Functions with special Walsh transforms

Bent functions, semi-bent functions, partially bent functions and related vectorial functions, such as vectorial bent functions, almost bent functions and perfect nonlinear functions, which all can be described in terms of their Walsh transform, attracted increasing attention in the last decade. One of the reasons are the rich connections to many areas and objects in number theory, algebra, combinatorics or geometry. Boolean bent functions for instance have highest possible nonlinearity, which is one of the major quality measure for Boolean functions used in cryptology, their nonlinearity corresponds to the covering radius of the first order Reed-Muller code. Bent functions in general correspond to relative difference sets. The research plan of this project can be divided into two areas, the construction and the analysis of bent functions and of related functions, and the analysis of quadratic functions (which are always partially bent) with applications in coding theory and in the construction of maximal curves. Some concrete research plans in the first topic are the construction and the analysis of bent functions which are affine on the elements of a semifield spread. These bent functions are interesting since they correspond to hyperovals of semifield planes; the construction of new strongly regular graphs obtained from a construction of bent functions; the analysis of the degree of bent functions and of the normality of bent functions in odd characteristic; the analysis of vectorial bent functions with respect to non-weak regularity or with respect to the duals of the component functions. Methods: Characters, Gaussian sums, finite fields arithmetic, finite geometry, combinatorics. Concrete plans at the analysis of quadratic functions are to obtain counting results for classes of quadratic functions which are related to the weight distribution of subcodes of the second order Reed-Muller code; the construction of maximal and minimal Artin-Schreier curves, or curves with many rational points. Methods: Discrete Fourier transform and number theoretical analysis, quadratic forms, character sum analysis.