MORE VECTORIAL BOOLEAN FUNCTIONS WITH UNBOUNDED NONLINEARITY PROFILE
The nonlinearity profile of Boolean functions is a generalization of the most important cryptographic criterion, called the (first order) nonlinearity. It is defined as the sequence of the minimum Hamming distances nl(r)(f) between a given Boolean function f and all Boolean functions in the same number of variables and of degrees at most r, for r >= 1. This parameter, which has a close relationship with the Gowers norm, quantifies the resistance to cryptanalyses by low degree approximations of stream ciphers using the Boolean function f as combiner or as filter. The nonlinearity profile can also be defined for vectorial functions: it is the sequence of the minimum Hamming distances between the component functions of the vectorial function and all Boolean functions of degrees at most r, for r >= 1. The nonlinearity 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 nonlinearity profile has been exhibited since then. In this paper, we lower bound the whole nonlinearity profile of the (simplest) Dillon bent function (x, y) -> xy(2n/2-2), x, y is an element of F(2n/2) and we exhibit another class of functions, for which bounding the whole profile of each of them comes down to bounding the first order nonlinearities of all functions.
boolean function covering radius higher-order nonlinearity reed-muller code s-box