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where the elements of F 1 are obtained with the (0, 1) → (1, −1) encoding. Its
Walsh spectrum is
S f 1 = [4, −4, −4, −4, −4, 4, 4, 4, −4, 4, −4, −4, −4, 4, −4, −4]
T
and since it is flat, the function is bent. Its dual function is
F d 1 = [1, −1, −1, −1, −1, 1, 1, 1, −1, 1, −1, −1, −1, 1, −1, −1]
T .
Notice that in this example, the initial function f 1 and its dual function f d 1 have 6
and 10 non-zero values, respectively.
The function f 2 specified by the function vector
F 2 = [1, 1, 1, 1, 1, 1, −1, −1, 1, −1, 1, −1, 1, −1, −1, 1]
T
is bent since its Walsh spectrum is
S f 2 = [4, 4, 4, 4, 4, 4, −4, −4, 4, −4, 4, −4, 4, −4, −4, 4]
T ,
and its dual function f d 2 is
F d 2 = [1, 1, 1, 1, 1, 1, −1, −1, 1, −1, 1, −1, 1, −1, −1, 1]
T
= F 2 .
This function has the same number of non-zero values as the initial function,
moreover, the Walsh spectrum is equal to the function vector multiplied by 4.
The function f 3 specified by the function vector
F 3 = [1, 1, 1, −1, 1, 1, 1, −1, −1, −1, −1, 1, 1, 1, 1, −1]
T
has the Walsh spectrum
S f 3 = [4, 4, 4, −4, −4, −4, −4, 4, 4, 4, 4, −4, 4, 4, 4, −4]
T ,
and its dual function f d 3 has the same number of non-zero elements but, it is not
identical to f 3
F d 3 = [1, 1, 1, −1, −1, −1, −1, 1, 1, 1, 1, −1, 1, 1, 1, −1]
T .
4.3 Gibbs Dyadic Derivative
For the sake of completeness, Sects. 4.3 and 4.4 review some aspects of Gibbs
derivatives, most of them already presented in [10–12].