The behavior of random reduced bases

TL;DR

This paper demonstrates that for high-dimensional lattices, the number of Siegel-reduced bases is highly concentrated around its mean, and most reduced bases resemble worst-case scenarios, with implications for lattice reduction outputs.

Contribution

It provides a rigorous concentration result for the count of Siegel-reduced bases and links typical reduced bases to worst-case behavior, using spectral theory and assumptions related to the Riemann hypothesis.

Findings

Number of Siegel-reduced bases concentrates around the mean as dimension grows

Most reduced bases resemble worst-case lattice reduction scenarios

Results suggest reduced bases are rarely outputs of typical lattice reduction algorithms

Abstract

We prove that the number of Siegel-reduced bases for a randomly chosen $n$-dimensional lattice becomes, for $n \rightarrow \infty$, tightly concentrated around its mean. We also show that most reduced bases behave as in the worst-case analysis of lattice reduction. Comparing with experiment, these results suggest that most reduced bases will, in fact, "very rarely" occur as an output of lattice reduction. The concentration result is based on an analysis of the spectral theory of Eisenstein series and uses (probably in a removable way) the Riemann hypothesis.