In this paper, we study the properties of the sum-of-squares indicator of vectorial Boolean functions. Firstly, we give the upper bound of $\sum_{u\in \mathbb{F}_2^n,v\in ...\mathbb{F}_2^m}\mathcal{W}_F^3(u,v)$. Secondly, based on the Walsh-Hadamard transform, we give a secondary construction of vectorial bent functions. Further, three kinds of sum-of-squares indicators of vectorial Boolean functions are defined by autocorrelation function and the lower and upper bounds of the sum-of-squares indicators are derived. Finally, we study the sum-of-squares indicators with respect to several equivalence relations, and get the sum-of-squares indicator which have the best cryptographic properties.
We prove new non-existence results for vectorial monomial Dillon type bent functions mapping the field of order 22m to the field of order 2m/3. When m is odd and m>3 we show that there are no such ...functions. When m is even we derive a condition for the bent coefficient. The latter result allows us to find examples of bent functions with m=6 in a simple way.
Pott et al. (2018) showed that <inline-formula> <tex-math notation="LaTeX">\mathcal {F}(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(x) </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">n ...= 2m </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">r\ge 1 </tex-math></inline-formula>, is a nontrivial example of a vectorial function with the maximal possible number <inline-formula> <tex-math notation="LaTeX">2^{n}-2^{m} </tex-math></inline-formula> of bent components. Mesnager et al. (2019) generalized this result by showing conditions on <inline-formula> <tex-math notation="LaTeX">\Lambda (x) = x + \sum _{j=1}^\sigma \alpha _{j}x^{2^{t_{j}}} </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">\alpha _{j}\in {\mathbb F} _{2^{m}} </tex-math></inline-formula>, under which <inline-formula> <tex-math notation="LaTeX">\mathcal {F}(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x)) </tex-math></inline-formula> has the maximal possible number of bent components. We simplify these conditions and further analyse this class of functions. For all related vectorial bent functions <inline-formula> <tex-math notation="LaTeX">F(x) = {\rm Tr^{n}_{m}}(\gamma \mathcal {F}(x)) </tex-math></inline-formula>, <inline-formula> <tex-math notation="LaTeX">\gamma \in {\mathbb F}_{2^{n}}\setminus {\mathbb F} _{2^{m}} </tex-math></inline-formula>, which as we will point out belong to the Maiorana-McFarland class, we describe the collection of the solution spaces for the linear equations <inline-formula> <tex-math notation="LaTeX">\mathcal {D}_{a}F(x) = F(x) + F(x+a) + F(a) = 0 </tex-math></inline-formula>, which forms a spread of <inline-formula> <tex-math notation="LaTeX">{\mathbb F}_{2^{n}} </tex-math></inline-formula>. Analysing these spreads, we can infer neat conditions for functions <inline-formula> <tex-math notation="LaTeX">H(x) = (F(x),G(x)) </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">{\mathbb F}_{2^{n}} </tex-math></inline-formula> to <inline-formula> <tex-math notation="LaTeX">{\mathbb F}_{2^{m}}\times {\mathbb F} _{2^{m}} </tex-math></inline-formula> to exhibit small differential uniformity (for instance for <inline-formula> <tex-math notation="LaTeX">\Lambda (x) = x </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">r=0 </tex-math></inline-formula> this fact is used in the construction of Carlet's, Pott-Zhou's, Taniguchi's APN-function). For some classes of <inline-formula> <tex-math notation="LaTeX">H(x) </tex-math></inline-formula> we determine differential uniformity and with a method based on Bezout's theorem nonlinearity.
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.
•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.
In this paper, we study the properties of the sum-of-squares indicator of vectorial Boolean functions. Firstly, we give the upper bound of $\sum_{u\in \mathbb{F}_2^n,v\in ...\mathbb{F}_2^m}\mathcal{W}_F^3(u,v)$. Secondly, based on the Walsh-Hadamard transform, we give a secondary construction of vectorial bent functions. Further, three kinds of sum-of-squares indicators of vectorial Boolean functions are defined by autocorrelation function and the lower and upper bounds of the sum-of-squares indicators are derived. Finally, we study the sum-of-squares indicators with respect to several equivalence relations, and get the sum-of-squares indicator which have the best cryptographic properties.
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.
A class of quadratic vectorial bent functions having the form F(x) = Trn m(ax2s1+1) + Trn1 (bx2s2+1) is investigated, where n; m; s1; s2 are positive integers and the coefficients a, b belong to the ...finite field F2n. Through some discussions on the permutation property of certain linearized polynomials over F2n, several classes of quadratic vectorial bent functions are presented for special cases of n, and it is also verified by computer that some vectorial bent functions proposed are extended affine inequivalent to all known quadratic vectorial bent functions.
The objective of this article is to broaden the understanding of the connections between bent functions and partial difference sets. Recently, the first two authors showed that the elements which a ...vectorial dual-bent function with certain additional properties maps to 0, form a partial difference set, which generalizes the connection between Boolean bent functions and Hadamard difference sets, and some later established connections between
p
-ary bent functions and partial difference sets to vectorial bent functions. We discuss the effects of coordinate transformations. As all currently known vectorial dual-bent functions
F
:
F
p
n
→
F
p
s
are linear equivalent to
l
-forms, i.e., to functions satisfying
F
(
β
x
)
=
β
l
F
(
x
)
for all
β
∈
F
p
s
, we investigate properties of partial difference sets obtained from
l
-forms. We show that they are unions of cosets of
F
p
s
∗
, which also can be seen as certain cyclotomic classes. We draw connections to known results on partial difference sets from cyclotomy. Motivated by experimental results, for a class of vectorial dual-bent functions from
F
p
n
to
F
p
s
, we show that the preimage set of the squares of
F
p
s
forms a partial difference set. This extends earlier results on
p
-ary bent functions.
In this article a construction of bent functions from an n-dimensional vector space Vn over Fp to Fp is presented for arbitrary primes p and dimensions n≥5. The construction can be seen as ...generalization of the Rothaus construction for Boolean bent functions. Since vectorial bent functions are used, we recall some classes of vectorial bent functions and employ them to obtain both, weakly regular and non-weakly regular bent functions. The suggested construction provides the second known procedure to design non-weakly regular bent functions.
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