Functional Inversion and Communication Complexity

In this paper, we study the relation between the multi-party communication complexity over various communication topologies and the complexity of inverting functions and/or permutations. In particular, we show that if a function has a ring-protocol or a tree-protocol of communication complexity bounded by H, then there is a circuit of size O(2Hn) which computes an inverse of the function. Consequently, we have proved, although inverting NC0 Boolean circuits is NP-complete, planar NC1 Boolean circuits can be inverted in NC, and hence in polynomial time. In general, NCk planar boolean circuits can be inverted in O(nlog(k-1)n) time. Also from the ring-protocol results, we derive an ?(n log n) lower bound on the VLSI area to layout any one-way functions. Our results on inverting boolean circuits can be extended to invert algebraic circuits over finite rings.One significant aspect of our result is that it enables us to compare the communication power of two topologies. We have proved that on some topologies, no one-way function nor its inverse can be computed with bounded communication complexity.