Generalized Bent Functions

Philippe Langevin

I3S, 250 Av. A. Einstein, 06560 Valbonne, France.

Let p be a prime, and let K be the field of order p. A function f from K^m into K is called a generalized bent function if there exists a non trivial additive character $\chi$ of K such that :

$$\vert \sum_{x\in K^m} \chi(f(x)+ax) \vert =\sqrt{p^m}$$

for any a in K^m. In the binary case p=2, there are bent functions if and only if m is even. In that case, a Boolean bent function has degree at most $m\over 2$ and is at maximal distance from the space of affine functions.

The goal of my talk is to compare the bent functions ( p=2) and the generalized bent functions (p>2). Namely, we will see that there are generalised bent functions for any m, and that the degree of a generalized bent function can be upperbounded like in the binary case. Surprisingly, for a given m the distance between the space of affine functions and a generalized bent function is not constant, leading to the question : are the generalized bent functions really bent ?