Computational and Combinatorial Group Theory and Cryptography: AMS Special Sessions: Computational Algebra, Groups, and Applications, April 30 - May ... Mathematical Aspec

210 Pages·2012·1.4 MB·English

by  Benjamin Fine

582
Computational and Combinatorial
Group Theory and Cryptography
AMS Special Sessions:
Computational Algebra, Groups, and Applications
April 30–May 1, 2011
University of Nevada, Las Vegas, NV
Mathematical Aspects of Cryptography and Cyber Security
September 10–11, 2011
Cornell University, Ithaca, NY
Benjamin Fine
Delaram Kahrobaei
Gerhard Rosenberger
Editors
AmericanMathematicalSociety
Computational and Combinatorial
Group Theory and Cryptography
AMS Special Sessions:
Computational Algebra, Groups, and Applications
April 30–May 1, 2011
University of Nevada, Las Vegas, NV
Mathematical Aspects of Cryptography and Cyber Security
September 10–11, 2011
Cornell University, Ithaca, NY
Benjamin Fine
Delaram Kahrobaei
Gerhard Rosenberger
Editors

582
Computational and Combinatorial
Group Theory and Cryptography
AMS Special Sessions:
Computational Algebra, Groups, and Applications
April 30–May 1, 2011
University of Nevada, Las Vegas, NV
Mathematical Aspects of Cryptography and Cyber Security
September 10–11, 2011
Cornell University, Ithaca, NY
Benjamin Fine
Delaram Kahrobaei
Gerhard Rosenberger
Editors
AmericanMathematicalSociety
Providence,RhodeIsland
EDITORIAL COMMITTEE
Dennis DeTurck, Managing Editor
Michael Loss Kailash Misra Martin J. Strauss
2000 Mathematics Subject Classification. Primary 20-XX, 68-XX.
Library of Congress Cataloging-in-Publication Data
Computational and combinatorial group theory and cryptography : AMS Special Session on
ComputationalAlgebra,Groups,andApplications,April30–May1,2011,UniversityofNevada,
LasVegas,Nevada: AMSSpecialSessiononMathematicalAspectsofCryptographyandCyber
Security,September10–11,2011,CornellUniversity,Ithaca,NewYork/BenjaminFine,Delaram
Kahrobaei,GerhardRosenberger,editors.
p.cm. –(Contemporarymathematics;volume582)
Includesbibliographicalreferences.
ISBN978-0-8218-7563-6(alk.paper)
1. Group theory–Congresses. 2. Cryptography–Congresses. I. Fine, Benjamin (1948–),
editorofcompilation. II.Kahrobaei,Delaram(1975–),editorofcompilation. III.Rosenberger,
Gerhard, editor of compilation. IV. AMS Special Session on Computational Algebra, Groups,
and Applications (2011 : University of Nevada). V. AMS Special Session on Mathematical
AspectsofCryptographyandCyberSecurity(2011: CornellUniversity)
QA174.C645 2012
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Contents
Preface vii
Weyl Gr¨obner Basis Cryptosystems
Rashid Ali and Martin Kreuzer 1
A New Look at Finitely Generated Metabelian Groups
Gilbert Baumslag, Roman Mikhailov, and Kent E. Orr 21
IA-Automorphisms of Groups with Almost Constant Upper Central Series
Marianna Bonanome, Margaret H. Dean, and Marcos
Zyman 39
A Proposed Alternative to the Shamir Secret Sharing Scheme
Chi Sing Chum, Benjamin Fine, Gerhard Rosenberger,
and Xiaowen Zhang 47
Improving Latin Square Based Secret Sharing Schemes
Chi Sing Chum and Xiaowen Zhang 51
A Hand-Computation Involving Surface Groups, the Reidemeister-Schreier
Rewriting Process and Kurosh Subgroup Theorem
Anthony E. Clement 65
Adjunction of Roots in Exponential A-Groups
Margaret H. Dean, Stephen Majewicz, and Marcos Zyman 71
Logspace Computations in Coxeter Groups and Graph Groups
Volker Diekert, Jonathan Kausch, and Markus Lohrey 77
Collection by Polynomials in Finite p-groups
Bettina Eick 95
All Finite Generalized Tetrahedron Groups II
Benjamin Fine, Alexander Hulpke, and Gerhard
Rosenberger 105
The Classification of One Relator Limit Groups and the Surface Group
Conjecture
Benjamin Fine and Gerhard Rosenberger 107
Discrimination and Separation in the Metabelian Variety
Anthony M. Gaglione, Seymour Lipschutz, and Dennis
Spellman 129
v
vi CONTENTS
A Secret Sharing Scheme Based on Group Presentations and the Word Problem
Maggie Habeeb, Delaram Kahrobaei, and Vladimir
Shpilrain 143
Authenticated Key Agreement with Key Re-Use in the Short Authenticated
Strings Model
Stanislaw Jarecki and Nitesh Saxena 151
Publicly Verifiable Secret Sharing Using Non-Abelian Groups
Delaram Kahrobaei and Elizabeth Vidaurre 175
A Note on the Hyperbolicity of Strict Pride Groups
Matthias Neumann-Brosig 181
An Algorithm to Express Words as a Product of Conjugates of Relators
Ellen Ziliak 187
Preface
This volume consists of contributions by participants and speakers at special
sessionsattwoAMSConferences. TheseconferencesconcernedComputationaland
Combinatorial Group Theory as well as various aspects of Group-Based Cryptog-
raphy. One of these sessions was at Cornell University in September of 2011 and
the other was the University of Nevada, Las Vegas in May of 2011.
Over the past twenty yearsCombinatorial and Infinite GroupTheory hasbeen
energized by three developments: the emergence of Geometric and Asymptotic
Group Theory, the development of Algebraic Geometry over Groups leading to the
solution of the Tarski problems, and the development of Group-Based Cryptogra-
phy. These three areas in turn have had an impact on computational algebra and
complexity theory. The papers in this volume, both survey and research, exhibit
the tremendous vitality that is at the heart of Group Theory in the beginning of
the twenty-first century as well as the diversity of interests in the field.
We are grateful to the American Mathematical Society for their help in publi-
cation of this volume. In particular, we thank Christine Thivierge for her patience
and diligence in assembling this volume.
Bejamin Fine
Delaram Kahrobaei
Gerhard Rosenberger
vii

