Multiple Parallel-Pollard's Rho Discrete Logarithm Algorithm

Title & Authors
Multiple Parallel-Pollard's Rho Discrete Logarithm Algorithm
Lee, Sang-Un;

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 $\small{{\alpha}^a{\beta}^b{\equiv}{\alpha}^A{\beta}^B}$ (modp) from the initial value a = b = 0, only to derive $\small{{\gamma}}$ from $\small{(a+b{\gamma})=(A+B{\gamma})}$, $\small{{\gamma}(B-b)=(a-A)}$. The basic Pollard's Rho algorithm computes $\small{x_i=(x_{i-1})^2,{\alpha}x_{i-1},{\beta}x_{i-1}}$ given $\small{{\alpha}^a{\beta}^b{\equiv}x}$(modp), and the general algorithm computes $\small{x_i=(x_{i-1})^2}$, $\small{Mx_{i-1}}$, $\small{Nx_{i-1}}$ for randomly selected $\small{M={\alpha}^m}$, $\small{N={\beta}^n}$. This paper proposes 4-model Pollard Rho algorithm that seeks $\small{{\beta}_{\gamma}={\alpha}^{\gamma},{\beta}_{\gamma}={\alpha}^{(p-1)/2+{\gamma}}}$, and $\small{{\beta}_{{\gamma}^{-1}}={\alpha}^{(p-1)-{\gamma}}}$) from $\small{m=n={\lceil}{\sqrt{n}{\rceil}}$, (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%.
Keywords
discrete logarithm;Euler's totient function;Pollard Rho algorithm;
Language
Korean
Cited by
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