Improving the Success Probability for Shor's Factoring Algorithm

TL;DR

This paper presents a method to improve the success probability of Shor's factoring algorithm by selectively choosing certain elements in Z_n^* to increase the likelihood of successful factorization from 50% to 75%.

Contribution

It introduces a strategy to select special elements in Z_n^* that enhances the probability of successful factorization in Shor's algorithm.

Findings

Success probability improved from 50% to 75%.

Reduces worst-case failure probability from 50% to 25%.

Provides an efficient selection method for better outcomes.

Abstract

Given n=p*q with p and q prim and y in Z_{p*q}^*. Shor's Algorithm computes the order r of y, i.e. y^r=1 (mod n). If r=2k is even and y^k \ne -1 (mod n) we can easily compute a non trivial factor of n: gcd(y^k-1,n). In the original paper it is shown that a randomly chosen y is usable for factoring with probabily {1/2}. In this paper we will show an efficient possibility to improve the lower bound of this probability by selecting only special y in Z_n^* to {3/4}, so we are able to reduce the fault probabilty in the worst case from {1/2} to {1/4}.