International Association for Cryptologic Research

International Association
for Cryptologic Research


Paper: The Twin Diffie-Hellman Problem and Applications

David Cash
Eike Kiltz
Victor Shoup
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Abstract: 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 ordinary Diffie-Hellman problem. In particular, we show how to build a certain “trapdoor test” that allows us to effectively answer such decision oracle queries 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.
  title={The Twin Diffie-Hellman Problem and Applications},
  booktitle={IACR Eprint archive},
  keywords={public-key cryptography / public-key encryption, identity-based encryption},
  note={Preliminary version to appear in EUROCRYPT 2008. This is the full version. 14041 received 8 Feb 2008, last revised 11 Jun 2008},
  author={David Cash and Eike Kiltz and Victor Shoup},