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Groups Complexity Cryptology

Managing Editor: Shpilrain, Vladimir / Weil, Pascal

Editorial Board: Ciobanu, Laura / Conder, Marston / Eick, Bettina / Elder, Murray / Fine, Benjamin / Gilman, Robert / Grigoriev, Dima / Ko, Ki Hyoung / Kreuzer, Martin / Mikhalev, Alexander V. / Myasnikov, Alexei / Perret, Ludovic / Roman'kov, Vitalii / Rosenberger, Gerhard / Sapir, Mark / Thomas, Rick / Tsaban, Boaz / Capell, Enric Ventura / Lohrey, Markus

CiteScore 2018: 0.80

SCImago Journal Rank (SJR) 2018: 0.368
Source Normalized Impact per Paper (SNIP) 2018: 1.061

Mathematical Citation Quotient (MCQ) 2017: 0.32

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Non-abelian analogs of lattice rounding

Evgeni Begelfor / Stephen D. Miller
  • Corresponding author
  • Department of Mathematics, Rutgers University, 110 Frelinghuysen Rd, Piscataway, NJ 08854-8019, USA
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/ Ramarathnam Venkatesan
Published Online: 2015-10-15 | DOI: https://doi.org/10.1515/gcc-2015-0010


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.

Keywords: Lattice rounding; matrix groups; norm concentration; Lyapunov exponents; word problems; inapproximability

MSC: 52C07; 37H15; 94A60; 68Q25

About the article

Received: 2015-01-18

Published Online: 2015-10-15

Published in Print: 2015-11-01

Funding Source: NSF

Award identifier / Grant number: DMS-1201362

Citation Information: Groups Complexity Cryptology, Volume 7, Issue 2, Pages 117–133, ISSN (Online) 1869-6104, ISSN (Print) 1867-1144, DOI: https://doi.org/10.1515/gcc-2015-0010.

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