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Non-abelian analogs of lattice rounding

Non-abelian analogs of lattice rounding 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

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Publisher
de Gruyter
Copyright
Copyright © 2015 by the
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/gcc-2015-0010
Publisher site
See Article on Publisher Site

Abstract

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.

Journal

Groups Complexity Cryptologyde Gruyter

Published: Nov 1, 2015

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