144 research outputs found
A new class of hyper-bent functions and Kloosterman sums
This paper is devoted to the characterization of hyper-bent functions.
Several classes of hyper-bent functions have been studied, such as
Charpin and Gong\u27s and Mesnager\u27s , where is a set of representations of the cyclotomic
cosets modulo of full size and .
In this paper, we generalize their results and consider a class of Boolean functions of the form , where , is odd, , and .
With the restriction of , we present the characterization of hyper-bentness of these functions with character sums. Further, we reformulate this characterization in terms of the number of points on
hyper-elliptic curves. For some special cases, with the help of Kloosterman sums and cubic sums, we determine the characterization for some hyper-bent functions including functions with four, six and ten traces terms. Evaluations of Kloosterman sums at three general points are used in the characterization. Actually, our results can generalized to the general
case: . And we explain this for characterizing binomial, trinomial and quadrinomial hyper-bent functions
On the Primary Constructions of Vectorial Boolean Bent Functions
Vectorial Boolean bent functions, which possess the maximal nonlinearity and the minimum differential uniformity, contribute to optimum resistance against linear cryptanalysis and differential cryptanalysis for the cryptographic algorithms that adopt them as nonlinear components. This paper is devoted to the new primary constructions of vectorial Boolean bent functions, including four types: vectorial monomial bent functions, vectorial Boolean bent functions with multiple trace terms, vectorial functions and -like vectorial functions. For vectorial monomial bent functions, this paper answers one open problem proposed by E. Pasalic et al. and characterizes the vectorial monomial bent functions corresponding to the five known classes of bent exponents. For the vectorial Boolean bent functions with multiple trace terms, this paper answers one open problem proposed by A. Muratović-Ribić et al., presents six new infinite classes of explicit constructions and shows the nonexistence of the vectorial Boolean bent functions from to of the form with and . Moreover, vectorial functions are further characterized. In addition, a new infinite class of vectorial Boolean bent function named as -like vectorial functions are derived, which includes vectorial functions as a subclass
A note on semi-bent functions with multiple trace terms and hyperelliptic curves
Semi-bent functions with even number of variables are a class of important Boolean
functions whose Hadamard transform takes three values. In this note we are interested
in the property of semi-bentness of Boolean functions defined on the Galois field (n
even) with multiple trace terms obtained via Niho functions and two Dillon-like functions
(the first one has been studied by Mesnager and the second one have been studied very
recently by Wang, Tang, Qi, Yang and Xu). We subsequently give a connection between the
property of semi-bentness and the number of rational points on some associated hyperelliptic
curves. We use the hyperelliptic curve formalism to reduce the computational complexity in
order to provide a polynomial time and space test leading to an efficient characterization of
semi-bentness of such functions (which includes an efficient characterization of the hyperbent
functions proposed by Wang et al.). The idea of this approach goes back to the recent work
of Lisonek on the hyperbent functions studied by Charpin and Gong
A note on constructions of bent functions from involutions
Bent functions are maximally nonlinear Boolean functions. They are important
functions introduced by Rothaus and studied rstly by Dillon and next by many researchers
for four decades. Since the complete classication of bent functions seems
elusive, many researchers turn to design constructions of bent functions. In this note,
we show that linear involutions (which are an important class of permutations) over
nite elds give rise to bent functions in bivariate representations. In particular, we
exhibit new constructions of bent functions involving binomial linear involutions whose
dual functions are directly obtained without computation
Relation between o-equivalence and EA-equivalence for Niho bent functions
Boolean functions, and bent functions in particular, are considered up to so-called EA-equivalence, which is the most general known equivalence relation preserving bentness of functions. However, for a special type of bent functions, so-called Niho bent functions there is a more general equivalence relation called o-equivalence which is induced from the equivalence of o-polynomials. In the present work we study, for a given o-polynomial, a general construction which provides all possible o-equivalent Niho bent functions, and we considerably simplify it to a form which excludes EA-equivalent cases. That is, we identify all cases which can potentially lead to pairwise EA-inequivalent Niho bent functions derived from o-equivalence of any given Niho bent function. Furthermore, we determine all pairwise EA-inequivalent Niho bent functions arising from all known o-polynomials via o-equivalence.publishedVersio
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