A Family of pp-ary Binomial Bent Functions

For a prime $p$ with $p\equiv 3\,({\rm mod}\, 4)$ and an odd number $m$, the Bentness of the $p$-ary binomial function $f_{a,b}(x)={\rm Tr}_{1}^n(ax^{p^m-1})+{\rm Tr}_{1}^2(bx^{\frac{p^n-1}{4}})$ is characterized, where $n=2m$, a\in \bF_{p^n}^*, and b\in \bF_{p^2}^*. The necessary and sufficient conditions of $f_{a,b}(x)$ being Bent are established respectively by an exponential sum and two sequences related to $a$ and $b$. For the special case of $p=3$, we further characterize the Bentness of the ternary function $f_{a,b}(x)$ by the Hamming weight of a sequence

Strongly Regular Graphs Constructed from pp-ary Bent Functions

In this paper, we generalize the construction of strongly regular graphs in [Y. Tan et al., Strongly regular graphs associated with ternary bent functions, J. Combin.Theory Ser. A (2010), 117, 668-682] from ternary bent functions to $p$-ary bent functions, where $p$ is an odd prime. We obtain strongly regular graphs with three types of parameters. Using certain non-quadratic $p$-ary bent functions, our constructions can give rise to new strongly regular graphs for small parameters.Comment: to appear in Journal of Algebraic Combinatoric

Dual and Hull code in the first two generic constructions and relationship with the Walsh transform of cryptographic functions

We contribute to the knowledge of linear codes from special polynomials and functions, which have been studied intensively in the past few years. Such codes have several applications in secret sharing, authentication codes, association schemes and strongly regular graphs. This is the first work in which we study the dual codes in the framework of the two generic constructions; in particular, we propose a Gram-Schmidt (complexity of $\mathcal{O}(n^3)$) process to compute them explicitly. The originality of this contribution is in the study of the existence or not of defining sets $D'$, which can be used as ingredients to construct the dual code $\mathcal{C}'$ for a given code $\mathcal{C}$ in the context of the second generic construction. We also determine a necessary condition expressed by employing the Walsh transform for a codeword of $\mathcal{C}$ to belong in the dual. This achievement was done in general and when the involved functions are weakly regularly bent. We shall give a novel description of the Hull code in the framework of the two generic constructions. Our primary interest is constructing linear codes of fixed Hull dimension and determining the (Hamming) weight of the codewords in their duals