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Bounds in Various Generalized Settings of the Discrete Logarithm Problem

机译：离散对数问题的各种广义设定下的界

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摘要

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.
机译：本文研究了广义多重离散对数问题的一般硬度，其中对于离散对数问题的各种设置，求解器必须解决n个实例中的k个问题。对于通用k和n，我们引入了两种技术来为这种计算复杂性确定下限。可以证明一种方法可以在经典设置中为小输入实现渐近紧边界。另一种方法可实现较大输入的界限，并且能够适应其他离散对数设置中的应用。在后者中，我们通过应用n的分区获得广义下界，并且进一步表明，我们选择的分区方法可以达到最佳边界。这项工作可以看作是Yun（EUROCRYPT '15）分析的多重离散对数问题的硬度的推广和扩展。还计算了相对于k的各种n的一些显式边界。

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