Non-abelian analogs of lattice rounding
-
Evgeni Begelfor
and
Ramarathnam Venkatesan
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.
Funding source: NSF
Award Identifier / Grant number: DMS-1201362
We would like to thank Anthony Bloch, Hillel Fürstenberg, Nathan Keller, Peter Sarnak, Adi Shamir, Boaz Tsaban, and Akshay Venkatesh for their helpful comments.
© 2015 by De Gruyter
Articles in the same Issue
- Frontmatter
- A combinatorial algorithm to compute presentations of mapping class groups of orientable surfaces with one boundary component
- Non-abelian analogs of lattice rounding
- Tree-based language complexity of Thompson's group F
- New probabilistic public-key encryption based on the RSA cryptosystem
- Algorithmic recognition of quasipositive 4-braids of algebraic length three
- Cryptanalysis of a system using matrices over group rings
- On transitive differentiable modulo pn functions
- On the generic complexity of the searching graph isomorphism problem
- Key agreement under tropical parallels
Articles in the same Issue
- Frontmatter
- A combinatorial algorithm to compute presentations of mapping class groups of orientable surfaces with one boundary component
- Non-abelian analogs of lattice rounding
- Tree-based language complexity of Thompson's group F
- New probabilistic public-key encryption based on the RSA cryptosystem
- Algorithmic recognition of quasipositive 4-braids of algebraic length three
- Cryptanalysis of a system using matrices over group rings
- On transitive differentiable modulo pn functions
- On the generic complexity of the searching graph isomorphism problem
- Key agreement under tropical parallels
