Abstract
AbstractDuring the last five decades, many different secondary constructions of bent functions were proposed in the literature. Nevertheless, apart from a few works, the question about the class inclusion of bent functions generated using these methods is rarely addressed. Especially, if such a “new” family belongs to the completed Maiorana–McFarland ($${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$
M
M
#
) class then there is no proper contribution to the theory of bent functions. In this article, we provide some fundamental results related to the inclusion in $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$
M
M
#
and eventually we obtain many infinite families of bent functions that are provably outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$
M
M
#
. The fact that a bent function f is in/outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$
M
M
#
if and only if its dual is in/outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$
M
M
#
is employed in the so-called 4-decomposition of a bent function on $${\mathbb {F}}_2^n$$
F
2
n
, which was originally considered by Canteaut and Charpin (IEEE Trans Inf Theory 49(8):2004–2019, 2003) in terms of the second-order derivatives and later reformulated in (Hodžić et al. in IEEE Trans Inf Theory 65(11):7554–7565, 2019) in terms of the duals of its restrictions to the cosets of an $$(n-2)$$
(
n
-
2
)
-dimensional subspace V. For each of the three possible cases of this 4-decomposition of a bent function (all four restrictions being bent, semi-bent, or 5-valued spectra functions), we provide generic methods for designing bent functions provably outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$
M
M
#
. For instance, for the elementary case of defining a bent function $$h(\textbf{x},y_1,y_2)=f(\textbf{x}) \oplus y_1y_2$$
h
(
x
,
y
1
,
y
2
)
=
f
(
x
)
⊕
y
1
y
2
on $${\mathbb {F}}_2^{n+2}$$
F
2
n
+
2
using a bent function f on $${\mathbb {F}}_2^n$$
F
2
n
, we show that h is outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$
M
M
#
if and only if f is outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$
M
M
#
. This approach is then generalized to the case when two bent functions are used. More precisely, the concatenation $$f_1||f_1||f_2||(1\oplus f_2)$$
f
1
|
|
f
1
|
|
f
2
|
|
(
1
⊕
f
2
)
also gives bent functions outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$
M
M
#
if $$f_1$$
f
1
or $$f_2$$
f
2
is outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$
M
M
#
. The cases when the four restrictions of a bent function are semi-bent or 5-valued spectra functions are also considered and several design methods of constructing infinite families of bent functions outside $${{{\mathcal {M}}}{{\mathcal {M}}}}^\#$$
M
M
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are provided.
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computer Science Applications
Reference30 articles.
1. Bapić A., Pasalic E.: Constructions of (vectorial) bent functions outside the completed Maiorana–McFarland class. Discret. Appl. Math. 314, 197–212 (2022).
2. Bapić A., Pasalic E., Zhang F., Hodžić S.: Constructing new superclasses of bent functions from known ones. Cryptogr. Commun. 4, 1–28 (2022).
3. Canteaut A., Charpin P.: Decomposing bent functions. IEEE Trans. Inf. Theory 49(8), 2004–2019 (2003).
4. Carlet C.: Two new classes of bent functions. Lect. Not. Comput. Sci. 765, 77–101 (1993).
5. Carlet C.: Partially bent functions. Des. Codes Cryptogr. 3(2), 135–145 (1993).
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