4 Permutation Matrices Associated to Bent Functions
99
Table 4.1 16 Bent functions that cannot be obtained by Gibbs permutation matrices from the three
basic bent functions
f
F and S rm
1
F 1 = [1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0] T
S rm1 = [1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 1 = 1 ⊕ x 4 ⊕ x 3 ⊕ x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
2
F 2 = [1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0] T
S rm2 = [1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 2 = 1 ⊕ x 4 ⊕ x 3 ⊕ x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
3
F 3 = [0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0] T
S rm3 = [0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 3 = x 4 ⊕ x 3 ⊕ x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
4
F 4 = [1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0] T
S rm4 = [1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 4 = 1 ⊕ x 4 ⊕ x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
5
F 5 = [0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0] T
S rm5 = [0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 5 = x 4 ⊕ x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
6
F 6 = [0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0] T
S rm6 = [0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 6 = x 4 ⊕ x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
7
F 7 = [1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0] T
S rm7 = [1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 7 = 1 ⊕ x 3 ⊕ x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
8
F 8 = [0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0] T
S rm8 = [0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 8 = x 3 ⊕ x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
9
F 9 = [0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0] T
S rm9 = [0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 9 = x 3 ⊕ x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
10
F 10 = [0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0] T
S rm10 = [0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 10 = x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
11
F 11 = [1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1] T
S rm11 = [1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 11 = 1 ⊕ x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
12
F 12 = [0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1] T
S rm12 = [0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 12 = x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
13
F 13 = [0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1] T
S rm13 = [0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 13 = x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
(continued)
99
Table 4.1 16 Bent functions that cannot be obtained by Gibbs permutation matrices from the three
basic bent functions
f
F and S rm
1
F 1 = [1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0] T
S rm1 = [1, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 1 = 1 ⊕ x 4 ⊕ x 3 ⊕ x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
2
F 2 = [1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0] T
S rm2 = [1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 2 = 1 ⊕ x 4 ⊕ x 3 ⊕ x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
3
F 3 = [0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0] T
S rm3 = [0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 3 = x 4 ⊕ x 3 ⊕ x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
4
F 4 = [1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0] T
S rm4 = [1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 4 = 1 ⊕ x 4 ⊕ x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
5
F 5 = [0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0] T
S rm5 = [0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 5 = x 4 ⊕ x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
6
F 6 = [0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0] T
S rm6 = [0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 6 = x 4 ⊕ x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
7
F 7 = [1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0] T
S rm7 = [1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 7 = 1 ⊕ x 3 ⊕ x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
8
F 8 = [0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 0] T
S rm8 = [0, 0, 1, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 8 = x 3 ⊕ x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
9
F 9 = [0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0] T
S rm9 = [0, 0, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 9 = x 3 ⊕ x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
10
F 10 = [0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0] T
S rm10 = [0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 10 = x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
11
F 11 = [1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1] T
S rm11 = [1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 11 = 1 ⊕ x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
12
F 12 = [0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1] T
S rm12 = [0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 0, 0] T
f 12 = x 3 x 4 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
13
F 13 = [0, 0, 0, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1] T
S rm13 = [0, 0, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0] T
f 13 = x 3 x 4 ⊕ x 2 ⊕ x 2 x 4 ⊕ x 2 x 3 ⊕ x 1 x 4 ⊕ x 1 x 3 ⊕ x 1 x 2
(continued)