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R. S. Stankovi´ c et al.
1. The Gibbs derivative does not change the sign of the function values of bent
functions in the (0, 1) → (1, −1) encoding. This feature can be easily seen for
the functions in Example 4.4 and their Gibbs dyadic derivatives.
2. The Gibbs coefficients of a function f and its logic complement f have identical
absolute values and the sign opposite to each other. That is, if the i-th Gibbs
coefficient of f has the value d i = r, then the value of the corresponding Gibbs
coefficient of f is d i = −r. This feature is illustrated again by the functions in
Example 4.4.
3. The sum of positive Gibbs coefficients is equal to the sum of negative Gibbs
coefficients and it is (2 n − 1)(2 n−2 ). This feature can be illustrated by the
functions in Examples 4.6 and 4.8 below for n = 4 and n = 6. It is easy to
verify that the sum of positive as well as the negative Gibbs coefficients in these
examples has magnitude 60 and 1008, for n = 4 and n = 6, respectively.
4.5 Gibbs Permutation Matrices
From the above considerations, it follows that the Gibbs coefficients are permuted
versions of the vector B n = [0, 1, 2, . . . , 2 n − 1] T . A different permutation of
these values corresponds to each bent function and, therefore, bent functions can be
characterized by the permutations determined by the Gibbs coefficients [10]. These
permutations are conveniently represented by the permutation matrices that we call
the Gibbs permutation matrices assigned to bent functions by the Gibbs derivative.
They are defined as follows.
Definition 4.6 The permutation matrix P assigned by the Gibbs derivative to a bent
function f in n variables is the (2 n ×2 n ) permutation matrix that permutes the vector
B n = [0, 1, 2, . . . , 2 n − 1] T into the vector of absolute values of Gibbs coefficients
of f .
Since for bent functions the Gibbs coefficients are eigenvalues of the Gibbs
derivative, it follows Definition 4.7.
Definition 4.7 The Gibbs permutation matrix is a (2 n × 2 n ) matrix P g = [p i,j ],
whose elements p i,j = 1 if the i-th Gibbs coefficient has the absolute value j ,
otherwise p i,j = 0.
Example 4.6 For function f 1 in Example 4.2 the Gibbs derivative is
D f 1 = [−10, −12, 11, −13, −3, 5, 2, 4, 14, 8, −15, 9, −7, 1, 6, 0]
T ,
R. S. Stankovi´ c et al.
1. The Gibbs derivative does not change the sign of the function values of bent
functions in the (0, 1) → (1, −1) encoding. This feature can be easily seen for
the functions in Example 4.4 and their Gibbs dyadic derivatives.
2. The Gibbs coefficients of a function f and its logic complement f have identical
absolute values and the sign opposite to each other. That is, if the i-th Gibbs
coefficient of f has the value d i = r, then the value of the corresponding Gibbs
coefficient of f is d i = −r. This feature is illustrated again by the functions in
Example 4.4.
3. The sum of positive Gibbs coefficients is equal to the sum of negative Gibbs
coefficients and it is (2 n − 1)(2 n−2 ). This feature can be illustrated by the
functions in Examples 4.6 and 4.8 below for n = 4 and n = 6. It is easy to
verify that the sum of positive as well as the negative Gibbs coefficients in these
examples has magnitude 60 and 1008, for n = 4 and n = 6, respectively.
4.5 Gibbs Permutation Matrices
From the above considerations, it follows that the Gibbs coefficients are permuted
versions of the vector B n = [0, 1, 2, . . . , 2 n − 1] T . A different permutation of
these values corresponds to each bent function and, therefore, bent functions can be
characterized by the permutations determined by the Gibbs coefficients [10]. These
permutations are conveniently represented by the permutation matrices that we call
the Gibbs permutation matrices assigned to bent functions by the Gibbs derivative.
They are defined as follows.
Definition 4.6 The permutation matrix P assigned by the Gibbs derivative to a bent
function f in n variables is the (2 n ×2 n ) permutation matrix that permutes the vector
B n = [0, 1, 2, . . . , 2 n − 1] T into the vector of absolute values of Gibbs coefficients
of f .
Since for bent functions the Gibbs coefficients are eigenvalues of the Gibbs
derivative, it follows Definition 4.7.
Definition 4.7 The Gibbs permutation matrix is a (2 n × 2 n ) matrix P g = [p i,j ],
whose elements p i,j = 1 if the i-th Gibbs coefficient has the absolute value j ,
otherwise p i,j = 0.
Example 4.6 For function f 1 in Example 4.2 the Gibbs derivative is
D f 1 = [−10, −12, 11, −13, −3, 5, 2, 4, 14, 8, −15, 9, −7, 1, 6, 0]
T ,