On Boolean Functions with Low Polynomial Degree and Higher Order Sensitivity

Abstract: Boolean functions are important primitives in different domains of cryptology, complexity and coding theory. In this paper, we connect the tools from cryptology and complexity theory in the domain of Boolean functions with low polynomial degree and high sensitivity. It is well known that the polynomial degree of of a Boolean function and its resiliency are directly connected. Using this connection we analyze the polynomial degree-sensitivity values through the lens of resiliency, demonstrating existence and non-existence results of functions with low polynomial degree and high sensitivity on small number of variables (upto 10). In this process, borrowing an idea from complexity theory, we show that one can implement resilient Boolean functions on a large number of variables with linear size and logarithmic depth. Finally, we extend the notion of sensitivity to higher order and note that the existing construction idea of Nisan and Szegedy (1994) can provide only constant higher order sensitivity when aiming for polynomial degree of $n-\omega(1)$. In this direction, we present a construction with low ($n-\omega(1)$) polynomial degree and super-constant $\omega(1)$ order sensitivity exploiting Maiorana-McFarland constructions, that we borrow from construction of resilient functions. The questions we raise identify novel combinatorial problems in the domain of Boolean functions.