98
R. S. Stankovi´ c et al.
obtained in this way and their Reed–Muller spectra and the corresponding functional
expressions are given in Table 4.1. In this case, 192 functions are generated a single
time, other 192 functions are generated two times, and 48 functions are obtained 16
times. It means, for each of these 48 functions there are 16 permutation matrices
that generate them.
The 16 functions that cannot be generated by the Gibbs permutation matrices
from the basic bent functions are defined as functions whose Reed–Muller expression consists of products of all possible pairs of variables and all possible linear
functions for n = 4, with the linear functions in more than two variables negated.
In Table 4.2 these 16 functions are written in terms of the function q defined in
terms of all possible pairs of variables as
q = x 1 x 2 ⊕ x 1 x 3 ⊕ x 1 x 4 ⊕ x 2 x 3 ⊕ x 2 x 4 ⊕ x 3 x 4 .
In this table, the second column shows integers whose binary representations are
the corresponding function vectors. In other words, the integer r f representation of
a function is determined as r f =
2 n −1
i=0 2 i f (i), where f (i) are elements of the
truth-vector F of f .
These 16 functions are obtained by the application of the Gibbs permutation
matrices to some bent functions other than these three basic functions.
The specific bent functions can be obtained from other bent functions by using
different Gibbs permutation matrices.
Table 4.3 shows 8 functions from which the specific function with integer
representation 6017 is obtained. In this table, the first column shows the integer
corresponding to the function the permutation matrix assigned to which is applied
to the function in the second column. The third column is the vector of Gibbs
coefficients determining the used permutation matrix and the function vector of the
function to which the permutation matrix is applied to get the function with the
integer representation 6017.
It is interesting to observe that the considered specific 16 functions are a
closed set with respect to the Gibbs permutation matrices in the sense that if
any Gibbs permutation matrix is applied to them, the same set of bent functions
is reproduced. Table 4.4 shows the correspondence between the considered 16
specific bent functions with respect to the Gibbs permutation matrix assigned to
the basic bent function f b 1 . The second and the third column show the integers
representing functions to which the Gibbs permutation matrix for f b 1 is applied
and the obtained functions, respectively. It should be observed that in the case
of functions enumerated as 1, 6, 11, 16 the application of the selected Gibbs
permutation matrix produces the initial functions. There are pairs of functions
reciprocal to each other, 2 and 3, 4 and 5, 7 and 10, 8 and 9, 12 and 13, 14 and 15.
R. S. Stankovi´ c et al.
obtained in this way and their Reed–Muller spectra and the corresponding functional
expressions are given in Table 4.1. In this case, 192 functions are generated a single
time, other 192 functions are generated two times, and 48 functions are obtained 16
times. It means, for each of these 48 functions there are 16 permutation matrices
that generate them.
The 16 functions that cannot be generated by the Gibbs permutation matrices
from the basic bent functions are defined as functions whose Reed–Muller expression consists of products of all possible pairs of variables and all possible linear
functions for n = 4, with the linear functions in more than two variables negated.
In Table 4.2 these 16 functions are written in terms of the function q defined in
terms of all possible pairs of variables as
q = x 1 x 2 ⊕ x 1 x 3 ⊕ x 1 x 4 ⊕ x 2 x 3 ⊕ x 2 x 4 ⊕ x 3 x 4 .
In this table, the second column shows integers whose binary representations are
the corresponding function vectors. In other words, the integer r f representation of
a function is determined as r f =
2 n −1
i=0 2 i f (i), where f (i) are elements of the
truth-vector F of f .
These 16 functions are obtained by the application of the Gibbs permutation
matrices to some bent functions other than these three basic functions.
The specific bent functions can be obtained from other bent functions by using
different Gibbs permutation matrices.
Table 4.3 shows 8 functions from which the specific function with integer
representation 6017 is obtained. In this table, the first column shows the integer
corresponding to the function the permutation matrix assigned to which is applied
to the function in the second column. The third column is the vector of Gibbs
coefficients determining the used permutation matrix and the function vector of the
function to which the permutation matrix is applied to get the function with the
integer representation 6017.
It is interesting to observe that the considered specific 16 functions are a
closed set with respect to the Gibbs permutation matrices in the sense that if
any Gibbs permutation matrix is applied to them, the same set of bent functions
is reproduced. Table 4.4 shows the correspondence between the considered 16
specific bent functions with respect to the Gibbs permutation matrix assigned to
the basic bent function f b 1 . The second and the third column show the integers
representing functions to which the Gibbs permutation matrix for f b 1 is applied
and the obtained functions, respectively. It should be observed that in the case
of functions enumerated as 1, 6, 11, 16 the application of the selected Gibbs
permutation matrix produces the initial functions. There are pairs of functions
reciprocal to each other, 2 and 3, 4 and 5, 7 and 10, 8 and 9, 12 and 13, 14 and 15.