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We prove that every key exchange 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 making O(n 2) queries to the oracle. This improves on the previous $\Tilde{\Omega}(n^6)$ query attack given by Impagliazzo and Rudich (STOC '89), and answers an open question posed by them. Our bound is optimal up to a constant factor since Merkle (CACM '78) gave a key exchange protocol that can easily be implemented in this model with n queries and cannot be broken by an adversary making o(n 2) queries.

Merkle Puzzles Are Optimal — An O(n2)-Query Attack on Any Key Exchange from a Random Oracle