Current Search: Combinatorial designs and configurations (x)
View All Items
 Title
 An Algorithmic Approach to Tran Van Trung's Basic Recursive Construction of tDesigns.
 Creator
 Lopez, Oscar A., Magliveras, Spyros S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

It was not until the 20th century that combinatorial design theory was studied as a formal subject. This field has many applications, for example in statistical experimental design, coding theory, authentication codes, and cryptography. Major approaches to the problem of discovering new tdesigns rely on (i) the construction of large sets of t designs, (ii) using prescribed automorphism groups, (iii) recursive construction methods. In 2017 and 2018, Tran Van Trung introduced new recursive...
Show moreIt was not until the 20th century that combinatorial design theory was studied as a formal subject. This field has many applications, for example in statistical experimental design, coding theory, authentication codes, and cryptography. Major approaches to the problem of discovering new tdesigns rely on (i) the construction of large sets of t designs, (ii) using prescribed automorphism groups, (iii) recursive construction methods. In 2017 and 2018, Tran Van Trung introduced new recursive techniques to construct t – (v, k, λ) designs. These methods are of purely combinatorial nature and require using "ingredient" tdesigns or resolutions whose parameters satisfy a system of nonlinear equations. Even after restricting the range of parameters in this new method, the task is computationally intractable. In this work, we enhance Tran Van Trung's "Basic Construction" by a robust and efficient hybrid computational apparatus which enables us to construct hundreds of thousands of new t – (v, k, Λ) designs from previously known ingredient designs. Towards the end of the dissertation we also create a new family of 2resolutions, which will be infinite if there are infinitely many Sophie Germain primes.
Show less  Date Issued
 2019
 PURL
 http://purl.flvc.org/fau/fd/FA00013233
 Subject Headings
 Combinatorial designs and configurations, Algorithms, tdesigns
 Format
 Document (PDF)
 Title
 On projected planes.
 Creator
 Caliskan, Cafer., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

This work was motivated by the wellknown question: "Does there exist a nondesarguesian projective plane of prime order?" For a prime p 1, determine all subplanes of order p up to collineations, and check whether one of these is nondesarguesian." In this manuscript we use a grouptheoretic methodology to determine the subplane structures of some nondesarguesian planes. In particular, we determine orbit representatives of all proper Qsubplanes both of a VeblenWedderburn (VW) plane of...
Show moreThis work was motivated by the wellknown question: "Does there exist a nondesarguesian projective plane of prime order?" For a prime p < 11, there is only the pappian plane of order p. Hence, such planes are indeed desarguesian. Thus, it is of interest to examine whether there are nondesarguesian planes of order 11. A suggestion by Ascher Wagner in 1985 was made to Spyros S. Magliveras: "Begin with a nondesarguesian plane of order pk, k > 1, determine all subplanes of order p up to collineations, and check whether one of these is nondesarguesian." In this manuscript we use a grouptheoretic methodology to determine the subplane structures of some nondesarguesian planes. In particular, we determine orbit representatives of all proper Qsubplanes both of a VeblenWedderburn (VW) plane of order 121 and of the Hughes plane of order 121, under their full collineation groups. In PI, there are 13 orbits of Baer subplanes, all of which are desarguesian, and approximately 3000 orbits of Fano subplanes. In Sigma , there are 8 orbits of Baer subplanes, all of which are desarguesian, 2 orbits of subplanes of order 3, and at most 408; 075 distinct Fano subplanes. In addition to the above results, we also study the subplane structures of some nondesarguesian planes, such as the Hall plane of order 25, the Hughes planes of order 25 and 49, and the Figueora planes of order 27 and 125. A surprising discovery by L. Puccio and M. J. de Resmini was the existence of a plane of order 3 in the Hughes plane of order 25. We generalize this result, showing that there are subplanes of order 3 in the Hughes planes of order q2, where q is a prime power and q 5 (mod 6). Furthermore, we analyze the structure of the full collineation groups of certain Veblen Wedderburn (VW) planes of orders 25, 49 and 121, and discuss how to recover the planes from their collineation groups.
Show less  Date Issued
 2010
 PURL
 http://purl.flvc.org/FAU/1927609
 Subject Headings
 Projected planes, Combinatorial designs and configurations, Surfaces, Algebraic, Manifolds (Mathematics)
 Format
 Document (PDF)
 Title
 Construction of combinatorial designs with prescribed automorphism groups.
 Creator
 Kolotoglu, Emre., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

