## Cryptology ePrint Archive: Report 2008/067

**The Twin Diffie-Hellman Problem and Applications**

*David Cash and Eike Kiltz and Victor Shoup*

**Abstract: **We propose a new computational problem called the \emph{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.

**Category / Keywords: **public-key cryptography / public-key encryption, identity-based encryption

**Publication Info: **Preliminary version to appear in EUROCRYPT 2008. This is the full version.

**Date: **received 8 Feb 2008, last revised 10 Feb 2009

**Contact author: **cdc at gatech edu

**Available format(s): **PDF | BibTeX Citation

**Note: **Appeared at EUROCRYPT 2008. This is the full version.

**Version: **20090210:204631 (All versions of this report)

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