3 Derivative Operations for Classes C N of Boolean Functions
71
0 0 1 0
0 0 0 1
1 1 0 1
0 0 0 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f0(x)
0 1 0 0
1 0 0 0
1 0 1 1
1 0 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f1(x)
0 0 0 1
0 0 1 0
1 1 1 0
0 0 1 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f2(x)
1 0 0 0
0 1 0 0
0 1 1 1
0 1 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f3(x)
0 0 0 1
1 1 0 1
0 0 0 1
0 0 1 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f4(x)
1 0 0 0
1 0 1 1
1 0 0 0
0 1 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f5(x)
0 0 1 0
1 1 1 0
0 0 1 0
0 0 0 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f6(x)
0 1 0 0
0 1 1 1
0 1 0 0
1 0 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f7(x)
0 0 0 1
0 0 1 0
0 0 0 1
1 1 0 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f8(x)
1 0 0 0
0 1 0 0
1 0 0 0
1 0 1 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f9(x)
0 0 1 0
0 0 0 1
0 0 1 0
1 1 1 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f10(x)
0 1 0 0
1 0 0 0
0 1 0 0
0 1 1 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f11(x)
1 1 0 1
0 0 0 1
0 0 1 0
0 0 0 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f12(x)
1 0 1 1
1 0 0 0
0 1 0 0
1 0 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f13(x)
1 1 1 0
0 0 1 0
0 0 0 1
0 0 1 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f14(x)
0 1 1 1
0 1 0 0
1 0 0 0
0 1 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f15(x)
Fig. 3.2 One class of 16 bent functions of four variables
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 } .
Besides the comparison of the alternative representations of this class C N of 16 bent
functions it is the aim of this example to calculate all vectorial maxima of this class
with regard to x 0 = (x 2 , x 4 ). Using the representative function
f
re (x) = f 0 (x) = x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4
and (3.16) we get the representative function of the vectorial maximum of the class
C N with regard to x 0 = (x 2 , x 4 )
g
re (x) = max
(x 2 ,x 4 )
f
re (x) = (x 1 (x 2 ⊕ x 4 )) ∨ x 3
71
0 0 1 0
0 0 0 1
1 1 0 1
0 0 0 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f0(x)
0 1 0 0
1 0 0 0
1 0 1 1
1 0 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f1(x)
0 0 0 1
0 0 1 0
1 1 1 0
0 0 1 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f2(x)
1 0 0 0
0 1 0 0
0 1 1 1
0 1 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f3(x)
0 0 0 1
1 1 0 1
0 0 0 1
0 0 1 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f4(x)
1 0 0 0
1 0 1 1
1 0 0 0
0 1 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f5(x)
0 0 1 0
1 1 1 0
0 0 1 0
0 0 0 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f6(x)
0 1 0 0
0 1 1 1
0 1 0 0
1 0 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f7(x)
0 0 0 1
0 0 1 0
0 0 0 1
1 1 0 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f8(x)
1 0 0 0
0 1 0 0
1 0 0 0
1 0 1 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f9(x)
0 0 1 0
0 0 0 1
0 0 1 0
1 1 1 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f10(x)
0 1 0 0
1 0 0 0
0 1 0 0
0 1 1 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f11(x)
1 1 0 1
0 0 0 1
0 0 1 0
0 0 0 1
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f12(x)
1 0 1 1
1 0 0 0
0 1 0 0
1 0 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f13(x)
1 1 1 0
0 0 1 0
0 0 0 1
0 0 1 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f14(x)
0 1 1 1
0 1 0 0
1 0 0 0
0 1 0 0
0
0
1
0
1
1
0
1 x1
x2
0
0
1
0
1
1
0
1
x3 x4
f15(x)
Fig. 3.2 One class of 16 bent functions of four variables
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 } .
Besides the comparison of the alternative representations of this class C N of 16 bent
functions it is the aim of this example to calculate all vectorial maxima of this class
with regard to x 0 = (x 2 , x 4 ). Using the representative function
f
re (x) = f 0 (x) = x 1 x 2 ⊕ x 3 x 4 ⊕ x 1 x 3 ⊕ x 1 x 4
and (3.16) we get the representative function of the vectorial maximum of the class
C N with regard to x 0 = (x 2 , x 4 )
g
re (x) = max
(x 2 ,x 4 )
f
re (x) = (x 1 (x 2 ⊕ x 4 )) ∨ x 3