On the complexity of two-dimensional discrete logarithm problem in a finite cyclic group with effective automorphism of order
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M. V. Nikolaev
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 [xxx] 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 [xxx] of group operations in G where (under standard heuristic assumptions) c ⋲ 2.43, N = 4N1N2, N → 1. 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 Gaudry- Schost algorithm for the two-dimensional discrete logarithm problem with P2 = φ(P1) and N1 = N2 is at most [xxx], where c ⋲ 0.9781.
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Articles in the same Issue
- Masthead
- Distance-regular graph with the intersection array {45, 30, 7; 1, 2, 27} does not exist
- The impact of the growth rate of the packing number of graphs on the computational complexity of the independent set problem
- Traveling salesman polytopes and cut polytopes. Affine reducibility
- Minors of the constraint matrix in multi-index transport problems
- Cycle structure of power mappings in a residue classes ring
- A criterion for the automata realizable functions to be boundedly deterministic
- On the complexity of two-dimensional discrete logarithm problem in a finite cyclic group with effective automorphism of order
- The completeness criterion for some systems containing P-sets of automaton functions
- On the synthesis of circuits admitting complete fault detection test sets of constant length under arbitrary constant faults at the outputs of the gates
- Factorially solvable rings
- Identities with permutations and quasigroups isotopic to groups and to Abelian groups
- Classes of projectively equivalent quadrics over local rings
