Paper 2009/577

On the nonlinearity profile of the Dillon function

Claude Carlet

Abstract

The nonlinearity profile of a Boolean function is the sequence of its minimum Hamming distances $nl_r(f)$ to all functions of degrees at most $r$, for $r\geq 1$. The nonlinearity profile of a vectorial function is the sequence of the minimum Hamming distances between its component functions and functions of degrees at most $r$, for $r\geq 1$.The profile of the multiplicative inverse functions has been lower bounded in a previous paper by the same author. No other example of an infinite class of functions with unbounded algebraic degree has been exhibited since then, whose nonlinearity profile could be efficiently lower bounded. In this preprint, we lower bound the whole nonlinearity profile of the simplest Dillon bent function $(x,y)\mapsto xy^{2^{n/2}-2}$, $x,y\in F_{2^{n/2}}$.