Theory Seminar: The Discrete-logarithm Problem with Preprocessing

Henry Corrigan-Gibbs (Stanford University)
Wednesday, 24.1.2018, 12:30
Taub 201

We study discrete-log algorithms that use preprocessing. In our model, an adversary may use a very large amount of precomputation to produce an "advice" string about a specific group (e.g., NIST P-256). In a subsequent online phase, the adversary's task is to use the preprocessed advice to quickly compute discrete logarithms in the group. Motivated by surprising recent preprocessing attacks on the discrete-log problem, we study the power and limits of such algorithms.

In particular, we focus on generic algorithms that operate in every cyclic group. We show a lower bound on the success probability of any generic discrete-log algorithm with preprocessing. Our lower bound, which is tight up to logarithmic factors, uses a synthesis of incompressibility techniques and classic methods for generic-group lower bounds. We apply our techniques to prove related lower bounds for the CDH, DDH, and multiple-discrete-log problems.

Finally, we demonstrate two new generic preprocessing attacks: one for the multiple-discrete-log problem and one for certain decisional-type problems in groups. This latter result demonstrates that, for generic algorithms with preprocessing, distinguishing tuples of the form [g, g^x, g^(x^2)] from random is much easier than the discrete-log problem.

This is joint work with Dima Kogan.

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