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discrete logarithm problem in non-abelian groups

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Date Issued:
2010
Summary:
This dissertation contains results of the candidate's research on the generalized discrete logarithm problem (GDLP) and its applications to cryptology, in non-abelian groups. The projective special linear groups PSL(2; p), where p is a prime, represented by matrices over the eld of order p, are investigated as potential candidates for implementation of the GDLP. Our results show that the GDLP with respect to specic pairs of PSL(2; p) generators is weak. In such cases the groups PSL(2; p) are not good candidates for cryptographic applications which rely on the hardness of the GDLP. Results are presented on generalizing existing cryptographic primitives and protocols based on the hardness of the GDLP in non-abelian groups. A special instance of a cryptographic primitive dened over the groups SL(2; 2n), the Tillich-Zemor hash function, has been cryptanalyzed. In particular, an algorithm for constructing collisions of short length for any input parameter is presented. A series of mathematical results are developed to support the algorithm and to prove existence of short collisions.
Title: The discrete logarithm problem in non-abelian groups.
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Name(s): Iliâc, Ivana.
Charles E. Schmidt College of Science
Department of Mathematical Sciences
Type of Resource: text
Genre: Electronic Thesis Or Dissertation
Date Issued: 2010
Publisher: Florida Atlantic University
Physical Form: electronic
Extent: vii, 94 p. : ill. (some col.)
Language(s): English
Summary: This dissertation contains results of the candidate's research on the generalized discrete logarithm problem (GDLP) and its applications to cryptology, in non-abelian groups. The projective special linear groups PSL(2; p), where p is a prime, represented by matrices over the eld of order p, are investigated as potential candidates for implementation of the GDLP. Our results show that the GDLP with respect to specic pairs of PSL(2; p) generators is weak. In such cases the groups PSL(2; p) are not good candidates for cryptographic applications which rely on the hardness of the GDLP. Results are presented on generalizing existing cryptographic primitives and protocols based on the hardness of the GDLP in non-abelian groups. A special instance of a cryptographic primitive dened over the groups SL(2; 2n), the Tillich-Zemor hash function, has been cryptanalyzed. In particular, an algorithm for constructing collisions of short length for any input parameter is presented. A series of mathematical results are developed to support the algorithm and to prove existence of short collisions.
Identifier: 702127124 (oclc), 3356783 (digitool), FADT3356783 (IID), fau:3978 (fedora)
Note(s): by Ivana Iliâc.
Thesis (Ph.D.)--Florida Atlantic University, 2010.
Includes bibliography.
Electronic reproduction. Boca Raton, Fla., 2010. Mode of access: World Wide Web. FboU
Subject(s): Data encryption (Computer science)
Computer security
Cryptography
Combinatorial group theory -- Data processing
Mapping (Mathematics)
Persistent Link to This Record: http://purl.flvc.org/FAU/3356783
Use and Reproduction: http://rightsstatements.org/vocab/InC/1.0/
Host Institution: FAU