Bent functions satisfying the dual bent condition and permutations with the $$(\mathcal {A}_m)$$ property

Abstract
The concatenation of four Boolean bent functions

$$f=f_1||f_2||f_3||f_4$$


f
=

f
1


|
|


f
2


|
|


f
3


|
|


f
4




is bent if and only if the dual bent condition

$$f_1^* + f_2^* + f_3^* + f_4^* =1$$



f
1
∗

+

f
2
∗

+

f
3
∗

+

f
4
∗

=
1



is satisfied. However, to specify four bent functions satisfying this duality condition is in general quite a difficult task. Commonly, to simplify this problem, certain relations between

$$f_i$$


f
i



are assumed, as well as functions

$$f_i$$


f
i



of a special shape are considered, e.g.,

$$f_i(x,y)=x\cdot \pi _i(y)+h_i(y)$$



f
i


(
x
,
y
)

=
x
·

π
i


(
y
)

+

h
i


(
y
)




are Maiorana-McFarland bent functions. In the case when permutations

$$\pi _i$$


π
i



of

$$\mathbb {F}_2^m$$


F
2
m



have the

$$(\mathcal {A}_m)$$


(

A
m

)



property and Maiorana-McFarland bent functions

$$f_i$$


f
i



satisfy the additional condition

$$f_1+f_2+f_3+f_4=0$$



f
1

+

f
2

+

f
3

+

f
4

=
0



, the dual bent condition is known to have a relatively simple shape allowing to specify the functions

$$f_i$$


f
i



explicitly. In this paper, we generalize this result for the case when Maiorana-McFarland bent functions

$$f_i$$


f
i



satisfy the condition

$$f_1(x,y)+f_2(x,y)+f_3(x,y)+f_4(x,y)=s(y)$$



f
1


(
x
,
y
)

+

f
2


(
x
,
y
)

+

f
3


(
x
,
y
)

+

f
4


(
x
,
y
)

=
s

(
y
)




and provide a construction of new permutations with the

$$(\mathcal {A}_m)$$


(

A
m

)



property from the old ones. Combining these two results, we obtain a recursive construction method of bent functions satisfying the dual bent condition. Moreover, we provide a generic condition on the Maiorana-McFarland bent functions

$$f_1,f_2,f_3,f_4$$



f
1

,

f
2

,

f
3

,

f
4




stemming from the permutations of

$$\mathbb {F}_2^m$$


F
2
m



with the

$$(\mathcal {A}_m)$$


(

A
m

)



property, such that the concatenation

$$f=f_1||f_2||f_3||f_4$$


f
=

f
1


|
|


f
2


|
|


f
3


|
|


f
4




does not belong, up to equivalence, to the Maiorana-McFarland class. Using monomial permutations

$$\pi _i$$


π
i



of

$$\mathbb {F}_{2^m}$$


F

2
m




with the

$$(\mathcal {A}_m)$$


(

A
m

)



property and monomial functions

$$h_i$$


h
i



on

$$\mathbb {F}_{2^m}$$


F

2
m




, we provide explicit constructions of such bent functions; a particular case of our result shows how one can construct bent functions from APN permutations, when m is odd. Finally, with our construction method, we explain how one can construct homogeneous cubic bent functions, noticing that only very few design methods of these objects are known.