A quantum version of Pollard's Rho of which Shor's Algorithm is a particular case

TL;DR

This paper introduces a quantum version of Pollard's Rho algorithm for integer factorization, demonstrating polynomial-time complexity and revealing it as a generalization of Shor's algorithm.

Contribution

It presents a quantum adaptation of Pollard's Rho with polynomial-time performance and establishes its relation as a generalization of Shor's algorithm.

Findings

Quantum Pollard's Rho runs in polynomial time.

Pollard's Rho is a generalization of Shor's algorithm.

New characterization of nontrivial factors in the sequence.

Abstract

Pollard's Rho is a method for solving the integer factorization problem. The strategy searches for a suitable pair of elements belonging to a sequence of natural numbers that given suitable conditions yields a nontrivial factor. In translating the algorithm to a quantum model of computation, we found its running time reduces to polynomial-time using a certain set of functions for generating the sequence. We also arrived at a new result that characterizes the availability of nontrivial factors in the sequence. The result has led us to the realization that Pollard's Rho is a generalization of Shor's algorithm, a fact easily seen in the light of the new result.