On the weightwise nonlinearity of weightwise perfectly balanced functions

In this article we perform a general study on the criterion of weightwise nonlinearity for the functions which are weightwise perfectly balanced (WPB). First, we investigate the minimal value this criterion can take over WPB functions, deriving theoretic bounds, and exhibiting the first values. We emphasize the link between this minimum and weightwise affine functions, and we prove that for n≥8 no n-variable WPB function can have this property. Then, we focus on the distribution and the maximum of this criterion over the set of WPB functions. We provide theoretic bounds on the latter and algorithms to either compute or estimate the former, together with the results of our experimental studies for n up to 8. Finally, we present two new constructions of WPB functions obtained by modifying the support of linear functions for each set of fixed Hamming weight. This provides a large corpus of WPB function with proven weightwise nonlinearity, and we compare the weightwise nonlinearity of these constructions to the average value, and to the parameters of former constructions in 8 and 16 variables.