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
|
#############################################################################
##
## This file is part of GAP, a system for computational discrete algebra.
## This file's authors include Thomas Breuer, Frank Celler.
##
## Copyright of GAP belongs to its developers, whose names are too numerous
## to list here. Please refer to the COPYRIGHT file for details.
##
## SPDX-License-Identifier: GPL-2.0-or-later
##
## This file deals with internal finite field elements.
##
#############################################################################
##
#V MAXSIZE_GF_INTERNAL . . . . . . . . . . . . maximal size of internal ffes
##
## Now set by the kernel.
#############################################################################
##
#F TYPE_FFE( <p> ) . . . . . . . . . . . type of a ffe in characteristic <p>
##
## <p> must be a small prime integer
## (see also `ffe.gi').
##
## Note that the `One' and `Zero' values of the family cannot be set
## in `TYPE_FFE' since this would need access to `One( Z(<p>) )' and
## `Zero( Z(<p>) )', respectively,
## which in turn would call `TYPE_FFE' and thus would lead to an infinite
## recursion.
##
BIND_GLOBAL( "TYPE_FFE", MemoizePosIntFunction(
function( p )
local fam;
fam:= NewFamily( "FFEFamily",
IS_FFE,CanEasilySortElements,CanEasilySortElements );
SetFilterObj( fam, IsUFDFamily );
SetCharacteristic( fam, p );
return NewType( fam, IS_FFE and IsInternalRep and HasDegreeFFE);
end, rec(flush := false) ));
#############################################################################
##
#F TYPE_FFE0( <p> ) . . . . . . . . .type of zero ffe in characteristic <p>
##
## see also "ffe.gi"
##
BIND_GLOBAL( "TYPE_FFE0", MemoizePosIntFunction(
function ( p )
local fam;
fam:= FamilyType(TYPE_FFE(p));
return NewType( fam, IS_FFE and IsInternalRep and IsZero and HasIsZero
and HasDegreeFFE );
end, rec(flush := false) ));
#############################################################################
##
#m DegreeFEE( <ffe> ) . . . . . . . . . . . . . . . . . . for internal ffe
##
InstallMethod( DegreeFFE,
"for internal FFE",
true,
[ IsFFE and IsInternalRep ], 0,
DEGREE_FFE_DEFAULT );
#############################################################################
##
#m Characteristic( <ffe> ) . . . . . . . . . . . . . . . for internal ffe
##
InstallMethod( Characteristic,
"for internal FFE",
true,
[ IsFFE and IsInternalRep ], 0,
CHAR_FFE_DEFAULT );
#############################################################################
##
#M LogFFE( <ffe>, <ffe> ) . . . . . . . . . . . . . . . . for internal ffe
##
InstallMethod( LogFFE,
"for two internal FFEs",
IsIdenticalObj,
[ IsFFE and IsInternalRep, IsFFE and IsInternalRep ], 0,
LOG_FFE_DEFAULT );
#############################################################################
##
#M IntFFE( <ffe> ) . . . . . . . . . . . . . . . . . . . . for internal ffe
##
InstallMethod( IntFFE,
"for internal FFE",
true,
[ IsFFE and IsInternalRep ], 0,
INT_FFE_DEFAULT );
#############################################################################
##
#m \*( <ffe>, <int> ) . . . . . . . . . . . . . for ffe and (large) integer
##
## Note that the multiplication of internally represented FFEs with small
## integers is handled by the kernel.
##
InstallOtherMethod( \*,
"internal ffe * (large) integer",
true,
[ IsFFE and IsInternalRep, IsInt ], 0,
function( ffe, int )
local char;
char:= Characteristic( ffe );
if IsSmallIntRep( char ) then
return ffe * ( int mod char );
else
return PROD_INT_OBJ( int, ffe );
fi;
end );
#############################################################################
##
#O SUM_FFE_LARGE
#O DIFF_FFE_LARGE
#O PROD_FFE_LARGE
#O QUO_FFE_LARGE
#O LOG_FFE_LARGE
##
## If the {\GAP} kernel cannot handle the addition, multiplication etc.
## of internally represented FFEs then it delegates to the library without
## checking the characteristic; therefore this check must be done here.
## (Note that `LogFFE' is an operation for which the kernel does not know
## a table of methods, so the check for equal characteristic is done by
## the method selection.
#T Note that `LogFFEHandler' would not need to call `LOG_FFE_DEFAULT';
#T if the two arguments <z>, <r> are represented w.r.t. incompatible fields
#T then either <z> can be represented in the field of <r> or the logarithm
#T does not exist.
##
DeclareOperation("SUM_FFE_LARGE", [IsFFE and IsInternalRep,
IsFFE and IsInternalRep]);
InstallOtherMethod(SUM_FFE_LARGE, [IsFFE,
IsFFE],
function( x, y )
if Characteristic( x ) <> Characteristic( y ) then
Error( "<x> and <y> have different characteristic" );
fi;
TryNextMethod();
end);
DeclareOperation("DIFF_FFE_LARGE", [IsFFE and IsInternalRep,
IsFFE and IsInternalRep]);
InstallOtherMethod(DIFF_FFE_LARGE, [IsFFE,
IsFFE],
function( x, y )
if Characteristic( x ) <> Characteristic( y ) then
Error( "<x> and <y> have different characteristic" );
fi;
TryNextMethod();
end);
DeclareOperation("PROD_FFE_LARGE", [IsFFE and IsInternalRep,
IsFFE and IsInternalRep]);
InstallOtherMethod(PROD_FFE_LARGE, [IsFFE,
IsFFE ],
function( x, y )
if Characteristic( x ) <> Characteristic( y ) then
Error( "<x> and <y> have different characteristic" );
fi;
TryNextMethod();
end);
DeclareOperation("QUO_FFE_LARGE", [IsFFE,
IsFFE]);
InstallOtherMethod(QUO_FFE_LARGE, [IsFFE and IsInternalRep,
IsFFE and IsInternalRep],
function( x, y )
if Characteristic( x ) <> Characteristic( y ) then
Error( "<x> and <y> have different characteristic" );
fi;
TryNextMethod();
end);
BIND_GLOBAL( "LOG_FFE_LARGE", function( x, y )
Error( "not supported yet -- this should never happen" );
end );
#############################################################################
##
#O ZOp -- operation to compute Z for large values of q
##
DeclareOperation("ZOp", [IsPosInt]);
|
