## Abstract

Lattice rounding in Euclidean space can be viewed as finding the nearest point in the orbit of an action by a discrete group, relative to the norm inherited from the ambient space. Using this point of view, we initiate the study of non-abelian analogs of lattice rounding involving matrix groups. In one direction, we consider an algorithm for solving a normed word problem when the inputs are random products over a basis set, and give theoretical justification for its success. In another direction, we prove a general inapproximability result which essentially rules out strong approximation algorithms (i.e., whose approximation factors depend only on dimension) analogous to LLL in the general case.

Original language | English (US) |
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Pages (from-to) | 117-133 |

Number of pages | 17 |

Journal | Groups, Complexity, Cryptology |

Volume | 7 |

Issue number | 2 |

DOIs | |

State | Published - Nov 1 2015 |

## All Science Journal Classification (ASJC) codes

- Computer Networks and Communications
- Computational Theory and Mathematics
- Computational Mathematics
- Applied Mathematics

## Keywords

- Lattice rounding
- Lyapunov exponents
- inapproximability
- matrix groups
- norm concentration
- word problems