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
|
// illl.cc: implementations of functions for integer LLL
//////////////////////////////////////////////////////////////////////////
//
// Copyright 1990-2023 John Cremona
//
// This file is part of the eclib package.
//
// eclib is free software; you can redistribute it and/or modify it
// under the terms of the GNU General Public License as published by the
// Free Software Foundation; either version 2 of the License, or (at your
// option) any later version.
//
// eclib is distributed in the hope that it will be useful, but WITHOUT
// ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or
// FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License
// for more details.
//
// You should have received a copy of the GNU General Public License
// along with eclib; if not, write to the Free Software Foundation,
// Inc., 51 Franklin St, Fifth Floor, Boston, MA 02110-1301, USA
//
//////////////////////////////////////////////////////////////////////////
#include <eclib/illl.h>
//#define DEBUG_LLL
//#define TRACE_LLL
bigint sdot(const vector<vec_m>& b, int i, int j)
{
bigint ans(0);
const vec_m& g=b[0], bi=b[i], bj=b[j];
int n=dim(g);
for(int k=1; k<=n; k++)
ans += (g[k]*bi[k]*bj[k]);
return ans;
}
void show(const int n, const vector<vec_m>& b, const vector<vector<bigint>>& lambda, const vector<bigint>& d)
{
int i=0, j;
cout<<"Vectors:\n";
for (const auto& bi : b)
{
if (i) cout<<bi<<endl;
i++;
}
cout<<endl;
cout<<"d: ";
for ( const auto& di : d) cout<<di<<"\t";
cout<<endl;
cout<<"Lambda matrix:\n";
for(i=1; i<=n; i++)
{
for(j=1; j<=i; j++)
if(j==i) cout<<d[i]<<"\t";
else cout<<lambda[i-1][j-1]<<"\t";
cout<<endl;
}
cout<<endl;
}
void redi(const int n, const int k, const int l,
vector<vec_m>& b, vector<vector<bigint>>& lambda, const vector<bigint>& d)
{
#ifdef TRACE_LLL
cout<<"R"<<k;
#endif
#ifdef DEBUG_LLL
cout<<"In redi with k="<<k<<", l="<<l<<endl;
show(n,b,lambda,d);
#endif
int i;
bigint lkl=lambda[k-1][l-1], dl=d[l], q;
nearest(q,lkl,dl); // nearest integer to lkl/dl
if(is_zero(q))
return;
b[k]-=q*b[l];
lambda[k-1][l-1]-=q*dl;
for(i=1; i<l; i++) lambda[k-1][i-1]-=q*lambda[l-1][i-1];
#ifdef DEBUG_LLL
cout<<"Leaving redi with k="<<k<<", l="<<l<<endl;
show(n,b,lambda,d);
#endif
}
void swapi(const int n, const int k, const int kmax,
vector<vec_m>& b, vector<vector<bigint>>& lambda, vector<bigint>& d)
{
#ifdef TRACE_LLL
cout<<"S"<<k<<endl;
#endif
#ifdef DEBUG_LLL
cout<<"In swapi with k="<<k<<", kmax="<<kmax<<endl;
show(n,b,lambda,d);
#endif
bigint t, lam, bb, dk=d[k], dk1=d[k-1];
int i, j;
std::swap(b[k-1],b[k]);
for(j=1; j<=k-2; j++)
{
t = lambda[k-1][j-1];
lambda[k-1][j-1]=lambda[k-2][j-1];
lambda[k-2][j-1]=t;
}
lam = lambda[k-1][k-2];
bb = (d[k-2]*dk+sqr(lam))/dk1;
for(i=k+1; i<=kmax; i++)
{
t = lambda[i-1][k-1];
lambda[i-1][k-1] = (dk*lambda[i-1][k-2]-lam*t)/dk1;
lambda[i-1][k-2] = (bb*t+lam*lambda[i-1][k-1])/dk;
}
d[k-1] = bb;
#ifdef DEBUG_LLL
cout<<"Leaving swapi with k="<<k<<", kmax="<<kmax<<endl;
show(n,b,lambda,d);
#endif
}
void step3(const int n, int& k, const int kmax,
vector<vec_m>& b, vector<vector<bigint>>& lambda, vector<bigint>& d)
{
redi(n,k,k-1,b,lambda,d);
bigint lhs = 4*(d[k]*d[k-2]+sqr(lambda[k-1][k-2]));
bigint rhs = 3*sqr(d[k-1]);
#ifdef DEBUG_LLL
cout<<"lhs="<<lhs<<", rhs="<<rhs<<endl;
#endif
if( lhs<rhs )
{
swapi(n,k,kmax,b,lambda,d);
k--; if(k<2) k=2;
step3(n,k,kmax,b,lambda,d);
}
else
{
int l;
for(l=k-2; l>0; l--) redi(n,k,l,b,lambda,d);
k++;
}
}
// b is an array of n+1 vectors indexed from 0 to n.
// b[1]...b[n] are the lattice basis, while b[0] holds the coefficients
// of the (diagonal) Gram matrix, so the inner product of b[i] and b[j]
// is sum(k,b[0][k]*b[i][k]*b[j][k]).
//
void lll_reduce(const int n, vector<vec_m>& b)
{
int i, j, k, kmax;
bigint u;
vector<bigint> d(n+1);
vector<vector<bigint>> lambda(n, vector<bigint>(n));
k=2; kmax=1;
d[0]=1; d[1]=sdot(b,1,1);
while(k<=n)
{
vector<bigint>& lambda_k = lambda[k-1];
if(k>kmax)
{
kmax=k;
for(j=1; j<=k; j++)
{
vector<bigint>& lambda_j = lambda[j-1];
u=sdot(b,k,j);
for(i=1; i<j; i++)
{
u=(d[i]*u-lambda_k[i-1]*lambda_j[i-1])/d[i-1];
}
if(j<k) lambda_k[j-1]=u;
else
{
if(u==0)
{
cout<<"lll_reduce(): input vectors dependent!\n";
return;
}
d[k]=u;
}
}
}
#ifdef DEBUG_LLL
cout<<"After step 2 with k="<<k<<", kmax="<<kmax<<endl;
show(n,b,lambda,d);
#endif
step3(n,k,kmax,b,lambda,d);
}
#ifdef TRACE_LLL
cout<<endl;
#endif
}
|
