Research on Attacking a Special Elliptic Curve Discrete Logarithm Problem

Cheon first proposed a novel algorithm for solving discrete logarithm problem with auxiliary inputs. Given some points P , α P , α 2 P , … , α d P ∈ G , an attacker can solve the secret key efficiently. In this paper, we propose a new algorithm to solve another form of elliptic curve discrete logarithm problem with auxiliary inputs. We show that if some points P , α P , α k P , α k 2 P , α k 3 P , … , α k φ ( d ) - 1 P ∈ G and a multiplicative cyclic group K = 〈 k 〉 are given, where d is a prime, φ ( d ) is the order of K . The secret key α ∈ F p ⁎ can be solved in O ( ( p - 1 ) / d + d ) group operations by using O ( ( p - 1 ) / d ) storage.