On the Algebraic Immunity of Weightwise Perfectly Balanced Functions

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 a WPB function with AI 2 by sampling at random. Finally, we address the problem of constructing WPB functions with bounded algebraic immunity, exploiting a construction from [12]. 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 [12] 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.