Progress in Cryptology – LATINCRYPT 2023

Abstract

In this article we study the Algebraic Immunity (AI) of Weightwise Perfectly Balanced (WPB) functions.After showing a lower bound on the AI of two classes of WPB functions from the previous literature, we prove that the minimal AI of a WPB n-variables function is constant, equal to 2 for n ≥ 4 . Then, we compute the distribution of the AI of WPB function in 4 variables, and estimate the one in 8 and 16 variables. For these values of n we observe that a large majority of WPB functions have optimal AI, and that we could not obtain an AI-2 WPB function by sampling at random. Finally, we address the problem of constructing WPB functions with bounded algebraic immunity, exploiting a construction from [GM22c]. In particular, we present a method to generate multiple WPB functions with minimal AI, and we prove that the WPB functions with high nonlinearity exhibited in [GM22c] also have minimal AI. We conclude with a construction giving WPB functions with lower bounded AI, and give as example a family with all elements with AI at least n/2 − log(n) + 1.