On the Complexity of Computing Discrete Logarithms over Algebraic Tori

Authors:
Shuji Isobe;Eisuke Koizumi;Yuji Nishigaki;Hiroki Shizuya
Affiliations:
Graduate School of Information Sciences, Tohoku University, Sendai, Japan 980-8576;Graduate School of Information Sciences, Tohoku University, Sendai, Japan 980-8576;Graduate School of Information Sciences, Tohoku University, Sendai, Japan 980-8576 and Presently with Development Planning Dept., Brother Industries, Ltd., Nagoya, Japan 467-8562;Graduate School of Information Sciences, Tohoku University, Sendai, Japan 980-8576
Venue:
CANS '09 Proceedings of the 8th International Conference on Cryptology and Network Security
Year:
2009

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Abstract

This paper studies the complexity of computing discrete logarithms over algebraic tori. We show that the order certified version of the discrete logarithm over general finite fields (OCDL, in symbols) reduces to the discrete logarithm over algebraic tori (TDL, in symbols) with respect to the polynomial-time Turing reducibility. This reduction means that if the integer factorization can be computed in polynomial time, then TDL is equivalent to the discrete logarithm DL over general finite fields with respect to the Turing reducibility.