4 Permutation Matrices Associated to Bent Functions
101
Table 4.3 An example of 8 bent functions from which the first specific bent function with the
integer representation 6017 is obtained by using different Gibbs permutation matrices
f
6017
D f and F
1
854
16,939 D f 1 = [0, 8, 4, 12, 2, 10, −6, −14, 1, −9, 5, −13, 3, −11, −7, 15] T
F 1 = [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1] T
2
857
13,834 D f 2 = [0, 8, 4, 12, 10, 2, −14, −6, 5, −13, 1, −9, −15, 7, 11, −3] T
F 2 = [0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0] T
3
869
7714 D f 3 = [0, 8, 12, 4, 2, 10, −14, −6, 3, −11, −15, 7, 1, −9, 13, −5] T
F 3 = [0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0] T
4
874
16,939 D f 4 = [8, 0, 4, 12, 2, 10, −14, −6, −15, 7, 3, −11, 5, −13, 9, −1] T
F 4 = [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1] T
5
23,648
45,604 D f 5 = [5, −1, 9, −13, −14, −10, 2, 6, 3, −7, −15, 11, 8, 12, 4, 0] T
F 5 = [1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0] T
6
59,416
36,376 D f 6 = [−8, −14, −13, 11, −3, 5, 6, 0, 12, 10, 9, −15, −7, 1, 2, 4] T
F 6 = [1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0] T
7
59,428
45,604 D f 7 = [−8, −14, −5, 3, −11, 13, 6, 0, 10, 12, −7, 1, 9, −15, 4, 2] T
F 7 = [1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0] T
8
59,458
54,338 D f 8 = [−8, −6, −13, 3, −11, 5, 14, 0, 9, −7, 12, 2, 10, 4, −15, 1] T
F 8 = [1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0] T
Table 4.4 Correspondence between 16 specific bent functions with respect to the Gibbs permutation matrix assigned to f b 1
Initial f
Derived f
Expression for derived f
1
33,047
33,047
f 1 = q ⊕ 1
2
36,376
33,256
f 2 = q ⊕ x 1
3
33,256
36,376
f 3 = q ⊕ x 2
4
54,338
45,604
f 4 = q ⊕ x 3
5
45,604
54,338
f 5 = q ⊕ x 4
6
28,952
28,952
f 6 = q ⊕ ⊕x 2 ⊕ x 1
7
9394
19,748
f 7 = q ⊕ x 3 ⊕ x 1
8
17,108
11,074
f 8 = q ⊕ x 4 ⊕ x 1
9
11,074
17,108
f 9 = q ⊕ x 3 ⊕ x 2
10
19,748
9394
f 10 = q ⊕ x 4 ⊕ x 2
11
6286
6286
f 11 = q ⊕ x 4 ⊕ x 3
12
9293
16,939
f 12 = q ⊕ 1 ⊕ x 3 ⊕ x 2 ⊕ x 1
13
16,939
9293
f 13 = q ⊕ 1 ⊕ x 4 ⊕ x 2 ⊕ x 1
14
6017
6257
f 14 = q ⊕ 1 ⊕ x 4 ⊕ x 3 ⊕ x 1
15
6257
6017
f 15 = q ⊕ 1 ⊕ x 4 ⊕ x 3 ⊕ x 2
16
59,521
59,521
f 16 = q ⊕ 1 ⊕ x 4 ⊕ x 3 ⊕ x 2 ⊕ x 1
101
Table 4.3 An example of 8 bent functions from which the first specific bent function with the
integer representation 6017 is obtained by using different Gibbs permutation matrices
f
6017
D f and F
1
854
16,939 D f 1 = [0, 8, 4, 12, 2, 10, −6, −14, 1, −9, 5, −13, 3, −11, −7, 15] T
F 1 = [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1] T
2
857
13,834 D f 2 = [0, 8, 4, 12, 10, 2, −14, −6, 5, −13, 1, −9, −15, 7, 11, −3] T
F 2 = [0, 0, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0] T
3
869
7714 D f 3 = [0, 8, 12, 4, 2, 10, −14, −6, 3, −11, −15, 7, 1, −9, 13, −5] T
F 3 = [0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0] T
4
874
16,939 D f 4 = [8, 0, 4, 12, 2, 10, −14, −6, −15, 7, 3, −11, 5, −13, 9, −1] T
F 4 = [0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1] T
5
23,648
45,604 D f 5 = [5, −1, 9, −13, −14, −10, 2, 6, 3, −7, −15, 11, 8, 12, 4, 0] T
F 5 = [1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0] T
6
59,416
36,376 D f 6 = [−8, −14, −13, 11, −3, 5, 6, 0, 12, 10, 9, −15, −7, 1, 2, 4] T
F 6 = [1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0] T
7
59,428
45,604 D f 7 = [−8, −14, −5, 3, −11, 13, 6, 0, 10, 12, −7, 1, 9, −15, 4, 2] T
F 7 = [1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0] T
8
59,458
54,338 D f 8 = [−8, −6, −13, 3, −11, 5, 14, 0, 9, −7, 12, 2, 10, 4, −15, 1] T
F 8 = [1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0] T
Table 4.4 Correspondence between 16 specific bent functions with respect to the Gibbs permutation matrix assigned to f b 1
Initial f
Derived f
Expression for derived f
1
33,047
33,047
f 1 = q ⊕ 1
2
36,376
33,256
f 2 = q ⊕ x 1
3
33,256
36,376
f 3 = q ⊕ x 2
4
54,338
45,604
f 4 = q ⊕ x 3
5
45,604
54,338
f 5 = q ⊕ x 4
6
28,952
28,952
f 6 = q ⊕ ⊕x 2 ⊕ x 1
7
9394
19,748
f 7 = q ⊕ x 3 ⊕ x 1
8
17,108
11,074
f 8 = q ⊕ x 4 ⊕ x 1
9
11,074
17,108
f 9 = q ⊕ x 3 ⊕ x 2
10
19,748
9394
f 10 = q ⊕ x 4 ⊕ x 2
11
6286
6286
f 11 = q ⊕ x 4 ⊕ x 3
12
9293
16,939
f 12 = q ⊕ 1 ⊕ x 3 ⊕ x 2 ⊕ x 1
13
16,939
9293
f 13 = q ⊕ 1 ⊕ x 4 ⊕ x 2 ⊕ x 1
14
6017
6257
f 14 = q ⊕ 1 ⊕ x 4 ⊕ x 3 ⊕ x 1
15
6257
6017
f 15 = q ⊕ 1 ⊕ x 4 ⊕ x 3 ⊕ x 2
16
59,521
59,521
f 16 = q ⊕ 1 ⊕ x 4 ⊕ x 3 ⊕ x 2 ⊕ x 1