Hadamard Matrices, $d$-Linearly Independent Sets and Correlation-Immune Boolean Functions with Minimum Hamming Weights

Abstract

It is known that correlation-immune (CI) Boolean functions used in the framework of side channel attacks need to have low Hamming weights. In 2013, Bhasin et al. studied the minimum Hamming weight of $d$-CI Boolean functions, and presented an open problem: the minimal weight of a $d$-CI function in $n$ variables might not increase with $n$. Very recently, Carlet and Chen proposed some constructions of low-weight CI functions, and gave a conjecture on the minimum Hamming weight of $3$-CI functions in $n$ variables. In this paper, we determine the values of the minimum Hamming weights of $d$-CI Boolean functions in $n$ variables for infinitely many $n$\u27s and give a negative answer to the open problem proposed by Bhasin et al. We then present a method to construct minimum-weight 2-CI functions through Hadamard matrices, which can provide all minimum-weight 2-CI functions in $4k-1$ variables. Furthermore, we prove that the Carlet-Chen conjecture is equivalent to the famous Hadamard conjecture. Most notably, we propose an efficient method to construct low-weight $n$-variable CI functions through $d$-linearly independent sets, which can provide numerous minimum-weight $d$-CI functions. Particularly, we obtain some new values of the minimum Hamming weights of $d$-CI functions in $n$ variables for $n\leq 13$. We conjecture that the functions constructed by us are of the minimum Hamming weights if the sets are of absolute maximum $d$-linearly independent. If our conjecture holds, then all the values for $n\leq 13$ and most values for general $n$ are determined