622 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 bent and hyper-bent functions
Bent functions are Boolean functions which have maximum possible nonlinearity i.e. maximal distance to the set of affine functions. They were introduced by Rothaus in 1976. In the last two decades, they have been studied widely due to their interesting combinatorial properties and their applications in cryptography. However the complete classification of bent functions has not been achieved yet. In 2001 Youssef and Gong introduced a subclass of bent functions which they called hyper-bent functions. The construction of hyper-bent functions is generally more difficult than the construction of bent functions. In this thesis we give a survey of recent constructions of infinite classes of bent and hyper-bent functions where the classification is obtained through the use of Kloosterman and cubic sums and Dickson polynomials
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
An Open Problem on the Bentness of Mesnager’s Functions
Let . In the present paper, we study the binomial Boolean functions of the form
where is an even positive integer, and .
We show that is a bent function if the Kloosterman sum
equals , thus settling an open problem of Mesnager. The proof employs tools including computing Walsh coefficients of Boolean functions via multiplicative
characters, divisibility properties of Gauss sums, and graph theory
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
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