">long</span> verbose=<span class="Constant">0</span>);
vec_ZZ_pX SFCanZass(<span class="Type">const</span> ZZ_pX& f, <span class="Type">long</span> verbose=<span class="Constant">0</span>);
<span class="Comment">// Assumes f is monic and square-free. returns list of factors of f.</span>
<span class="Comment">// Uses "Cantor/Zassenhaus" approach, using the routines NewDDF and</span>
<span class="Comment">// EDF above.</span>
<span class="Type">void</span> CanZass(vec_pair_ZZ_pX_long& factors, <span class="Type">const</span> ZZ_pX& f,
<span class="Type">long</span> verbose=<span class="Constant">0</span>);
vec_pair_ZZ_pX_long CanZass(<span class="Type">const</span> ZZ_pX& f, <span class="Type">long</span> verbose=<span class="Constant">0</span>);
<span class="Comment">// returns a list of factors, with multiplicities. f must be monic.</span>
<span class="Comment">// Calls SquareFreeDecomp and SFCanZass.</span>
<span class="Comment">// NOTE: In most situations, you should use the CanZass factoring</span>
<span class="Comment">// routine, rather than Berlekamp: it is faster and uses less space.</span>
<span class="Type">void</span> mul(ZZ_pX& f, <span class="Type">const</span> vec_pair_ZZ_pX_long& v);
ZZ_pX mul(<span class="Type">const</span> vec_pair_ZZ_pX_long& v);
<span class="Comment">// multiplies polynomials, with multiplicities</span>
<span class="Comment">/*</span><span class="Comment">*************************************************************************\</span>
<span class="Comment"> Irreducible Polynomials</span>
<span class="Comment">\*************************************************************************</span><span class="Comment">*/</span>
<span class="Type">long</span> ProbIrredTest(<span class="Type">const</span> ZZ_pX& f, <span class="Type">long</span> iter=<span class="Constant">1</span>);
<span class="Comment">// performs a fast, probabilistic irreduciblity test. The test can</span>
<span class="Comment">// err only if f is reducible, and the error probability is bounded by</span>
<span class="Comment">// p^{-iter}. This implements an algorithm from [Shoup, J. Symbolic</span>
<span class="Comment">// Comp. 17:371-391, 1994].</span>
<span class="Type">long</span> DetIrredTest(<span class="Type">const</span> ZZ_pX& f);
<span class="Comment">// performs a recursive deterministic irreducibility test. Fast in</span>
<span class="Comment">// the worst-case (when input is irreducible). This implements an</span>
<span class="Comment">// algorithm from [Shoup, J. Symbolic Comp. 17:371-391, 1994].</span>
<span class="Type">long</span> IterIrredTest(<span class="Type">const</span> ZZ_pX& f);
<span class="Comment">// performs an iterative deterministic irreducibility test, based on</span>
<span class="Comment">// DDF. Fast on average (when f has a small factor).</span>
<span class="Type">void</span> BuildIrred(ZZ_pX& f, <span class="Type">long</span> n);
ZZ_pX BuildIrred_ZZ_pX(<span class="Type">long</span> n);
<span class="Comment">// Build a monic irreducible poly of degree n.</span>
<span class="Type">void</span> BuildRandomIrred(ZZ_pX& f, <span class="Type">const</span> ZZ_pX& g);
ZZ_pX BuildRandomIrred(<span class="Type">const</span> ZZ_pX& g);
<span class="Comment">// g is a monic irreducible polynomial. Constructs a random monic</span>
<span class="Comment">// irreducible polynomial f of the same degree.</span>
<span class="Type">long</span> ComputeDegree(<span class="Type">const</span> ZZ_pX& h, <span class="Type">const</span> ZZ_pXModulus& F);
<span class="Comment">// f is assumed to be an "equal degree" polynomial; h = X^p mod f.</span>
<span class="Comment">// The common degree of the irreducible factors of f is computed This</span>
<span class="Comment">// routine is useful in counting points on elliptic curves</span>
<span class="Type">long</span> ProbComputeDegree(<span class="Type">const</span> ZZ_pX& h, <span class="Type">const</span> ZZ_pXModulus& F);
<span class="Comment">// Same as above, but uses a slightly faster probabilistic algorithm.</span>
<span class="Comment">// The return value may be 0 or may be too big, but for large p</span>
<span class="Comment">// (relative to n), this happens with very low probability.</span>
<span class="Type">void</span> TraceMap(ZZ_pX& w, <span class="Type">const</span> ZZ_pX& a, <span class="Type">long</span> d, <span class="Type">const</span> ZZ_pXModulus& F,
<span class="Type">const</span> ZZ_pX& h);
ZZ_pX TraceMap(<span class="Type">const</span> ZZ_pX& a, <span class="Type">long</span> d, <span class="Type">const</span> ZZ_pXModulus& F,
<span class="Type">const</span> ZZ_pX& h);
<span class="Comment">// w = a+a^q+...+^{q^{d-1}} mod f; it is assumed that d >= 0, and h =</span>
<span class="Comment">// X^q mod f, q a power of p. This routine implements an algorithm</span>
<span class="Comment">// from [von zur Gathen and Shoup, Computational Complexity 2:187-224,</span>
<span class="Comment">// 1992].</span>
<span class="Type">void</span> PowerCompose(ZZ_pX& w, <span class="Type">const</span> ZZ_pX& h, <span class="Type">long</span> d, <span class="Type">const</span> ZZ_pXModulus& F);
ZZ_pX PowerCompose(<span class="Type">const</span> ZZ_pX& h, <span class="Type">long</span> d, <span class="Type">const</span> ZZ_pXModulus& F);
<span class="Comment">// w = X^{q^d} mod f; it is assumed that d >= 0, and h = X^q mod f, q</span>
<span class="Comment">// a power of p. This routine implements an algorithm from [von zur</span>
<span class="Comment">// Gathen and Shoup, Computational Complexity 2:187-224, 1992]</span>
</pre>
</body>
</html>
<!-- vim: set foldmethod=manual : -->
|
