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B. Steinbach and C. Posthoff
and
rank(IDM(g)) = rank(IDM(f )) + 1.
If Condition (3.13) is not satisfied, we have
s min = MIDC(IDM(f ), x 0 ) = 0
and all functions of the given class C N do not depend on the simultaneous change
of x 0 . In this case the mark functions of the vectorial derivative operations of the
given class C N
f re (x), f id (x)
are
g
re
1 (x) = 0,
IDM(g 1 ) = I n ,
(3.18)
g
re
2 (x) = f
re (x), IDM(g 2 ) = IDM(f ),
(3.19)
g
re
3 (x) = f
re (x), IDM(g 3 ) = IDM(f ),
(3.20)
where I n is the identity matrix of the size n.
From (3.18) follows that the vectorial derivatives with regard to x 0 of all
functions f (x) of the given class C N
f re (x), f id (x)
, which do not depend on
the simultaneous change of x 0 , i.e., s min = 0, are equal to the constant function
f (x) = 0(x).
Example 3.8 Figure 3.2 shows the Karnaugh-maps of the 16 Boolean functions of
one class C N of bent functions of four variables.
These 16 functions depend on all 2 4 − 1 = 15 directions of change, i.e., all
vectorial derivatives of these functions are not equal to 0; hence, the independence
matrix of this class C N is an empty matrix of the size 4 and the independence
function f id (x) = 0.
Obviously, it is more convenient to represent this class of bent functions by
C N x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 , 0
instead of the enumeration of all 16 functions:
C N = { x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 , x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 ,
x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 , x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 ,
x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 , x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 ,
x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 , x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 ,
x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 , x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 ,
x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 , x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4 ,