Bounds in Various Generalized Settings of the Discrete Logarithm Problem
This paper examines the generic hardness of the generalized multiple discrete logarithm problem, where the solver has to solve k out of n instances for various settings of the discrete logarithm problem. For generic k and n, we introduce two techniques to establish the lower bounds for this computational complexity. One method can be shown to achieve asymptotically tight bounds for small inputs in the classical setting. The other method achieves bounds for larger inputs as well as being able to adapt for applications in other discrete logarithm settings. In the latter, we obtain the generalized lower bounds by applying partitions of n and furthermore show that our chosen method of partition achieves the best bounds. This work can be regarded as a generalization and extension on the hardness of the multiple discrete logarithm problem analyzed by Yun (EUROCRYPT ’15). Some explicit bounds for various n with respect to k are also computed.
KeywordsDiscrete logarithm Generalized multiple discrete logarithm Chebyshev’s inequality Optimization Gaussian elimination
The authors wish to thank Phong Nguyen for valuable discussions and all anonymous reviewers for their helpful comments. This research was partially supported by JST CREST Grant Number JPMJCR14D6, Japan and JSPS KAKENHI Grant Number 16H02780.
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