4 Permutation Matrices Associated to Bent Functions
87
Gibbs dyadic derivative is defined as a differential operator that has in the Walsh
dyadic analysis a role corresponding to that of the Newton–Leibniz derivative in
classical mathematical analysis [2].
Definition 4.2 For a function f on G n , equally on B n , the Gibbs dyadic derivative
is defined as
f
[1] (x) =
1
2
n−1
r=0
(f (x ⊕ 2
r ) − f (x))2
r , x ∈ B n ,
and the operator D n f = f [1] is called the Gibbs differentiator [2].
This operator can be equally applied to either complex-valued functions or
Boolean functions, but in this case the logic values 0 and 1 are interpreted as the
corresponding integers due to the arithmetic operations in Definition 4.2. Further,
the abovementioned encoding of Boolean values (0, 1) → (1, −1) is assumed. It
follows that the values of the Gibbs dyadic derivative are in the general case complex
numbers, and integers in the case of Boolean functions.
In matrix notation, the Gibbs dyadic derivative is defined by the (2 n × 2 n ) matrix
D(n) = [d ξ,η ], ξ, η ∈ {0, 1, . . . , 2 n − 1}, whose entries are defined as
d ξ,η =
1
2
(2
n
− 1)δ(ξ ⊕ η, 0) −
n−1
r=0
2
r δ(ξ ⊕ η, 2
r )
,
where the function δ is the Kronecker delta.
The matrix D(n) is called the Gibbs matrix, and the Gibbs dyadic derivative
D f = [d(0), d(1), . . . , d(2 n − 1)] T of a function f specified by the function vector
F is determined as
D f = D(n)F.
The elements d(i), i = 0, 1, . . . , 2 n − 1, of D f , which in the case of Boolean
functions are integers, are called the Gibbs coefficients.
It is shown already in [2] that the Gibbs matrix can be written as
D(n) = −
1
2 n W(n)G(n)W(n),
where G(n) = diag(0, 1, . . . , 2 n − 1) is the diagonal matrix whose elements are
non-negative integers smaller than 2 n − 1.
Example 4.3 For n = 3, the matrix G(n) = diag(0, 1, 2, 3, 4, 5, 6, 7), and
W(3) =
1 1
1 −1
⊗3
,
87
Gibbs dyadic derivative is defined as a differential operator that has in the Walsh
dyadic analysis a role corresponding to that of the Newton–Leibniz derivative in
classical mathematical analysis [2].
Definition 4.2 For a function f on G n , equally on B n , the Gibbs dyadic derivative
is defined as
f
[1] (x) =
1
2
n−1
r=0
(f (x ⊕ 2
r ) − f (x))2
r , x ∈ B n ,
and the operator D n f = f [1] is called the Gibbs differentiator [2].
This operator can be equally applied to either complex-valued functions or
Boolean functions, but in this case the logic values 0 and 1 are interpreted as the
corresponding integers due to the arithmetic operations in Definition 4.2. Further,
the abovementioned encoding of Boolean values (0, 1) → (1, −1) is assumed. It
follows that the values of the Gibbs dyadic derivative are in the general case complex
numbers, and integers in the case of Boolean functions.
In matrix notation, the Gibbs dyadic derivative is defined by the (2 n × 2 n ) matrix
D(n) = [d ξ,η ], ξ, η ∈ {0, 1, . . . , 2 n − 1}, whose entries are defined as
d ξ,η =
1
2
(2
n
− 1)δ(ξ ⊕ η, 0) −
n−1
r=0
2
r δ(ξ ⊕ η, 2
r )
,
where the function δ is the Kronecker delta.
The matrix D(n) is called the Gibbs matrix, and the Gibbs dyadic derivative
D f = [d(0), d(1), . . . , d(2 n − 1)] T of a function f specified by the function vector
F is determined as
D f = D(n)F.
The elements d(i), i = 0, 1, . . . , 2 n − 1, of D f , which in the case of Boolean
functions are integers, are called the Gibbs coefficients.
It is shown already in [2] that the Gibbs matrix can be written as
D(n) = −
1
2 n W(n)G(n)W(n),
where G(n) = diag(0, 1, . . . , 2 n − 1) is the diagonal matrix whose elements are
non-negative integers smaller than 2 n − 1.
Example 4.3 For n = 3, the matrix G(n) = diag(0, 1, 2, 3, 4, 5, 6, 7), and
W(3) =
1 1
1 −1
⊗3
,