Paper 2022/861

Genus Distribution of Random q-ary Lattices

Peter J. Bruin, Leiden University
Léo Ducas, Centrum Wiskunde & Informatica, Leiden University
Shane Gibbons, Leiden University, Centrum Wiskunde & Informatica

The genus is an efficiently computable arithmetic invariant for lattices up to isomorphism. Given the recent proposals of basing cryptography on the lattice isomorphism problem, it is of cryptographic interest to classify relevant families of lattices according to their genus. We propose such a classification for q-ary lattices, and also study their distribution. In particular, for an odd prime q, we show that random q-ary lattices are mostly concentrated on two genera. Because the genus is local, this also provides information on the distribution for general odd q. The case of q a power of 2 is also studied, although we only achieve a partial classification.

Available format(s)
Public-key cryptography
Publication info
Lattice Genus Quadratic Form Distribution Lattice Isomorphism Problem
Contact author(s)
p j bruin @ math leidenuniv nl
leo ducas @ cwi nl
shane gibbons @ cwi nl
2022-07-01: approved
2022-07-01: received
See all versions
Short URL
Creative Commons Attribution


      author = {Peter J. Bruin and Léo Ducas and Shane Gibbons},
      title = {Genus Distribution of Random q-ary Lattices},
      howpublished = {Cryptology ePrint Archive, Paper 2022/861},
      year = {2022},
      note = {\url{}},
      url = {}
Note: In order to protect the privacy of readers, does not use cookies or embedded third party content.