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A PTIME solution to the restricted conjugacy problem in generalized Heisenberg groups

A PTIME solution to the restricted conjugacy problem in generalized Heisenberg groups Abstract We examine the Anshel–Anshel–Goldfeld key exchange protocol with a generalized Heisenberg group, H m , as a platform. We show that subgroup-restricted simultaneous conjugacy search problem in H m can be solved in quasi-quintic time, which allows the computation of the private keys of the parties. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Groups Complexity Cryptology de Gruyter

A PTIME solution to the restricted conjugacy problem in generalized Heisenberg groups

A PTIME solution to the restricted conjugacy problem in generalized Heisenberg groups


We examine the Anshel­Anshel­Goldfeld key exchange protocol with a generalized Heisenberg group, H m , as a platform. We show that subgroup-restricted simultaneous conjugacy search problem in H m can be solved in quasi-quintic time, which allows the computation of the private keys of the parties. Keywords: Anshel­Anshel­Goldfeld protocol, key establishment, conjugacy problem, generalized Heisenberg group MSC 2010: 20F10, 20F18, 94A60 1 Introduction The Anshel­Anshel­Goldfeld (AAG) key establishment protocol was first introduced in 1999 [1] as a robust system for exchanging private encryption keys through public channels. An implementation of AAG uses a predetermined platform group to exchange this information between two parties. This choice of platform group has a major impact on the security of the key exchange. The security of AAG under some selections of platform groups is known. Some examples include braid groups by Ko, Lee, Cheon, Han, Kang and Park [6], Thompson's group by Shpilrain and Ushakov [13] and polycyclic groups by Eick and Kahrobaei [3]. There have been a number of attack methods applied to AAG [4, 7, 10]. An attack method which has seen some success in braid groups and the Thompson group is the length based attack (LBA) [11, 12]. Tests from Kahrobaei and Lam have suggested for a given length function that AAG with a generalized Heisenberg group as a platform is resistant to LBA [5]. We determine here that the computational complexity of computing the private key from public information in this case is polynomial time despite this resistance. In this paper we first examine AAG in Section 2 and the major algebraic questions involved in breaking AAG. In Section 3 we state the properties of generalized Heisenberg groups. Next, in Section 4 we lay out the foundation for our centralizer attack and then examine it piece by piece. Finally, in Section 5 we explicitly state our main theorem and suggest qualities for platform...
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Publisher
de Gruyter
Copyright
Copyright © 2016 by the
ISSN
1867-1144
eISSN
1869-6104
DOI
10.1515/gcc-2016-0003
Publisher site
See Article on Publisher Site

Abstract

Abstract We examine the Anshel–Anshel–Goldfeld key exchange protocol with a generalized Heisenberg group, H m , as a platform. We show that subgroup-restricted simultaneous conjugacy search problem in H m can be solved in quasi-quintic time, which allows the computation of the private keys of the parties.

Journal

Groups Complexity Cryptologyde Gruyter

Published: May 1, 2016

References