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On the Complexity of Generalized Discrete Logarithm Problem

Dec 2022

Generalized Discrete Logarithm Problem (GDLP) is an extension of the DiscreteLogarithm Problem where the goal is to find $x\in\mathbb{Z}_s$ such $g^x\mods=y$ for a given $g,y\in\mathbb{Z}_s$. Generalized discrete logarithm issimilar but instead of a single base element, uses a number of base elementswhich does not necessarily commute with each other. In this paper, we provethat GDLP is NP-hard for symmetric groups. Furthermore, we prove that GDLPremains NP-hard even when the base elements are permutations of at most 3elements. Lastly, we discuss the implications and possible implications of ourproofs in classical and quantum complexity theory.

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