A New Construction of Weightwise Perfectly Balanced Functions with High Weightwise Nonlinearity

Abstract

The FLIP cipher was proposed at Eurocrypt 2016 for the purpose of meliorating the efficiency of fully homomorphic cryptosystems. Weightwise perfectly balanced Boolean functions meet the balancedness requirement of the filter function in FLIP ciphers, and the construction of them has attracted serious attention from researchers. Nevertheless, the literature is still thin. Modifying the supports of functions with a low degree is a general construction technique whose key problem is to find a class of available low-degree functions. We first seek out a class of quadratic functions and then, based on these functions, present the new construction of weightwise perfectly balanced Boolean functions by adopting an iterative approach. It is worth mentioning that the functions we construct have good performance in weightwise nonlinearity. In particular, some p-weight nonlinearities achieve the highest values in the literature for a small number of variables.