Results on cubic bent and weakly regular bent $p$-ary functions leading to a class of cubic ternary non-weakly regular bent functions

Abstract

Much work has been devoted to bent functions in odd characteristic, but there still remains a gap between our knowledge of binary and nonbinary bent functions. In the first part of this paper, we attempt to partially bridge this gap by generalizing to any characteristic important properties known in characteristic two concerning the Walsh transform of derivatives of bent functions. Some of these properties generalize to all bent functions, while others appear to apply only to weakly regular bent functions. We deduce a method to obtain a bent function by adding a quadratic function to a weakly regular bent function. We also identify a particular class of bent functions possessing the property that every first-order derivative in a nonzero direction has a derivative (which is then a second-order derivative of the function) equal to a nonzero constant. We show that this property implies bentness and is shared in particular by all cubic bent functions. This generalizes to the odd characteristic the notion of cubic-like bent function, that was introduced and studied for binary functions by Irene Villa and the first author. In the second part of the paper, we provide (for the first time) a primary construction leading to an infinite class of cubic ternary vectorial bent functions that have only not weakly regular components. We show the bentness of the component functions by two approaches: by calculating the Walsh transform directly and by considering the second-order derivatives (and applying the results from the first part of the paper). We prove that they are not weakly regular by showing they do not have one of the properties that we proved in the first part of the paper for weakly regular bent functions.