A class of weightwise almost perfectly balanced Boolean functions

Abstract

Constructing Boolean functions with good cryptographic properties over a subset of vectors with fixed Hamming weight $ E_{n,k} \subset {{\rm{I\!F}}}_2^n $ is significant in lightweight stream ciphers like FLIP [14]. In this article, we have given a construction for a class of $ n $-variable weightwise almost perfectly balanced (WAPB) Boolean functions from known support of an $ n_0 $-variable WAPB Boolean function where $ n_0 < n $. This is a generalization of constructing a weightwise perfectly balanced (WPB) Boolean function by Mesnager and Su [16]. We have studied some cryptographic properties like ANF, nonlinearity, weightwise nonlinearities, and algebraic immunity of the functions. The ANF of this function is obtained recursively, which would be a low-cost implementation in a lightweight stream cipher. Further, we have presented another class of WAPB Boolean functions by modifying the earlier function, and we studied some of its cryptographic properties. The nonlinearity and weightwise nonlinearities of the modified functions improve substantially.