Conjugacy in Garside Groups III: Periodic braids

Journal of Algebra (Impact Factor: 0.6). 10/2006; DOI: 10.1016/j.jalgebra.2007.02.002
Source: arXiv

ABSTRACT An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B_n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms. This paper is the third in a series of papers by the same authors about the conjugacy problem in Garside groups. They have a unified goal: the development of a polynomial algorithm for the conjugacy decision and search problems in B_n, which generalizes to other Garside groups whenever possible. It is our hope that the methods introduced here will allow the generalization of the results in this paper to all Artin-Tits groups of spherical type.

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    ABSTRACT: In the last decade, a number of public key cryptosys-tems based on combinatorial group theoretic problems in braid groups have been proposed. Our tutorial is aimed at presenting these cryptosystems and some known attacks on them. We start with some basic facts on braid groups and on the Gar-side normal form of its elements. We then present some known algorithms for solving the word problem in the braid group. After that, we present the major public-key cryptosystems based on the braid group. We then discuss some of the known attacks on these cryptosystems. We finish with a discussion of future directions.
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    ABSTRACT: We propose a new public-key cryptosystem named conjugacy search problem-based Diffie–Hellman integrated encryption scheme (CSP-DHIES), by using conjugation-related assumptions for a special monoid of matrices of truncated multi-variable polynomials over the ring ℤ12 where the CSP is assumed to be intractable. Our construction can be viewed as the first noncommunicative variant of the well-known DHIES cryptosystem. Under the assumptions of the intractability of the CSP-based hash Diffie–Hellman problem and the CSP-based oracle Diffie–Hellman problem, our scheme is provably secure against both chosen-plaintext attacks and secure against chosen-ciphertext attacks. Our proofs are constructed in the standard model. We also discuss the possibility of implementing our proposal using braid groups. Copyright © 2011 John Wiley & Sons, Ltd.
    Security and Communication Networks 10/2011; 5(7):809 - 822. · 0.43 Impact Factor
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