Abstract: In Public Key Cryptography system, separate keys are used to encode and decode the data. Public key being distributed publicly, the strength of security depends on large key size. The discrete logarithm for mathematical base in Public Key Cryptography systems. Unlike the finite field Discrete Logarithm Problem; there are no general purpose sub exponential algorithms to solve the Elliptic Curve Discrete Logarithm Problem. Though good algorithms are known for certain specific types of elliptic curves, all known algorithms that apply to general curves take fully exponential time. As a result, elliptic curves are gaining popularity for building cryptosystems. The absence of sub exponential algorithms implies that smaller fields can be chosen compared to those needed for cryptosystems. Elliptic curve based cryptosystems are popular because they provide good security at key sizes much smaller than number theoretical Public Key Schemes like RSA cryptosystem. Solving Elliptic Curve Discrete Logarithm Problem using Pollardís Rho algorithm provide efficiency in terms of time and storage. Using various parallel architectures like MPI, GP-GPU and FPGA increase accessing precision and efficiency of solving ECDLP.This article covers Elliptic Curve Cryptography, Elliptic Curve Discrete Logarithm Problem and Pollard's Rho algorithms to solve ECDLP.

Keywords: Elliptic curve, ECC, DLP, ECDLP, Pollardís Rho Algorithm, Parallel Architectures.