Browsing Combinatorics and Optimization by Issue Date
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Counting Bases
(University of Waterloo, 2004)A theorem of Edmonds characterizes when a pair of matroids has a common basis. Enumerating the common bases of a pair of matroid is a much harder problem, and includes the #Pcomplete problem of counting the number of ... 
Combinatorial Constructions for Transitive Factorizations in the Symmetric Group
(University of Waterloo, 2004)We consider the problem of counting <i>transitive factorizations</i> of permutations; that is, we study tuples (σ<i>r</i>,. . . ,σ1) of permutations on {1,. . . ,<i>n</i>} such that (1) the product ... 
Regularization Using a Parameterized Trust Region Subproblem
(University of Waterloo, 2004)We present a new method for regularization of illconditioned problems that extends the traditional trustregion approach. Illconditioned problems arise, for example, in image restoration or mathematical processing of ... 
Packing Directed Joins
(University of Waterloo, 2004)Edmonds and Giles conjectured that the maximum number of directed joins in a packing is equal to the minimum weight of a directed cut, for any weighted directed graph. This is a generalization of Woodall's Conjecture ... 
Independent Sets and Eigenspaces
(University of Waterloo, 2004)The problems we study in this thesis arise in computer science, extremal set theory and quantum computing. The first common feature of these problems is that each can be reduced to characterizing the independent sets ... 
Efficient Integer Representations for Cryptographic Operations
(University of Waterloo, 2004)Every positive integer has a unique radix 2 representation which uses the digits {0,1}. However, if we allow digits other than 0 and 1, say {0,1,1}, then a positive integer has many representations. Of these ... 
Resolution of Ties in Parametric Quadratic Programming
(University of Waterloo, 2004)We consider the convex parametric quadratic programming problem when the end of the parametric interval is caused by a multiplicity of possibilities ("ties"). In such cases, there is no clear way for the proper active ... 
Colouring Subspaces
(University of Waterloo, 2005)This thesis was originally motivated by considering vector space analogues of problems in extremal set theory, but our main results concern colouring a graph that is intimately related to these vector space analogues. ... 
A Survey of Attacks on Multivariate Cryptosystems
(University of Waterloo, 2005)This thesis provides a survey of the attacks on multivariate cryptosystems. We begin by providing an outline of the general multivariate cryptosystem. Proceeding from there, we show that even with this level of detail, ... 
Algebraic Tori in Cryptography
(University of Waterloo, 2005)Communicating bits over a network is expensive. Therefore, cryptosystems that transmit as little data as possible are valuable. This thesis studies several cryptosystems that require significantly less bandwidth than ... 
Transitive Factorizations of Permutations and Eulerian Maps in the Plane
(University of Waterloo, 2005)The problem of counting ramified covers of a Riemann surface up to homeomorphism was proposed by Hurwitz in the late 1800's. This problem translates combinatorially into factoring a permutation with a specified cycle ... 
Security Models and Proofs for Key Establishment Protocols
(University of Waterloo, 2005)In this thesis we study the problem of secure key establishment, motivated by the construction of secure channels protocols to protect information transmitted over an open network. In the past, the purported security of ... 
Design and Analysis of RC4like Stream Ciphers
(University of Waterloo, 2005)RC4 is one of the most widely used ciphers in practical software applications. In this thesis we examine security and design aspects of RC4. First we describe the functioning of RC4 and present previously published ... 
Character Polynomials and Lagrange Inversion
(University of Waterloo, 2005)In this thesis, we investigate two expressions for symmetric group characters: Kerov?s universal character polynomials and Stanley?s character polynomials. We give a new explicit form for Kerov?s polynomials, which ... 
Colouring Cayley Graphs
(University of Waterloo, 2005)We will discuss three ways to bound the chromatic number on a Cayley graph. 1. If the connection set contains information about a smaller graph, then these two graphs are related. Using this information, we will ... 
A Fundamentally Topological Perspective on Graph Theory
(University of Waterloo, 2005)We adopt a novel topological approach for graphs, in which edges are modelled as points as opposed to arcs. The model of classical <i>topologized graphs</i> translates graph isomorphism into topological homeomorphism, ... 
On the Evolutionary Design of Quantum Circuits
(University of Waterloo, 2005)The goal of this work is to understand the application of the evolutionary programming approach to the problem of quantum circuit design. This problem is motivated by the following observations: <ul> <li>In order to ... 
Digital Signcryption
(University of Waterloo, 2005)Signcryption is a new cryptographic primitive which simultaneously provides both confidentiality and authenticity. Previously, these two goals had been considered separately, with encryption schemes providing confidentiality ... 
Genus one partitions
(University of Waterloo, 2006)We obtain a tight upper bound for the genus of a partition, and calculate the number of partitions of maximal genus. The generating series for genus zero and genus one rooted hypermonopoles is obtained in closed form ... 
On the Crossing Numbers of Complete Graphs
(University of Waterloo, 2006)In this thesis we prove two main results. The Triangle Conjecture asserts that the convex hull of any optimal rectilinear drawing of <em>K<sub>n</sub></em> must be a triangle (for <em>n</em> ≥ 3). We prove that, ...