8 research outputs found
On Exponential Sums, Nowton identities and Dickson Polynomials over Finite Fields
Let be a finite field, be an extension of
, let be a polynomial of degree
with . We present a recursive formula for evaluating the
exponential sum . Let and
be two elements in with , be a positive integer. We
obtain an estimate for the exponential sum , where is the lifting
of an additive character of . Some properties of the
sequences constructed from these exponential sums are provided also.Comment: 18 page
A conjecture about Gauss sums and bentness of binomial Boolean functions
In this note, the polar decomposition of binary fields of even extension degree is used to reduce the evaluation of the Walsh transform of binomial Boolean functions to that of Gauss sums. In the case of extensions of degree four times an odd number, an explicit formula involving a Kloosterman sum is conjectured, proved with further restrictions, and supported by extensive experimental data in the general case. In particular, the validity of this formula is shown to be equivalent to a simple and efficient characterization for bentness previously conjectured by Mesnager
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