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.