Interplay between resiliency and polynomial degree — Recursive amplification, higher order sensitivity and beyond

Abstract

Boolean functions are important primitives in different domains of cryptology, complexity and coding theory, and far beyond in different areas of science and technology. In this paper we connect the tools of cryptology and complexity theory in the domain of resilient Boolean functions. It is well known that the resiliency of a Boolean function and its polynomial degree are directly connected. We first show that borrowing an idea from complexity theory, one can implement resilient Boolean functions on a large number of variables with little amount of circuit. Further, we also look into the search techniques used in the construction of resilient Boolean functions to show the existence and non-existence results of functions with low polynomial degree and high sensitivity on small number of variables. In the process, we settle some previously open problems. Finally, we extend the notion of sensitivity to higher order and present a construction with low polynomial degree and higher order sensitivity exploiting Maiorana-McFarland functions. The questions we raise identify novel combinatorial problems in the domain of Boolean functions.