Analysis of Pollard Rho Attacks Over ECDLP

Elliptic Curve Cryptography (ECC) is a popular kind of public key encryption technique that offers a significant advantage over cryptographic systems like RSA. ECC has attracted a lot of interest lately as it offers higher security with reduced key sizes. Security of ECC is based on a hard problem known as Elliptic Curve Discrete Logarithm Problem (ECDLP). Computation of private scalar integer from public key is computationally difficult. Solving ECDLP using Pollard rho method provides higher efficiency than baby-step giant-step and index calculus methods. Performance of Pollard rho method depends on the cycle detection techniques and iteration function used in sequential approach. In this study, survey of several proposed variants of Pollard rho method in sequential and paralleled architectures is presented. Analysis of Pollard method in sequential architecture for different cycle detection techniques and iteration function is discussed. Experimental analysis of the Floyd and Brent cycle over different prime curves is presented. Further, proposed techniques for paralleled Pollard method using distinguished point property are discussed using CPU, Graphics Processing Unit (GPU), and Field-Programmable Gate Array (FPGA) clusters for prime and binary curves.