Functional inversion and communication complexity

Abstract

We study the relationship between the multiparty communication complexity of functions over certain communication topologies and the complexity of inverting those functions. We show that if a function of n variables has a ringprotocol or a tree-protocol of communication complexity bounded by q~, then there is a circuit of size 2°~n that computes an inverse of the function. Consequently, we prove that although inverting NC ° Boolean circuits is NP-hard, planar NC 1 Boolean circuits can be inverted in NC, and hence in polynomial time. From the ring-protocol theorem, we derive an ~(n 10g n) lower bound on the VLSI area required to lay out any one-way function. Our results on inverting boolean circuits can be extended to algebraic circuits over finite rings. We prove that on certain topologies no one-way function can be computed with low communication complexity.