1. Counting Co-Cyclic Lattices
- Author
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Phong Q. Nguyen, Igor E. Shparlinski, Institute for Advanced Study [Tsinghua], Tsinghua University [Beijing] (THU), Cryptanalyse (CRYPT), Laboratoire Franco-Chinois d'Informatique, d'Automatique et de Mathématiques Appliquées (LIAMA), Centre de Coopération Internationale en Recherche Agronomique pour le Développement (Cirad)-Institut National de la Recherche Agronomique (INRA)-Chinese Academy of Sciences [Changchun Branch] (CAS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institute of Automation - Chinese Academy of Sciences-Centre National de la Recherche Scientifique (CNRS)-Centre de Coopération Internationale en Recherche Agronomique pour le Développement (Cirad)-Institut National de la Recherche Agronomique (INRA)-Chinese Academy of Sciences [Changchun Branch] (CAS)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institute of Automation - Chinese Academy of Sciences-Centre National de la Recherche Scientifique (CNRS)-Inria Paris-Rocquencourt, Institut National de Recherche en Informatique et en Automatique (Inria), University of New South Wales [Sydney] (UNSW), Tsinghua University [Beijing], Inria de Paris, Japanese French Laboratory for Informatics (JFLI), and National Institute of Informatics (NII)-Université Pierre et Marie Curie - Paris 6 (UPMC)-The University of Tokyo (UTokyo)-Centre National de la Recherche Scientifique (CNRS)
- Subjects
FOS: Computer and information sciences ,Discrete Mathematics (cs.DM) ,High Energy Physics::Lattice ,General Mathematics ,0102 computer and information sciences ,01 natural sciences ,Combinatorics ,[INFO.INFO-CR]Computer Science [cs]/Cryptography and Security [cs.CR] ,Integer ,FOS: Mathematics ,Natural density ,Asymptotic formula ,Number Theory (math.NT) ,[MATH]Mathematics [math] ,Mathematics ,Discrete mathematics ,homogeneous congruences ,Mathematics - Number Theory ,cyclic lattices ,Group (mathematics) ,Lattice problem ,Order (ring theory) ,[MATH.MATH-NT]Mathematics [math]/Number Theory [math.NT] ,multiplicative functions ,010201 computation theory & mathematics ,Computer Science - Discrete Mathematics - Abstract
International audience; There is a well-known asymptotic formula, due to W. M. Schmidt [Duke Math. J., 35 (1968), pp. 327--339], 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 $(\zeta(6) \prod_{k=4}^n \zeta(k))^{-1} \approx 85\%$. The problem is motivated by complexity theory, namely worst-case to average-case reductions for lattice problems.
- Published
- 2016
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