Cryptography Reference
In-Depth Information
Table 1.
Cryptographic Properties of the Equivalence Class
f
GP
VA R
AI DEG
NL
10110001
8
4
7
106
11000110
5
3
4
80
10001110
8
4
6
104
10111000
8
4
7
102
10110100
8
4
7
106
10010110
8
4
7
106
11110011
8
4
7
102
11100111
8
4
6
100
10010101
8
4
7
106
11010100
8
4
6
108
10110010
8
4
7
100
10100110
8
4
7
104
10101111
8
4
6
104
11111010
8
4
7
108
11100001
8
4
7
104
11000011
8
4
6
104
to construct equivalence classes satisfying all the necessary criteria. Anyway,
the approach described above may be promising, which can be seen from the
following example.
Example 2.
Let
f
∈
B
8
. We want to construct an equivalence class
f
such that
deg(
f
)
58 (this is the lowest possible nonlinearity
of a 8-variable function with the optimum algebraic immunity). Moreover, we
want
var
(
f
)
≥
4,
AI
(
f
)
≥
3and
nl
(
f
)
≥
5 (that is, there is a function of the equivalence class such that
its ANF contains at most 5 variables). Let 1
f
≤
M
i
b
0
|
,where
M
is the
generator matrix and
b
0
= (10000000)
T
. There are many sets
E
such that the
matrices in (2) are of rank 2. We choose
=
{
i
∈
E
}
E
=[ 159141517192021293031323638394041424345464751
52 58 59 60 61 62 63 64 66 67 70 71 72 73 76 77 82 83 84 85 88 90 95
96 97 98 99 103 105 109 111 112 113 114 115 116 126 127 133 136
137 138 140 142 144 145 146 158 159 160 164 165 166 167 168 169
170 171 173 176 177 178 179 180 181 182 183 184 185 188 189 190
191 193 194 196 198 201 202 203 206 207 208 209 210 211 212 213
214 215 219 220 221 222 224 237 244 245 246 249 250 251 252 253]
.
There are exactly
φ
(255)
/
8 = 16 classes of
g
f
, and every
class is characteristiced by a generator polynomial. We can find some crypto-
graphic properties of these classes from Table 1. In the table, GP denotes gener-
ator polynomial (we represent
f
(
x
)=
x
8
+
a
8
x
7
+
a
7
x
6
+
a
6
x
5
+
a
5
x
4
+
a
4
x
3
+
∈
B
8
such that
g
∼
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