Weightwise almost perfectly balanced functions: secondary constructions for all n and better weightwise nonlinearities.

The design of FLIP stream cipher presented at Eurocrypt 2016 motivates the study of Boolean function with good cryptographic criteria when restricted to subsets of F-2(n). Since the security of FLIP relies on properties of functions restricted to subsets of constant Hamming weight, called slices, several studies investigate functions with good properties on the slices, i.e. weightwise properties. A major challenge is to build functions balanced on each slice, from which we get the notion of Weightwise Almost Perfectly Balanced (WAPB) functions. Although various constructions ofWAPB functions have been exhibited since 2017, building WAPB functions with high weightwise nonlinearities remains a difficult task. Lower bounds on the weightwise nonlinearities of WAPB functions are known for very few families, and exact values were computed only for functions in at most 16 variables. In this article, we introduce and study two new secondary constructions of WAPB functions. This new strategy allows us to bound the weightwise nonlinearities from those of the parent functions enabling us to produce WAPB functions with high weightwise nonlinearities. As a practical application, we build several novel WAPB functions in up to 16 variables by taking parent functions from two different known families. Moreover, combining these outputs, we also produce the 16-variable WAPB function with the highest weightwise nonlinearities known so far.