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

Tatsuaki Okamoto, Kouichi Sakurai, Hiroki Shizuya

    Research output: Contribution to journalArticle

    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 languageEnglish
    Pages (from-to)61-65
    Number of pages5
    JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
    VolumeE79-A
    Issue number1
    Publication statusPublished - 1996 Jan 1

    Keywords

    • Computational complexity
    • Cryptography
    • Discrete logarithm

    ASJC Scopus subject areas

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

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