Cryptographic systems using linear groups.
Item
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Title
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Cryptographic systems using linear groups.
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Identifier
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AAI3204959
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identifier
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3204959
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Creator
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Xu, Xiaowei.
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Contributor
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Adviser: Gilbert Baumslag
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Date
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2006
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Language
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English
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Publisher
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City University of New York.
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Subject
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Computer Science
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Abstract
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The objective of this thesis is to propose and discuss some new public-key encoding and zero-knowledge protocols based on matrix groups over polynomial rings. One of the key aspects of these protocols is the use of a rewriting system, known as the method of Reidemeister and Schreier, which is common in group theory but not used until now in cryptography. In this thesis, we will explore how to use groups of matrices over polynomial rings in cryptography. This will involve solving various group-theoretic problems and the difficulty of solving various polynomial equations. This will allow us to propose various new public-key encoding and zero-knowledge protocols.;Instead of using protocols based on number theory, we will devise protocols based on groups of matrices, whose entries consist of polynomials in a number of variables. The groups involved will be free groups and we use the free generators as letters in an alphabet which permit the encoding of messages in terms of matrices. We then transform these matrices into 2 x 2 integral matrices of determinant 1. The Euclidean algorithm and the method of Reidemeister-Schreier alluded to above, then enable us to decode the given messages.;We describe two cryptography schemes based on combinatorial group theory. We will discuss the underlying theory and provide some explicit examples. We suggest that combinatorial group theory could be the next generation platform in the development of cryptography because it is more general and more flexible to take advantage of dynamic algebraic structures instead of static arithmetic. Our various schemes are based on more than one secret element and breaking one doesn't break the whole scheme.
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Type
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dissertation
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Source
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PQT Legacy CUNY.xlsx
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degree
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Ph.D.
