Breaking One-Round Key-Agreement Protocols in the Random Oracle Model

In this paper we study one-round key-agreement protocols analogous to Merkle's puzzles in the random oracle model. The players Alice and Bob are allowed to query a random permutation oracle $n$ times and upon their queries and communication, they both output the same key with high probability. We prove that Eve can always break such a protocol by querying the oracle $O(n^2)$ times. The long-time unproven optimality of the quadratic bound in the fully general, multi-round scenario has been shown recently by Barak and Mahmoody-Ghidary. The results in this paper have been found independently of their work.