Dissertation/Thesis Abstract

Orthomorphisms of Boolean Groups
by Schimanski, Nichole Louise, Ph.D., Portland State University, 2016, 99; 10154830
Abstract (Summary)

An orthomorphism, π, of a group, (G, +), is a permutation of G with the property that the map &khgr; → −&khgr; + (&khgr;) is also a permutation. In this paper, we consider orthomorphisms of the additive group of binary n-tuples, Zn 2. We use known orthomorphism preserving functions to prove a uniformity in the cycle types of orthomorphisms that extend certain partial orthomorphisms, and prove that extensions of particular sizes of partial orthomorphisms exist. Further, in studying the action of conjugating orthomorphisms by automorphisms, we find several symmetries within the orbits and stabilizers of this action, and other orthomorphism-preserving functions. In addition, we prove a lower bound on the number of orthomorphisms of Zn2 using the equivalence of orthomorphisms to transversals in Latin squares. Lastly, we present a Monte Carlo method for generating orthomorphisms and discuss the results of the implementation.

Indexing (document details)
Advisor: Caughman, John
Commitee: Garton, Derek, Massey, Bart, O'Halloran, Joyce, Shrimpton, Thomas
School: Portland State University
Department: Mathematics
School Location: United States -- Oregon
Source: DAI-B 78/01(E), Dissertation Abstracts International
Subjects: Mathematics
Keywords: Cryptography, Davies-meyer mode, Metropolis-hastings, Orthomorphism, Permutation, Transversal
Publication Number: 10154830
ISBN: 978-1-369-09922-5
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