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A new approach to the discrete logarithm problem with auxiliary inputs

Part of: Theory of computing Computational number theory

Published online by Cambridge University Press:  01 January 2016

Jung Hee Cheon  and
Taechan Kim
Jung Hee Cheon
Affiliation:
Department of Mathematical Sciences, Seoul National University, GwanAkRo 1, Gwanak-Gu, Seoul 151-747, Korea email jhcheon@snu.ac.kr
Taechan Kim
Affiliation:
NTT Secure Platform Laboratories, 3-9-11, Midori-cho, Musashino-Shi, Tokyo 180-8585, Japan email taechan.kim@lab.ntt.co.jp

Abstract

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The aim of the discrete logarithm problem with auxiliary inputs is to solve for ${\it\alpha}$, given the elements $g,g^{{\it\alpha}},\ldots ,g^{{\it\alpha}^{d}}$ of a cyclic group $G=\langle g\rangle$, of prime order $p$. The best-known algorithm, proposed by Cheon in 2006, solves for ${\it\alpha}$ in the case where $d\mid (p\pm 1)$, with a running time of $O(\sqrt{p/d}+d^{i})$ group exponentiations ($i=1$ or $1/2$ depending on the sign). There have been several attempts to generalize this algorithm to the case of ${\rm\Phi}_{k}(p)$ where $k\geqslant 3$. However, it has been shown by Kim, Cheon and Lee that a better complexity cannot be achieved than that of the usual square root algorithms.

We propose a new algorithm for solving the DLPwAI. We show that this algorithm has a running time of $\widetilde{O}(\sqrt{p/{\it\tau}_{f}}+d)$ group exponentiations, where ${\it\tau}_{f}$ is the number of absolutely irreducible factors of $f(x)-f(y)$. We note that this number is always smaller than $\widetilde{O}(p^{1/2})$.

In addition, we present an analysis of a non-uniform birthday problem.

MSC classification

Primary: 68Q25: Analysis of algorithms and problem complexity
Secondary: 11Y16: Algorithms; complexity
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Issue 1 , 2016 , pp. 1 - 15
Copyright
© The Author(s) 2016 

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