ContemporaryMathematics
Volume582,2012
http://dx.doi.org/10.1090/conm/582/11554
Weyl Gro¨bner Basis Cryptosystems
Rashid Ali and Martin Kreuzer
Abstract. The main purpose of this paper is to propose ways of construct-
ing Gro¨bner basis cryptosystems based on left ideals or two-sided ideals in
Weyl algebras. The main advantage of using Weyl algebras over commuta-
tivepolynomialringsisthatWeylmultiplicationincreasesthesupportofthe
standardformofthepolynomials,slowsdowntheGr¨obnerbasiscomputation,
and changes many coefficients in the ciphertext polynomials. Special care is
takentoprovideconcretedirectionsforconstructingsecureinstancesofWeyl
Gr¨obner basis cryptosystems (WGBC). Their security and possible defences
against chosenciphertext attacks, linear algebraattacks and partial Gr¨obner
basis attacks are analysed. Based on these guidelines, we construct several
actual instances of left and two-sided WGBC of which we believe that they
arereasonablysecureandefficient.
1. Introduction
In1994,M.FellowsandN.Koblitzintroducedin[7]anewtypeofcryptosystem
which became known as Polly Cracker cryptosystem and was based on (commu-
tative) polynomials and the difficulty of computing Gro¨bner bases for polynomial
ideals. Although these cryptosystems could, in principle, encode NP-hard prob-
lems, it turned out to be very difficult to construct hard instances. Many methods
of attacking Polly Cracker cryptosystems were proposed and carried out success-
fully. To counter these attacks, it was suggested in [22] and [1] to use Gro¨bner
basesfornon-commutative algebraicstructures. Inthispaperweconstructspecific
instancesofthegeneralGro¨bnerbasiscryptosystemdefinedin[1]. WeuseGro¨bner
bases for left and two-sided ideals in Weyl algebras. We analyse the resistance of
these cryptosystems to some standard attacks and provide computational evidence
that secure instances can be built using left and two-sided Gro¨bner bases in Weyl
algebras.
Recall that the Weyl algebra A of index n over a field K is the associa-
n
tive algebra A = K(cid:3)x ,...,x ,∂ ,...,∂ (cid:4) such that [x ,x ] = [∂ ,∂ ] = 0 and
n 1 n 1 n i j i j
[∂ ,x ] = δ . For a variety of reasons, it appears beneficial to use a finite base
i j ij
field K. Since Weyl algebras in characteristic p have properties which differ from
the well-known case char(K) = 0, we start in Section 2 by recalling them (see
2010MathematicsSubjectClassification. Primary94A60;Secondary14G15,16Z05,68W30.
Keywordsandphrases. Algebraiccryptography,Groebnerbasiscryptosystem,Weylalgebra,
non-commutativecryptosystem.
ThefirstauthorwassupportedbytheHEC-DAADFellowshipProgramme.
(cid:3)c2012 American Mathematical Society
1