Factorization with Exponential Sums

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.