Constructing Low-Weight $d$ th-Order Correlation-Immune Boolean Functions Through the Fourier-Hadamard Transform

Abstract

The correlation immunity of Boolean functions is a property related to cryptography, to error correcting codes, to orthogonal arrays (in combinatorics), and in a slightly looser way to sequences. Correlation-immune Boolean functions (in short, CI functions) have the property of keeping the same output distribution when some input variables are fixed. They have been widely used as combiners in stream ciphers to allow resistance to the Siegenthaler correlation attack. Very recently, a new use of CI functions has appeared in the framework of side channel attacks (SCA). To reduce the cost overhead of counter-measures to SCA, CI functions need to have low Hamming weights. This actually poses new challenges since the known constructions which are based on properties of the Walsh-Hadamard transform, do not allow to build unbalanced CI functions. In this paper, we propose constructions of low-weight d th-order CI functions based on the Fourier-Hadamard transform, while the known constructions of resilient functions are based on the Walsh-Hadamard transform. These two transforms are closely related but the resulting constructions are very different. We first prove a simple but powerful result, which makes that one only need to consider the case where d is odd in further research. Then, we investigate how constructing low Hamming weight CI functions through the Fourier-Hadamard transform (which behaves well with respect to the multiplication of Boolean functions). We use the characterization of CI functions by the Fourier-Hadamard transform and introduce a related general construction of CI functions by multiplication. By using the Kronecker product of vectors, we obtain more constructions of low-weight d-CI Boolean functions. Furthermore, we present a method to construct low-weight d-CI Boolean functions by making additional restrictions on the supports built from the Kronecker product.