Complexity of short generating functions

TL;DR

This paper analyzes the computational complexity of short generating functions, showing that certain operations can significantly increase their complexity and that some functions are inherently hard within this class.

Contribution

It provides the first complexity analysis of short generating functions, demonstrating their limitations under common operations and establishing hardness results for specific functions.

Findings

Operations like intersections, unions, projections increase GF complexity super-polynomially

Short generating functions are not closed under these operations

Truncated theta functions are proven to be computationally hard within this class

Abstract

We give complexity analysis of the class of short generating functions (GF). Assuming $\#P \not\subseteq FP/poly$, we show that this class is not closed under taking many intersections, unions or projections of GFs, in the sense that these operations can increase the bitlength of coefficients of GFs by a super-polynomial factor. We also prove that truncated theta functions are hard in this class.