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

Abstract

Two-dimensional discrete logarithm problem in a finite group G with addition as the group operation is a generalization of the classical discrete logarithm problem and consists in solving the equation Q = n1P1 + n2P2 with respect to n1, n2 for the specified P1, P2, Q ∈ G, 0 < N1, N2 < |G| under the assumption that there exists solution with −N1 ≤ n1 ≤ N1, −N2 ≤ n2 ≤ N2. In 2004, Gaudry and Schost have proposed an algorithm solving this problem with the average complexity (c + o(1)) √ N of group operations in G where (under standard heuristic assumptions) c ≈ 2.43, N = 4N1N2, N → ∞. In 2009, Galbraith and Ruprai improved this algorithm to obtain c ≈ 2.36.In this paper we prove that for the group of points of an elliptic curve over a finite prime field having an effective automorphism ϕ of order 6 the average complexity of the GaudrySchost algorithm for the two-dimensional discrete logarithm problem with P2 = ϕ(P1) and N1 = N2 is at most (c + o(1)) √ N , where c ≈ 0.9781.

Modified Gaudry-Schost algorithm for the two-dimensional discrete logarithm problem

“…Then average complexity measured in group operations does not exceed the total number Z N of values (x i , y i ) and (z j , w j ) chosen before the appearance of (x l , y l ) and (z m , w m ) such that C(x l , y l ) = C(z m , w m ). Further we will estimate average number of steps of algorithm Z N as it was done in [13]. Under the conditions of the Galbraith-Holmes theorem, we have the following equalities:…”

“…1) In the case of subgroup of an elliptic curve y 2 = x 3 + Ax + B over a finite prime field with p > 3 elements (it has an efficient automorphism of order 2) with N 1 = N 2 there exists an algorithm with average 4 ) group operations [13], where…”

“…2) In the case of a subgroup of an elliptic curve y 2 = x 3 + Ax over a finite prime field with p ≡ 1 mod 4 elements, P 2 = ϕ(P 1 ) and N 1 = N 2 , where ϕ is an efficient automorphism of order 4, there exists an algorithm with average complexity (1 + ε)0.8862 4 ) group operations [13], where N = 4N 1 N 2 .…”