In this dissertation, we study some open problems concerning the existence or nonexistence of some combinatorial designs. We give the construction or proof of nonexistence of some Steiner systems, large sets of designs, and graph designs, with prescribed automorphism groups.
 Date Issued
 2013
 PURL
 http://purl.flvc.org/fcla/dt/3360795
 Subject Headings
 Combinatorial designs and configurations, Finite geometries, Curves, Algebraic, Automorphisms, Mathematical optimization, Steiner systems
 Format
 Document (PDF)
 Title
 New Results in Group Theoretic Cryptology.
 Creator
 Sramka, Michal, Florida Atlantic University, Magliveras, Spyros S., Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

With the publication of Shor's quantum algorithm for solving discrete logarithms in finite cyclic groups, a need for new cryptographic primitives arose; namely, for more secure primitives that would prevail in the postquantum era. The aim of this dissertation is to exploit some hard problems arising from group theory for use in cryptography. Over the years, there have been many such proposals. We first look at two recently proposed schemes based on some form of a generalization of the...
Show moreWith the publication of Shor's quantum algorithm for solving discrete logarithms in finite cyclic groups, a need for new cryptographic primitives arose; namely, for more secure primitives that would prevail in the postquantum era. The aim of this dissertation is to exploit some hard problems arising from group theory for use in cryptography. Over the years, there have been many such proposals. We first look at two recently proposed schemes based on some form of a generalization of the discrete logari thm problem (DLP), identify their weaknesses, and cryptanalyze them. By applying the exper tise gained from the above cryptanalyses, we define our own generalization of the DLP to arbitrary finite groups. We show that such a definition leads to the design of signature schemes and pseudorandom number generators with provable security under a security assumption based on a group theoretic problem. In particular, our security assumption is based on the hardness of factorizing elements of the projective special linear group over a finite field in some representations. We construct a oneway function based on this group theoretic assumption and provide a security proof.
Show less  Date Issued
 2006
 PURL
 http://purl.flvc.org/fau/fd/FA00000878
 Subject Headings
 Group theory, Mathematical statistics, Cryptography, Combinatorial designs and configurations, Data encryption (Computer science), Coding theory
 Format
 Document (PDF)
 Title
 Low rank transitive representations, primitive extensions, and the collision problem in PSL (2, q).
 Creator
 Thapa Magar, Krishna B., Magliveras, Spyros S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Mathematical Sciences
 Abstract/Description

Every transitive permutation representation of a finite group is the representation of the group in its action on the cosets of a particular subgroup of the group. The group has a certain rank for each of these representations. We first find almost all rank3 and rank4 transitive representations of the projective special linear group P SL(2, q) where q = pm and p is an odd prime. We also determine the rank of P SL (2, p) in terms of p on the cosets of particular given subgroups. We then...
Show moreEvery transitive permutation representation of a finite group is the representation of the group in its action on the cosets of a particular subgroup of the group. The group has a certain rank for each of these representations. We first find almost all rank3 and rank4 transitive representations of the projective special linear group P SL(2, q) where q = pm and p is an odd prime. We also determine the rank of P SL (2, p) in terms of p on the cosets of particular given subgroups. We then investigate the construction of rank3 transitive and primitive extensions of a simple group, such that the extension group formed is also simple. In the latter context we present a new, group theoretic construction of the famous HoffmanSingleton graph as a rank3 graph.
Show less  Date Issued
 2015
 PURL
 http://purl.flvc.org/fau/fd/FA00004471, http://purl.flvc.org/fau/fd/FA00004471
 Subject Headings
 Combinatorial designs and configurations, Cryptography, Data encryption (Computer science), Finite geometries, Finite groups, Group theory, Permutation groups
 Format
 Document (PDF)