Effectiveness of a Method to Eliminate Fruitless Cycles for Pollard's Rho Method

In this research, the authors focus on an attack on a kind of pairing-friendly curves called the Barreto-Naehring curve. Accelerated attacks involve evaluating the security level concerning the elliptic curve discrete logarithm problem (ECDLP). The authors confirm the effectiveness of a method to eliminate a fruitless cycle in a random-walk path for parallel Pollard's rho method with skew Frobenius mapping for the curve. Though the rho method is known to solve the ECDLP efficiently, a random-walk path sometimes induces the unsolvable cycle, called a fruitless cycle, then the random-walk must restart with yet another starting point. In a previous work, the authors proposed a method to eliminate the fruitless cycle for a random-walk path. In this research, the authors implement a parallel rho method and confirm the effectiveness of the proposed method by several experiments. The results show that the proposed method effectively eliminate the fruitless cycles of length two and three, but increase the fruitless cycles of length four.