Functional Inversion and Communication Complexity

Shang-Hua Teng

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 ofn variables has aring-protocol or atree-protocol of communication complexity bounded by ϕ, then there is a circuit of size\(2^{0(\phi )} n\) that computes an inverse of the function. Consequently, we prove that although invertingNC 0 Boolean circuits isNP-hard, planarNC 1 Boolean circuits can be inverted inNC, and hence in polynomial time. From the ring-protocol theorem, we derive an ω(n logn) 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.