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arXiv:1210.6474 · New Journal of Physics

Factorization of numbers with Gauss sums: I. Mathematical background

Sabine Wölk, W. Merkel, Wolfgang P. Schleich, Ilya Sh. Averbukh, B. Girard

Published Oct 7, 2011• 23 citations
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Abstract

We use the periodicity properties of generalized Gauss sums to factor numbers. Moreover, we derive rules for finding the factors and illustrate this factorization scheme for various examples. This algorithm relies solely on interference and scales exponentially.
DOI
10.1088/1367-2630/13/10/103007

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Comment on "NMR Experiment Factors Numbers with Gauss Sums"

Julia A. Wei Xiaohua Jones

Mehring et al. have recently described an elegant nuclear magnetic resonance (NMR) experiment implementing an algorithm to factor numbers based on the properties of Gauss sums. Similar experiments have also been described by Mahesh et al. In fact these algorithms do not factor numbers directly, but rather check whether a trial integer $\ell$ is a factor of a given integer $N$. Here I show that these NMR schemes cannot be used for factor checking without first implicitly determining whether or not $\ell$ is a factor of $N$.

arXiv:0704.2065•Apr 16, 2007•2 citations
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.
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.
Mehring et al. have recently described an elegant nuclear magnetic resonance (NMR) experiment implementing an algorithm to factor numbers based on the properties of Gauss sums. Similar experiments have also been described by Mahesh et al. In fact these algorithms do not factor numbers directly, but rather check whether a trial integer ℓ\ellℓ is a factor of a given integer NNN. Here I show that these NMR schemes cannot be used for factor checking without first implicitly determining whether or not ℓ\ellℓ is a factor of NNN.