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Counting Co-Cyclic Lattices

Phong Q. Nguyen 1, 2 Igor E. Shparlinski 3 
2 CRYPT - Cryptanalyse
LIAMA - Laboratoire Franco-Chinois d'Informatique, d'Automatique et de Mathématiques Appliquées, Inria Paris-Rocquencourt
Abstract : 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.
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Preprints, Working Papers, ...
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Submitted on : Thursday, January 14, 2016 - 11:45:04 AM
Last modification on : Friday, January 21, 2022 - 3:15:40 AM

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  • HAL Id : hal-01256022, version 1
  • ARXIV : 1505.06429



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



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