Weightwise Perfectly Balanced Functions and Nonlinearity

Abstract

In this article we realize a general study on the nonlinearity of weightwise perfectly balanced (WPB) functions. First, we derive upper and lower bounds on the nonlinearity from this class of functions for all n. Then, we give a general construction that allows us to provably provide WPB functions with nonlinearity as low as 2 n/2−1 and WPB functions with high nonlinearity, at least 2 n−1 − 2 n/2 . We provide concrete examples in 8 and 16 variables with high nonlinearity given by this construction. In 8 variables we experimentally obtain functions reaching a nonlinearity of 116 which corresponds to the upper bound of Dobbertin's conjecture, and it improves upon the maximal nonlinearity of WPB functions recently obtained with genetic algorithms. Finally, we study the distribution of nonlinearity over the set of WPB functions. We examine the exact distribution for n = 4 and provide an algorithm to estimate the distributions for n = 8 and 16, together with the results of our experimental studies for n = 8 and 16.

Progress in Cryptology – LATINCRYPT 2023

“…To conclude this section we perform an experimental investigation on the algebraic immunity distribution for WPB functions in a small number of variables, following the same principle as in [GM22a,GM22c]. Exhausting WPB 2 , we found that all the WPB function in 4 variables have algebraic immunity 2, it is indeed coherent with Corollary 1.…”

“…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.…”

“…The main cryptographic properties that have been studied on WPB functions so far are the weightwise nonlinearity (i.e. nonlinearity restricted on the slices) such as in [LM19, MSLZ22,GM22a], and more recently the (global) nonlinearity such as in [MKCL22,GM22c]. Other relevant cryptographic properties on Boolean functions have not been studied deeply on the set of WPB functions, such as the algebraic immunity.…”

“…Then, we address the problem of constructing WPB functions with bounded algebraic immunity. We use the construction from [GM22c] to build functions with upper bounded AI. 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 use the construction from [GM22c] to build functions with upper bounded AI. 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 finish 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.…”