Counting Co-Cyclic Lattices - Inria - Institut national de recherche en sciences et technologies du numérique Access content directly
Preprints, Working Papers, ... Year : 2015

Counting Co-Cyclic Lattices


There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the number of full-rank integer lattices of index at most $V$ in $\mathbb{Z}^n$. This set of lattices $L$ can naturally be partitioned with respect to the factor group $\mathbb{Z}^n/L$. Accordingly, we count the number of full-rank integer lattices $L \subseteq \mathbb{Z}^n$ such that $\mathbb{Z}^n/L$ is cyclic and of order at most $V$, and deduce that these co-cyclic lattices are dominant among all integer lattices: their natural density is $\left(\zeta(6) \prod_{k=4}^n \zeta(k)\right)^{-1} \approx 85\%$. The problem is motivated by complexity theory, namely worst-case to average-case reductions for lattice problems.

Dates and versions

hal-01256022 , version 1 (14-01-2016)



Phong Q. Nguyen, Igor E. Shparlinski. Counting Co-Cyclic Lattices. 2015. ⟨hal-01256022⟩
176 View
0 Download



Gmail Facebook X LinkedIn More