4 Permutation Matrices Associated to Bent Functions
89
4.3.1 Partial Gibbs Dyadic Derivatives
In some applications, and especially for computation purposes, it is convenient
to consider the definition of the Gibbs derivatives in terms of the partial Gibbs
derivatives with respect to all the variables in a function f (x 1 , x 2 , . . . , x n ).
Definition 4.3 The partial Gibbs derivative with respect to the variable x i , i =
1, 2, . . . , n, is defined as
D i f = f (x 1 , . . . , x i ⊕ 1, . . . , x n ) − f (x 1 , . . . , x n ).
Definition 4.4 In matrix notation, the partial Gibbs derivative is defined as
D i =
n−1
j =0
A j , A j =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−1 1
1 −1
, (j = i),
1 0
0 1
,
(j = i).
Definition 4.5 In matrix notation, the Gibbs derivative is defined in terms of the
partial Gibbs derivatives as
D = −
1
2
n−1
i=0
2
n−i−1 D i .
4.4 Gibbs Dyadic Derivative of Bent Functions
The Gibbs derivative of a bent function can be represented as a vector whose
elements are all elements in the set B n [10]. The i-th Gibbs coefficient d(i) has
the same sign as the function value f (i). In other words, the Gibbs derivative does
not change the sign of function values.
Lemma 4.1 Absolute values of Gibbs coefficients of bent functions are nonnegative integers 0, 1, . . . , 2 n − 1, i.e., eigenvalues of the Gibbs dyadic derivative.
We present here the proof of this lemma for the case of quadratic bent functions.
Recall that a function is called quadratic if its positive polarity Reed–Muller
expression consists of the sum of pairs of variables.
Example 4.4 For n = 4, the function f = x 1 x 2 ⊕ x 3 x 4 is a quadratic bent function,
and it is often considered as the basic bent function in the space of four variable bent
functions. Its function vector in the (0, 1) → (1, −1) encoding is
F = [1, 1, 1, −1, 1, 1, 1, −1, 1, 1, 1, −1, −1, −1, −1, 1]
T ,
89
4.3.1 Partial Gibbs Dyadic Derivatives
In some applications, and especially for computation purposes, it is convenient
to consider the definition of the Gibbs derivatives in terms of the partial Gibbs
derivatives with respect to all the variables in a function f (x 1 , x 2 , . . . , x n ).
Definition 4.3 The partial Gibbs derivative with respect to the variable x i , i =
1, 2, . . . , n, is defined as
D i f = f (x 1 , . . . , x i ⊕ 1, . . . , x n ) − f (x 1 , . . . , x n ).
Definition 4.4 In matrix notation, the partial Gibbs derivative is defined as
D i =
n−1
j =0
A j , A j =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎩
−1 1
1 −1
, (j = i),
1 0
0 1
,
(j = i).
Definition 4.5 In matrix notation, the Gibbs derivative is defined in terms of the
partial Gibbs derivatives as
D = −
1
2
n−1
i=0
2
n−i−1 D i .
4.4 Gibbs Dyadic Derivative of Bent Functions
The Gibbs derivative of a bent function can be represented as a vector whose
elements are all elements in the set B n [10]. The i-th Gibbs coefficient d(i) has
the same sign as the function value f (i). In other words, the Gibbs derivative does
not change the sign of function values.
Lemma 4.1 Absolute values of Gibbs coefficients of bent functions are nonnegative integers 0, 1, . . . , 2 n − 1, i.e., eigenvalues of the Gibbs dyadic derivative.
We present here the proof of this lemma for the case of quadratic bent functions.
Recall that a function is called quadratic if its positive polarity Reed–Muller
expression consists of the sum of pairs of variables.
Example 4.4 For n = 4, the function f = x 1 x 2 ⊕ x 3 x 4 is a quadratic bent function,
and it is often considered as the basic bent function in the space of four variable bent
functions. Its function vector in the (0, 1) → (1, −1) encoding is
F = [1, 1, 1, −1, 1, 1, 1, −1, 1, 1, 1, −1, −1, −1, −1, 1]
T ,