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We prove that every key agreement protocol in the random oracle model inwhich the honest users make at most $n$ queries to the oracle can be broken byan adversary who makes $O(n^2)$ queries to the oracle. This improves on theprevious $\widetilde{\Omega}(n^6)$ query attack given by Impagliazzo and Rudich(STOC '89) and resolves an open question posed by them.  Our bound is optimal up to a constant factor since Merkle proposed a keyagreement protocol in 1974 that can be easily implemented with $n$ queries to arandom oracle and cannot be broken by any adversary who asks $o(n^2)$ queries.

Merkle's Key Agreement Protocol is Optimal: An $O(n^2)$ Attack on any Key Agreement from Random Oracles