On the complexity of two-dimensional discrete logarithm problem in a finite cyclic group with efficient automorphism

Abstract

Two-dimensional discrete logarithm problem in a finite additive group G consists in solving the equation Q = n 1 P 1 + n 2 P 2 with respect to n 1 , n 2 for specified P 1 , P 2 , Q ∈ G, 0 < N 1 , N 2 < |G| such that there exists solution with |n 1 | ≤ N 1 , |n 2 | ≤ N 2 . In 2004, Gaudry and Schost proposed an algorithm to solve this problem with average complexity (c + o(1)) √ N of group operations in G where c ≈ 2.43, N = 4N 1 N 2 , N → ∞. In 2009, Galbraith and Ruprai improved this algorithm to obtain c ≈ 2.36. We show that the constant c may be reduced if the group G has an automorphism computable faster than the group operation.