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

TL;DR

This paper introduces a novel collision-based enhancement to Pollard's Rho algorithm for elliptic curve discrete logarithm problems, reducing iterations and computational effort by leveraging new collision strategies.

Contribution

It proposes a new collision technique relying on values ai, bi that improves Pollard's Rho efficiency for elliptic curve discrete logarithm computations.

Findings

Fewer iterations needed to reach collision.

Reduced number of mathematical operations.

Compatibility with previous improvements.

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 ai , bi 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.