### Abstract

GDL is the language whose membership problem is polynomial-time Turing equivalent to the discrete logarithm problem for a general finite group G. This paper gives a characterization of GDL from the viewpoint of computational complexity theory. It is shown that GDL ε NP ∩ co-AM, assuming that G is in NP ∩ co-NP, and that the group law operation of G can be executed in 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 ε MA ∩ co-AM if G ε MA ∩ co-MA. As a consequence, we show that GDL is not NP-complete unless the polynomial time hierarchy collapses to the second level.

Original language | English |
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Pages (from-to) | 61-65 |

Number of pages | 5 |

Journal | IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences |

Volume | E79-A |

Issue number | 1 |

Publication status | Published - 1996 |

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### All Science Journal Classification (ASJC) codes

- Signal Processing
- Computer Graphics and Computer-Aided Design
- Electrical and Electronic Engineering
- Applied Mathematics

### Cite this

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*,

*E79-A*(1), 61-65.

**On the complexity of the discrete logarithm for a general finite group.** / Okamoto, Tatsuaki; Sakurai, Kouichi; Shizuya, Hiroki.

Research output: Contribution to journal › Article

*IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences*, vol. E79-A, no. 1, pp. 61-65.

}

TY - JOUR

T1 - On the complexity of the discrete logarithm for a general finite group

AU - Okamoto, Tatsuaki

AU - Sakurai, Kouichi

AU - Shizuya, Hiroki

PY - 1996

Y1 - 1996

N2 - GDL is the language whose membership problem is polynomial-time Turing equivalent to the discrete logarithm problem for a general finite group G. This paper gives a characterization of GDL from the viewpoint of computational complexity theory. It is shown that GDL ε NP ∩ co-AM, assuming that G is in NP ∩ co-NP, and that the group law operation of G can be executed in 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 ε MA ∩ co-AM if G ε MA ∩ co-MA. As a consequence, we show that GDL is not NP-complete unless the polynomial time hierarchy collapses to the second level.

AB - GDL is the language whose membership problem is polynomial-time Turing equivalent to the discrete logarithm problem for a general finite group G. This paper gives a characterization of GDL from the viewpoint of computational complexity theory. It is shown that GDL ε NP ∩ co-AM, assuming that G is in NP ∩ co-NP, and that the group law operation of G can be executed in 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 ε MA ∩ co-AM if G ε MA ∩ co-MA. As a consequence, we show that GDL is not NP-complete unless the polynomial time hierarchy collapses to the second level.

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UR - http://www.scopus.com/inward/citedby.url?scp=0029732406&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0029732406

VL - E79-A

SP - 61

EP - 65

JO - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

JF - IEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences

SN - 0916-8508

IS - 1

ER -