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Group factorizations in cryptography

Charles Anthony Cusack, University of Nebraska - Lincoln

Abstract

Logarithmic signatures of permutation groups and their applications in private-key cryptography have been studied since the early 1980s. More recently, Magliveras, Stinson, and Trung have done some preliminary work in creating two new public-key cryptosystems, MST1, based on logarithmic signatures, and MST2, based on another type of group coverings called (r, s)-meshes. ^ In this thesis, we discuss implementation and security issues relating to both cryptosystems. Our discussion of MST2 is rudimentary. We give an elementary proof that factoring with respect to an (s, r)-mesh is at least as hard as the discrete logarithm problem, and discuss what is known about the subgroup intersection problem and its relationship to MST2. ^ The bulk of the thesis is devoted to logarithmic signatures and MST1. In order to implement MST1, we need two distinct types of logarithmic signatures; those for which factorization is easy, and those for which it is hard. In addition, we need a method of constructing hard-to-factor logarithmic signatures from easy-to-factor logarithmic signatures, so that the former can serve as public keys, and the latter as private keys. ^ We provide a thorough analysis of several transformations that can be performed on logarithmic signatures. Every logarithmic signature of a group induces a permutation on SG. Furthermore, a consequence of the analysis of transformations in this thesis is the discovery that the set of permutations resulting from several classes of logarithmic signatures of a group is the union of cosets of several different groups. ^ We show that a class of logarithmic signatures called transversal are easy to factor, and define and discuss the class of permutably transversal logarithmic signatures, which may help provide the trap-door needed for MST1. ^ We define the class of canonical logarithmic signatures. We show that the permutations generated by logarithmic signatures are generated by just the canonical logarithmic signatures, and give bounds on the number of logarithmic signatures and induced permutations of certain classes. ^

Subject Area

Mathematics|Computer Science

Recommended Citation

Cusack, Charles Anthony, "Group factorizations in cryptography" (2000). ETD collection for University of Nebraska - Lincoln. AAI9991985.
http://digitalcommons.unl.edu/dissertations/AAI9991985

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