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 $F_{2^n}$ (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 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 $\sum\limits_{r\in R}\mathrm{Tr}_{1}^{n} (a_{r}x^{r(2^m-1)})$ and Mesnager\u27s $\sum\limits_{r\in R}\mathrm{Tr}_{1}^{n}(a_{r}x^{r(2^m-1)}) +\mathrm{Tr}_{1}^{2}(bx^{\frac{2^n-1}{3}})$, where $R$ is a set of representations of the cyclotomic cosets modulo $2^m+1$ of full size $n$ and $a_{r}\in \mathbb{F}_{2^m}$. In this paper, we generalize their results and consider a class of Boolean functions of the form $\sum_{r\in R}\sum_{i=0}^{2}Tr^n_1(a_{r,i}x^{r(2^m-1)+\frac{2^n-1}{3}i}) +Tr^2_1(bx^{\frac{2^n-1}{3}})$, where $n=2m$, $m$ is odd, $b\in\mathbb{F}_4$, and $a_{r,i}\in \mathbb{F}_{2^n}$. With the restriction of $a_{r,i}\in \mathbb{F}_{2^m}$, 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: $a_{r,i}\in \mathbb{F}_{2^n}$. And we explain this for characterizing binomial, trinomial and quadrinomial hyper-bent functions

An Open Problem on the Bentness of Mesnager’s Functions

Let $n=2m$. In the present paper, we study the binomial Boolean functions of the form $$f_{a,b}(x) = \mathrm{Tr}_1^{n}(a x^{2^m-1 }) +\mathrm{Tr}_1^{2}(bx^{\frac{2^n-1}{3} }),$$ where $m$ is an even positive integer, $a\in \mathbb{F}_{2^n}^*$ and $b\in \mathbb{F}_4^*$. We show that $f_{a,b}$ is a bent function if the Kloosterman sum $$K_{m}\left(a^{2^m+1}\right)=1+ \sum_{x\in \mathbb{F}_{2^m}^*} (-1)^{\mathrm{Tr}_1^{m}(a^{2^m+1} x+ \frac{1}{x})}$$ equals $4$, 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

Cifrado de datos e intercambio de claves utilizando polinomios de Dickson

La aparición de la informática y el uso masivo de las comunicaciones digitales han producido un número creciente de problemas de seguridad. El objetivo de la criptografía es el de proporcionar comunicaciones seguras (y secretas) sobre canales inseguros. Algunos de los problemas que la criptografía trata de resolver son el cifrado de datos y el intercambio de claves seguro a través de un medio inseguro. En este trabajo se estudia un sistema de cifrado de datos basado en un tipo especial de polinomios, llamados polinomios de Dickson, con coeficientes en los enteros modulo n. Se analizan criterios para polinomios de permutación, con énfasis en las propiedades de los polinomios de Dickson. Se dan las condiciones necesarias para que se pueda transmitir información de manera segura usando este sistema de cifrado. Se incluye además un análisis detallado sobre la seguridad del mismo. También se describe un protocolo para el intercambio de claves basado en polinomios de Dickson, que permite acordar la clave a utilizar en cada comunicación. Se examina la seguridad del protocolo comparando con el protocolo más usado en la actualidad que es el de Diffie-Hellman, mismo que basa su seguridad en el problema del logaritmo discreto (análogamente se define el problema de Dickson discreto)

Dickson polynomials, hyperelliptic curves and hyper-bent functions

In this paper, we study the action of Dickson polynomials on subsets of finite fields of even characteristic related to the trace of the inverse of an element and provide an alternate proof of a not so well-known result. Such properties are then applied to the study of a family of Boolean functions and a characterization of their hyper-bentness in terms of exponential sums recently proposed by Wang et al. Finally, we extend previous works of Lisoněk and Flori and Mesnager to reformulate this characterization in terms of the number of points on hyperelliptic curves and present some numerical results leading to an interesting problem.