New factorization algorithm based on a continuous representation of truncated Gauss sums

TL;DR

This paper introduces a novel factorization algorithm utilizing continuous Gauss sums, capable of identifying all factors of a number simultaneously without prior ratio calculations, and proposes optical implementations for experimental realization.

Contribution

The paper presents a new factorization method based on continuous Gauss sums that generalizes to higher orders and enables single-run factor detection without precomputations.

Findings

Allows factorization of numbers in a single run

Distinguishes factors from non-factors effectively

Proposes optical interferometer setups for implementation

Abstract

In this paper, we will describe a new factorization algorithm based on the continuous representation of Gauss sums, generalizable to orders j>2. Such an algorithm allows one, for the first time, to find all the factors of a number N in a single run without precalculating the ratio N/l, where l are all the possible trial factors. Continuous truncated exponential sums turn out to be a powerful tool for distinguishing factors from non-factors (we also suggest, with regard to this topic, to read an interesting paper by S. Woelk et al. also published in this issue [Woelk, Feiler, Schleich, J. Mod. Opt. in press]) and factorizing different numbers at the same time. We will also describe two possible M-path optical interferometers, which can be used to experimentally realize this algorithm: a liquid crystal grating and a generalized symmetric Michelson interferometer.