Merkle's Key Agreement Protocol is Optimal: An O(n²) Attack on any Key Agreement from Random Oracles

We prove that every key agreement protocol in the random oracle model in which the honest users make at most $n$ queries to the oracle can be broken by an adversary who makes $O(n^2)$ queries to the oracle. This improves on the previous $\widetilde{\Omega}(n^6)$ 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 key agreement protocol in 1974 that can be easily implemented with $n$ queries to a random oracle and cannot be broken by any adversary who asks $o(n^2)$ queries.