### Abstract

We introduce a new cryptosystem with trapdoor decryption based on the difficulty of computing discrete logarithms in the class group of the nonmaximal imaginary quadratic order N Δq, where δq = δq2, δ square-free and q prime. The trapdoor information is the conductor q. Knowledge of this trapdoor information enables one to switch to and from the class group of the maximal order N Δ, where the representatives of the ideal classes have smaller coefficients. Thus, the decryption procedure may be performed in the class group of N Δ rather than in the class group of the public N Δq, which is much more efficient. We show that inverting our proposed cryptosystem is computationally equivalent to factoring the non-fundamental discriminant δq, which is intractable for a suitable choice of δ and q. We also describe how signature schemes in N Δq may be set up using this trapdoor information. Furthermore, we illustrate how one may embed key escrow capability into classical imaginary quadratic field cryptosystems.

Original language | English |
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Title of host publication | Advances in Cryptology — EUROCRYPT 1998, International Conference on the Theory and Application of Cryptographic Techniques, Proceedings |

Editors | Kaisa Nyberg |

Publisher | Springer Verlag |

Pages | 294-307 |

Number of pages | 14 |

ISBN (Print) | 3540645187, 9783540645184 |

DOIs | |

Publication status | Published - Jan 1 1998 |

Event | International Conference on the Theory and Application of Cryptographic Techniques, EUROCRYPT 1998 - Espoo, Finland Duration: May 31 1998 → Jun 4 1998 |

### 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 | 1403 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | International Conference on the Theory and Application of Cryptographic Techniques, EUROCRYPT 1998 |
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Country | Finland |

City | Espoo |

Period | 5/31/98 → 6/4/98 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Advances in Cryptology — EUROCRYPT 1998, International Conference on the Theory and Application of Cryptographic Techniques, Proceedings*(pp. 294-307). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 1403). Springer Verlag. https://doi.org/10.1007/BFb0054134

**A cryptosystem based on non-maximal imaginary quadratic orders with fast decryption.** / Hühnlein, Detlef; Jacobson, Michael J.; Paulus, Sachar; Takagi, Tsuyoshi.

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

*Advances in Cryptology — EUROCRYPT 1998, International Conference on the Theory and Application of Cryptographic Techniques, Proceedings.*Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics), vol. 1403, Springer Verlag, pp. 294-307, International Conference on the Theory and Application of Cryptographic Techniques, EUROCRYPT 1998, Espoo, Finland, 5/31/98. https://doi.org/10.1007/BFb0054134

}

TY - GEN

T1 - A cryptosystem based on non-maximal imaginary quadratic orders with fast decryption

AU - Hühnlein, Detlef

AU - Jacobson, Michael J.

AU - Paulus, Sachar

AU - Takagi, Tsuyoshi

PY - 1998/1/1

Y1 - 1998/1/1

N2 - We introduce a new cryptosystem with trapdoor decryption based on the difficulty of computing discrete logarithms in the class group of the nonmaximal imaginary quadratic order N Δq, where δq = δq2, δ square-free and q prime. The trapdoor information is the conductor q. Knowledge of this trapdoor information enables one to switch to and from the class group of the maximal order N Δ, where the representatives of the ideal classes have smaller coefficients. Thus, the decryption procedure may be performed in the class group of N Δ rather than in the class group of the public N Δq, which is much more efficient. We show that inverting our proposed cryptosystem is computationally equivalent to factoring the non-fundamental discriminant δq, which is intractable for a suitable choice of δ and q. We also describe how signature schemes in N Δq may be set up using this trapdoor information. Furthermore, we illustrate how one may embed key escrow capability into classical imaginary quadratic field cryptosystems.

AB - We introduce a new cryptosystem with trapdoor decryption based on the difficulty of computing discrete logarithms in the class group of the nonmaximal imaginary quadratic order N Δq, where δq = δq2, δ square-free and q prime. The trapdoor information is the conductor q. Knowledge of this trapdoor information enables one to switch to and from the class group of the maximal order N Δ, where the representatives of the ideal classes have smaller coefficients. Thus, the decryption procedure may be performed in the class group of N Δ rather than in the class group of the public N Δq, which is much more efficient. We show that inverting our proposed cryptosystem is computationally equivalent to factoring the non-fundamental discriminant δq, which is intractable for a suitable choice of δ and q. We also describe how signature schemes in N Δq may be set up using this trapdoor information. Furthermore, we illustrate how one may embed key escrow capability into classical imaginary quadratic field cryptosystems.

UR - http://www.scopus.com/inward/record.url?scp=84957614726&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84957614726&partnerID=8YFLogxK

U2 - 10.1007/BFb0054134

DO - 10.1007/BFb0054134

M3 - Conference contribution

AN - SCOPUS:84957614726

SN - 3540645187

SN - 9783540645184

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

SP - 294

EP - 307

BT - Advances in Cryptology — EUROCRYPT 1998, International Conference on the Theory and Application of Cryptographic Techniques, Proceedings

A2 - Nyberg, Kaisa

PB - Springer Verlag

ER -