Multiple Parallel-Pollard's Rho Discrete Logarithm Algorithm

Abstract

This paper proposes a discrete logarithm algorithm that remarkably reduces the execution time of Pollard’s Rho algorithm. Pollard’s Rho algorithm computes congruence or collision of α a β b ≡ α A β B (mod ρ) from the initial value a = b = o, only to derive γ from (a+bγ) = (A+Bγ), γ(B-b) = (a-A). The basic Pollard’s Rho algorithm computes χ i = (χ i-1 )², αχ i-1 , βχ i-1 given α a β b ≡ χ(mod ρ), and the general algorithm computes χ i = (χ i-1 )², Mχ i-1 , Nχ i-1 for randomly selected M=α m , N=β n . This paper proposes 4-model Pollard Rho algorithm that seeks βγ = αγ, βγ′ = α(p-1)/2+γ and βγ-1 = α(p-1)-γ) from m = n = ?√n?, (a,b) = (0.0), (1,1). algorithm that seeks β γ = α γ , β γ′ = α (p-1)/2+γ and β γ-1 = α (p-1)-γ ) from m = n = ?√n?, (a,b) = (0.0), (1,1). The proposed algorithm has proven to improve the performance of the (0,0)-basic Pollard’s Rho algorithm by 71.70%.