On the Modulo Degree Complexity of Boolean Functions

Abstract

For each integer m\ge 2, every Boolean function f can be expressed as a unique multilinear polynomial modulo m, and the degree of this multilinear polynomial is called its modulo m degree. In this paper we investigate the modulo degree complexity of total Boolean functions initiated by Parikshit Gopalan et al. [8], in which they asked the following question: whether the degree complexity of a Boolean function is polynomially related with its modulo m degree. For m be a power of primes, it is already known that the module m degree can be arbitrarily smaller compare to the degree complexity When m has at least two distinct prime factors, the question remains open. Towards this question, our results include: (1) we obtain some nontrivial equivalent forms of this question; (2) we affirm this question for some special classes of functions; (3) we prove a no-go theorem, explaining why this problem is difficult to attack from the computational complexity point of view; (4) we show a super-linear separation between the degree complexity and the modulo m degree.