Recursions for modified Walsh transforms of some families of Boolean functions

Abstract

We show that, under certain conditions, restricted and biased exponential sums and Walsh transforms of symmetric and rotation symmetric Boolean functions are, as in the case of nonbiased domain, $C$-finite sequences. We also prove that under other conditions, these sequences are $P$-finite, which is a somewhat different behavior than their nonbiased counterparts. We further show that exponential sums and Walsh transforms of a family of rotation symmetric monomials over the restricted domain $E_{n,j}=\{x\in {\mathbb{𝔽}}_2^n:wt(x)=j\}$ ($wt(x)$ is the weight of the vector $x$) are given by polynomials of degree at most $j$, and so, they are also $C$-finite sequences. Finally, we also present a study of the behavior of symmetric Boolean functions under these biased transforms.