Reducing Fruitless Cycles in Pollard's Rho Method with SFM for Efficient ECDLP Attacks on BN Curves

Pollard's rho method is a widely used probabilistic algorithm for solving the Elliptic Curve Discrete Logarithm Problem (ECDLP). To enhance its efficiency, the skew Frobenius mapping (SFM) has been incorporated into the method, reducing the average number of steps required to find a collision. However, the combination of Pollard's rho method with SFM introduces the occurrence of fruitless cycles, which are computational loops that do not contribute to solving the ECDLP and hinder the overall attacking efficiency. This paper presents a comprehensive analysis of fruitless cycles in Pollard's rho method with SFM, focusing on their impact when applied to Barreto-Naehrig (BN) curves. We propose a scheme to mitigate the occurrence of fruitless cycles by optimizing the seed point selection to be adaptive according to situation. To validate the effectiveness of our proposed method, we perform comprehensive experiments and compare the results with the conventional Pollard's rho method with SFM. The results indicate that the proposed approach significantly reduces the occurrence of fruitless cycles, leading to faster and more efficient ECDLP attacks on BN curves.