### 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" which 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.

Original language | English (US) |
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Title of host publication | Advances in Cryptology - EUROCRYPT 2008 - 27th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings |

Pages | 127-145 |

Number of pages | 19 |

Volume | 4965 LNCS |

DOIs | |

State | Published - 2008 |

Event | 27th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2008 - Istanbul, Turkey Duration: Apr 13 2008 → Apr 17 2008 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 4965 LNCS |

ISSN (Print) | 03029743 |

ISSN (Electronic) | 16113349 |

### Other

Other | 27th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2008 |
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Country | Turkey |

City | Istanbul |

Period | 4/13/08 → 4/17/08 |

### Fingerprint

### ASJC Scopus subject areas

- Computer Science(all)
- Biochemistry, Genetics and Molecular Biology(all)
- Theoretical Computer Science

### Cite this

*Advances in Cryptology - EUROCRYPT 2008 - 27th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings*(Vol. 4965 LNCS, pp. 127-145). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 4965 LNCS). https://doi.org/10.1007/978-3-540-78967-3_8

**The twin diffie-hellman problem and applications.** / Cash, David; Kiltz, Eike; Shoup, Victor.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Advances in Cryptology - EUROCRYPT 2008 - 27th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings.*vol. 4965 LNCS, Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 4965 LNCS, pp. 127-145, 27th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2008, Istanbul, Turkey, 4/13/08. https://doi.org/10.1007/978-3-540-78967-3_8

}

TY - GEN

T1 - The twin diffie-hellman problem and applications

AU - Cash, David

AU - Kiltz, Eike

AU - Shoup, Victor

PY - 2008

Y1 - 2008

N2 - 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" which 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.

AB - 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" which 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.

UR - http://www.scopus.com/inward/record.url?scp=44449176564&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=44449176564&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-78967-3_8

DO - 10.1007/978-3-540-78967-3_8

M3 - Conference contribution

SN - 3540789669

SN - 9783540789666

VL - 4965 LNCS

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 127

EP - 145

BT - Advances in Cryptology - EUROCRYPT 2008 - 27th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Proceedings

ER -