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Journal of Physics A Mathematical and Theoretical

Factorization with exponential sums

M. Štefaňák, D Haase, W. Merkel, Muhammad Suhail Zubairy, Wolfgang P. Schleich

Published Jul 15, 2008• 15 citations
Publisher

Abstract

We generalize the concept of factorization using truncated Gauss sums to exponential sums where the phase increases with the jth power of the summation index. For such sums the number of terms needed to suppress ghost factors of N scales as . Unfortunately, this advantageous scaling law is accompanied by a disadvantage: the gap between factors and non-factors decreases rapidly with increasing power j and as a consequence it gets more difficult to identify factors. This feature serves as our motivation to study sums with an exponential phase. Our numerical simulations indicate that in this case the scaling law is logarithmic and that we retain a significant gap between factors and non-factors.

DOI
10.1088/1751-8113/41/30/304024

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We generalize the concept of factorization using truncated Gauss sums to exponential sums where the phase increases with the jth power of the summation index. For such sums the number of terms needed to suppress ghost factors of N scales as . Unfortunately, this advantageous scaling law is accompanied by a disadvantage: the gap between factors and non-factors decreases rapidly with increasing power j and as a consequence it gets more difficult to identify factors. This feature serves as our motivation to study sums with an exponential phase. Our numerical simulations indicate that in this case the scaling law is logarithmic and that we retain a significant gap between factors and non-factors.
Recent experiments have shown that truncated Gauss sums allow us to find the factors of an integer N . This method relies on the fact that for a factor the absolute value of the Gauss sum is unity. However, for every integer N there exist integers which are not factors, but where the Gauss sum reaches a value which is arbitrarily close to unity. In order to distinguish such ghost factors from real factors we need to amplify this difference. We show that a proper choice of the truncation parameter of the Gauss sum suppresses the ghost factors below a threshold value. We derive the scaling law of the truncation parameter on the number to be factored. Moreover, we show that this scaling law is also necessary for the success of our factorization scheme, even if we relax the threshold or allow limited error tolerance.
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. Wölk et al. also published in this issue [Wölk, 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.
Finding the factors of an integer can be achieved by various experimental techniques, based on an algorithm developed by Schleich et al . ( Fortschr. Phys. , 54 (2006) 856), which uses specific properties of Gauß sums. Experimental limitations usually require truncation of these series, but if the truncation parameter is too small, it is no longer possible to distinguish between factors and so-called “ghost" factors. Here, we discuss two techniques for distinguishing between true factors and ghost factors while keeping the number of terms in the sum constant or only slowly increasing. We experimentally test these modified algorithms in a nuclear spin system, using NMR.
We report on a new implementation of the factorisation of numbers using Gauss sums which improves tremendously the efficiency to eliminate all "ghost" factors. We show that by choosing randomly the terms in the Gauss sum, the required number of terms varies as ln N instead of . As an illustration, we present experimental results obtained by interfering thirty ultrashort laser pulses where we factorise 1340333404807. This new approach is totally general and can be implemented for all the experiments based on the Gauss sum.