Multiparty Communication Complexity of Collision Finding

TL;DR

This paper establishes a new lower bound on the communication complexity for collision-finding in multiparty settings, which also implies stronger lower bounds for certain proof systems related to the pigeonhole principle.

Contribution

It introduces a novel lower bound on the k-party communication complexity of collision-finding, improving previous bounds and connecting to proof complexity.

Findings

New lower bound of (n^{1-1/k}  log k / 2^k) for collision-finding

Implication of a 2^{n^{1-o(1)}} lower bound on cutting-planes proofs

Improves upon previous lower bounds of 2^{\u221a n}

Abstract

We prove an $\Omega(n^{1-1/k} \log k \ /2^k)$ lower bound on the $k$-party number-in-hand communication complexity of collision-finding. This implies a $2^{n^{1-o(1)}}$ lower bound on the size of tree-like cutting-planes proofs of the bit pigeonhole principle, a compact and natural propositional encoding of the pigeonhole principle, improving on the best previous lower bound of $2^{\Omega(\sqrt{n})}$.