This article is devoted to Boolean and vectorial bent functions and their duals. Our ultimate objective is to increase such functions' corpus by designing new ones covering many previous bent ...functions' constructions. To this end, we provide several new infinite families of bent functions, including idempotent bent functions of any algebraic degree, bent functions in univariate trace form, and self-dual bent functions. Those bent functions are of great theoretical and practical interest because of their special structures and relationship with self-dual codes. In particular, many well-known bent functions are special cases of our bent functions. Moreover, we extend our results to vectorial bent functions and obtain three new infinite classes of vectorial bent functions of any possible degree by determining the explicit duals of three classes of well-known bent functions.
Cryptographic Boolean functions play an important role in the design of symmetric ciphers. Many cryptographic criteria such as balancedness, nonlinearity, correlation immunity and transparency order ...are connected with the Walsh support of a Boolean function. However, we still know little about the possible structure of the Walsh supports of Boolean functions. In 2005, Carlet and Mesnager studied the Walsh supports of Boolean functions and constructed a class of n-variable Boolean functions whose Walsh support is F2n∖{0}, for n≥10. For n≤6, it can be verified using the computer that there is no Boolean function with the Walsh Support F2n∖{0}. However, concerning the values of n=7,8,9, it has been an open problem for many years. In this paper, we construct two classes of balanced Boolean functions with the maximum possible Walsh support F2n∖{0}, and partially solve this problem. The first class of functions are of odd variables with n≥9, and the second class of functions are constructed based on the Maiorana–McFarland bent functions, which are of even variables with n≥8. As a result, the above open problem has been settled for n=8,9, and the only unsolved case is n=7.
In this letter, we give a characterization for a generic construction of bent functions. This characterization enables us to obtain another efficient construction of bent functions and to give a ...positive answer on a problem of bent functions.
Bent partitions Anbar, Nurdagül; Meidl, Wilfried
Designs, codes, and cryptography,
2022/4, Volume:
90, Issue:
4
Journal Article
Peer reviewed
Open access
Spread and partial spread constructions are the most powerful bent function constructions. A large variety of bent functions from a 2
m
-dimensional vector space
V
2
m
(
p
)
over
F
p
into
F
p
can be ...generated, which are constant on the sets of a partition of
V
2
m
(
p
)
obtained with the subspaces of the (partial) spread. Moreover, from spreads one obtains not only bent functions between elementary abelian groups, but bent functions from
V
2
m
(
p
)
to
B
, where
B
can be any abelian group of order
p
k
,
k
≤
m
. As recently shown (Meidl, Pirsic 2021), partitions from spreads are not the only partitions of
V
2
m
(
2
)
, with these remarkable properties. In this article we present first such partitions—other than (partial) spreads—which we call bent partitions, for
V
2
m
(
p
)
,
p
odd. We investigate general properties of bent partitions, like number and cardinality of the subsets of the partition. We show that with bent partitions we can construct bent functions from
V
2
m
(
p
)
into a cyclic group
Z
p
k
. With these results, we obtain the first constructions of bent functions from
V
2
m
(
p
)
into
Z
p
k
,
p
odd, which provably do not come from (partial) spreads.
For each non-constant Boolean function q, Klapper introduced the notion of q-transforms of Boolean functions. The q-transform of a Boolean function f is related to the Hamming distances from f to the ...functions obtainable from q by nonsingular linear change of basis.
In this work, we discuss the existence of q-nearly bent functions, a new family of Boolean functions characterized by the q-transform. We prove that any balanced Boolean functions (linear or non-linear) are q-nearly bent if q has weight one, which gives a positive answer to an open question (whether there exist non-affine q-nearly bent functions) proposed by Klapper. We also prove a necessary condition for checking when a function is not q-nearly bent.
The autocorrelation properties of Boolean functions are closely related to the Shannon’s concept of diffusion and can be accompanied with other cryptographic criteria (such as high nonlinearity and ...algebraic degree) for ensuring an overall robustness to various cryptanalytic methods. In a series of recent articles 14,9,15, the design methods of n-variable balanced Boolean functions n is strictly even) with small absolute indicator Δf < 2n/2 have been considered. Whereas the two first articles managed to solve this problem for relatively large n⩾46, a recent approach 15 has introduced a generic design framework achieving Δf < 2n/2 for even n⩾22. Based on a suitable modification of the method of Rothaus, used to construct new bent functions from known ones, we provide a generic iterative framework for designing balanced functions satisfying the condition Δf < 2n/2 and having overall good cryptographic properties for any even n⩾12. Even though the problem of specifying functions having Δf < 2n/2 for smaller n has been considered in 14,9,15 using various search algorithms, our method for the first time provides relatively simple iterative framework for variable spaces of more practical interest. Moreover, our approach can be efficiently applied to certain classes of initial functions (derived from partial spread bent functions) for deriving balanced functions with Δf < 2n/2 for relatively large n, namely for n⩾48 satisfying n≡0 mod 4 and n⩾54 with n≡2 mod 4. In the latter case, our nonlinearity bound is better than the one presented in 14.
In 2017, Tang et al. provided a complete characterization of generalized bent functions from ℤn2 to ℤq(q = 2m) in terms of their component functions (IEEE Trans. Inf. Theory. vol.63, no.7, ...pp.4668-4674). In this letter, for a general even q, we aim to provide some characterizations and more constructions of generalized bent functions with flexible coefficients. Firstly, we present some sufficient conditions for a generalized Boolean function with at most three terms to be gbent. Based on these results, we give a positive answer to a remaining question proposed by Hodžić in 2015. We also prove that the sufficient conditions are also necessary in some special cases. However, these sufficient conditions whether they are also necessary, in general, is left as an open problem. Secondly, from a uniform point of view, we provide a secondary construction of gbent function, which includes several known constructions as special cases.
•Bent functions have important applications in cryptography, coding and combinatorics.•Bent functions from the Partial Spreads class and their duals have the same form.•An answer to an open problem ...of Mesnager is from the Partial Spreads class.
In 2014, Mesnager proposed two open problems in 4 on the construction of bent functions. One problem has been settled by Tang et al. in 2017. However, the other is still outstanding, which is solved in this letter by considering a class of PS− vectorial bent functions.
Several new classes of binary and p -ary regular bent functions are obtained in this paper. The bentness of all these functions is determined by some exponential sums over finite fields, most of ...which have close relations with the well-known Kloosterman sums.
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