New Collisions to Improve Pollard’s Rho Method of Solving the Discrete Logarithm Problem on Elliptic Curves

Abstract

It is true that different approaches have been utilised to accelerate the computation of discrete logarithm problem on elliptic curves with Pollard's Rho method. However, trapping in cycles fruitless will be obtained by using the random walks with Pollard's Rho. An efficient alternative approach that is based on new collisions which are reliant on the values a i , b i to solve this problem is proposed. This may requires less iterations than Pollard's Rho original in reaching collision. Thus, the performance of Pollard's Rho method is more efficiently because the improved method not only reduces the number of mathematical operations but these collisions can also applied on previous improvements which reported in the literature.

Modification of Pollard Rho Algorithm Using Negation Mapping

“…However, to prove the security of our cryptosystem, we must try to crack the system. One of the fastest algorithms to attack ECDLP is the Pollard Rho algorithm [11], which will be explained in Chapter 2. This research modified the Pollard Rho algorithm by applying negation mapping to reduce iteration steps.…”