### Abstract

GDL is the discrete logarithm problem for a general finitc group G. This paper gives a characterization for the intractability of GDL from the viewpoint of computational complexity theory. It is shown that GDL ∈ NP ∩ co-AM, assuming that G is in NP ∩ co-NP, and that the group law operation of G can be exccuted in a polynomial time of the element size. Furthermore, as a natural probabilistic extension, the complexity of GDL is investigated under the assumption that the group law operation is executed in an expected polynomial time of the element size. In this case, it is shown that GDL ∈ MA ∩ co-AM if G ∈ NP ∩ co-NP. Finally, we show that GDL is less intractable than NP-complete problems unless the polynomial time hierarchy collapses to the second level.

Original language | English |
---|---|

Title of host publication | Advances in Cryptology – EUROCRYPT 1992 - Workshop on the Theory and Application of Cryptographic Techniques, Proceedings |

Editors | Rainer A. Rueppel |

Publisher | Springer Verlag |

Pages | 420-428 |

Number of pages | 9 |

ISBN (Print) | 9783540564133 |

DOIs | |

Publication status | Published - Jan 1 1993 |

Externally published | Yes |

Event | Workshop on the Theory and Application of Cryptographic Technique, EUROCRYPT 1992 - Balatonfured, Hungary Duration: May 24 1992 → May 28 1992 |

### Publication series

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

Volume | 658 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Other

Other | Workshop on the Theory and Application of Cryptographic Technique, EUROCRYPT 1992 |
---|---|

Country | Hungary |

City | Balatonfured |

Period | 5/24/92 → 5/28/92 |

### Fingerprint

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Computer Science(all)

### Cite this

*Advances in Cryptology – EUROCRYPT 1992 - Workshop on the Theory and Application of Cryptographic Techniques, Proceedings*(pp. 420-428). (Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics); Vol. 658 LNCS). Springer Verlag. https://doi.org/10.1007/3-540-47555-9_34

**How intractable is the discrete logarithm for a general finite group?** / Okamoto, Tatsuaki; Sakurai, Kouichi; Shizuya, Hiroki.

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

*Advances in Cryptology – EUROCRYPT 1992 - Workshop 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. 658 LNCS, Springer Verlag, pp. 420-428, Workshop on the Theory and Application of Cryptographic Technique, EUROCRYPT 1992, Balatonfured, Hungary, 5/24/92. https://doi.org/10.1007/3-540-47555-9_34

}

TY - GEN

T1 - How intractable is the discrete logarithm for a general finite group?

AU - Okamoto, Tatsuaki

AU - Sakurai, Kouichi

AU - Shizuya, Hiroki

PY - 1993/1/1

Y1 - 1993/1/1

N2 - GDL is the discrete logarithm problem for a general finitc group G. This paper gives a characterization for the intractability of GDL from the viewpoint of computational complexity theory. It is shown that GDL ∈ NP ∩ co-AM, assuming that G is in NP ∩ co-NP, and that the group law operation of G can be exccuted in a polynomial time of the element size. Furthermore, as a natural probabilistic extension, the complexity of GDL is investigated under the assumption that the group law operation is executed in an expected polynomial time of the element size. In this case, it is shown that GDL ∈ MA ∩ co-AM if G ∈ NP ∩ co-NP. Finally, we show that GDL is less intractable than NP-complete problems unless the polynomial time hierarchy collapses to the second level.

AB - GDL is the discrete logarithm problem for a general finitc group G. This paper gives a characterization for the intractability of GDL from the viewpoint of computational complexity theory. It is shown that GDL ∈ NP ∩ co-AM, assuming that G is in NP ∩ co-NP, and that the group law operation of G can be exccuted in a polynomial time of the element size. Furthermore, as a natural probabilistic extension, the complexity of GDL is investigated under the assumption that the group law operation is executed in an expected polynomial time of the element size. In this case, it is shown that GDL ∈ MA ∩ co-AM if G ∈ NP ∩ co-NP. Finally, we show that GDL is less intractable than NP-complete problems unless the polynomial time hierarchy collapses to the second level.

UR - http://www.scopus.com/inward/record.url?scp=59049098563&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=59049098563&partnerID=8YFLogxK

U2 - 10.1007/3-540-47555-9_34

DO - 10.1007/3-540-47555-9_34

M3 - Conference contribution

AN - SCOPUS:59049098563

SN - 9783540564133

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

SP - 420

EP - 428

BT - Advances in Cryptology – EUROCRYPT 1992 - Workshop on the Theory and Application of Cryptographic Techniques, Proceedings

A2 - Rueppel, Rainer A.

PB - Springer Verlag

ER -