New Construction Method for Weightwise Perfectly Balanced Boolean Functions

In federated learning, to strengthen privacy protection, model updates can be encrypted using Fully Homomorphic Encryption (FHE) before aggregation. Combining FLIP with specific FHE schemes to form a hybrid fully homomorphic encryption scheme can improve the efficiency of FHE. The FLIP stream cipher requires inputs of constant Hamming weight, which motivates the study of Weightwise Perfectly Balanced (WPB) Boolean functions. This paper constructs a new class of WPB functions through an iterative approach. We have proven their algebraic degree and calculated their algebraic immunity and weightwise nonlinearity. Compared to the existing literature, the weightwise nonlinearity of the functions presented here is found to be the highest for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k=7, k=8$</tex>, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$k=9$</tex> in the 16-variable case.