The Twin Diffie–Hellman Problem and Applications

We propose a new computational problem called the twin Diffie–Hellman problem. This problem is closely related to the usual (computational) Diffie–Hellman problem and can be used in many of the same cryptographic constructions that are based on the Diffie–Hellman problem. Moreover, the twin Diffie–Hellman problem is at least as hard as the ordinary Diffie–Hellman problem. However, we are able to show that the twin Diffie–Hellman problem remains hard, even in the presence of a decision oracle that recognizes solutions to the problem—this is a feature not enjoyed by the Diffie–Hellman problem, in general. Specifically, we show how to build a certain “trapdoor test” that allows us to effectively answer decision oracle queries for the twin Diffie–Hellman problem without knowing any of the corresponding discrete logarithms. Our new techniques have many applications. As one such application, we present a new variant of ElGamal encryption with very short ciphertexts, and with a very simple and tight security proof, in the random oracle model, under the assumption that the ordinary Diffie–Hellman problem is hard. We present several other applications as well, including a new variant of Diffie and Hellman’s non-interactive key exchange protocol; a new variant of Cramer–Shoup encryption, with a very simple proof in the standard model; a new variant of Boneh–Franklin identity-based encryption, with very short ciphertexts; a more robust version of a password-authenticated key exchange protocol of Abdalla and Pointcheval. This paper was solicted by the Editors-in-Chief as one of the best papers from EUROCRYPT 2008, based on the recommendation of the program committee. Part of this work completed while at CWI. Supported by the research program Sentinels. Supported by NSF award number CNS-0716690.