Combinatorics and Optimizationhttp://hdl.handle.net/10012/99282018-09-25T19:48:49Z2018-09-25T19:48:49ZOrdinary and Generalized Circulation Algebras for Regular MatroidsOlson-Harris, Nicholashttp://hdl.handle.net/10012/138262018-09-20T02:31:14Z2018-09-19T00:00:00ZOrdinary and Generalized Circulation Algebras for Regular Matroids
Olson-Harris, Nicholas
Let E be a finite set, and let R(E) denote the algebra of polynomials in indeterminates (x_e)_{e in E}, modulo the squares of these indeterminates. Subalgebras of R(E) generated by homogeneous elements of degree 1 have been studied by many authors and can be understood combinatorially in terms of the matroid represented by the linear equations satisfied by these generators. Such an algebra is related to algebras associated to deletions and contractions of the matroid by a short exact sequence, and can also be written as the quotient of a polynomial algebra by certain powers of linear forms.
We study such algebras in the case that the matroid is regular, which we term circulation algebras following Wagner. In addition to surveying the existing results on these algebras, we give a new proof of Wagner's result that the structure of the algebra determines the matroid, and construct an explicit basis in terms of basis activities in the matroid. We then consider generalized circulation algebras in which we mod out by a fixed power of each variable, not necessarily equal to 2. We show that such an algebra is isomorphic to the circulation algebra of a "subdivided" matroid, a variation on a result of Nenashev, and derive from this generalized versions of many of the results on ordinary circulation algebras, including our basis result. We also construct a family of short exact sequences generalizing the deletion-contraction decomposition.
2018-09-19T00:00:00ZNon-Constructivity in Security ProofsSoundararajan, Priyahttp://hdl.handle.net/10012/137702018-09-11T02:31:14Z2018-09-10T00:00:00ZNon-Constructivity in Security Proofs
Soundararajan, Priya
In the field of cryptography, one generally obtains assurances for the
security of a cryptographic protocol by giving a reductionist security
proof, which is comprised of a reduction from breaking a mathematical
problem (that is well-studied and widely believed to be intractable)
to the breaking of the cryptographic protocol. While such reductions
are generally constructive, some authors give non-constructive
reductions (also called non-uniform reductions) in order to reduce
the tightness gap of the reduction. However, in order to assess the
concrete security that the proof provides, one also needs to assess
the intractability of the underlying mathematical problem against
non-constructive attacks. Unfortunately, there has been very little
work in the literature on non-constructive attacks on these problems,
and sometimes non-constructive attacks are found that are much faster
than their constructive counterparts. Thus, it is sometimes very
difficult to obtain meaningful security assurances about a cryptographic
protocol from a non-constructive reductionist security proof.
In this thesis, we examine three instances of non-constructive security
proofs for cryptographic protocols in the literature:
(1) a password-based key derivation function; (2) an HMAC-related message
authentication code scheme; and (3) a
round-optimal blind signature scheme.
2018-09-10T00:00:00ZAcyclic Colouring of Graphs on SurfacesRedlin, Shaylahttp://hdl.handle.net/10012/137322018-09-05T02:31:44Z2018-09-04T00:00:00ZAcyclic Colouring of Graphs on Surfaces
Redlin, Shayla
An acyclic k-colouring of a graph G is a proper k-colouring of G with no bichromatic cycles. In 1979, Borodin proved that planar graphs are acyclically 5-colourable, an analog of the Four Colour Theorem. Kawarabayashi and Mohar proved in 2010 that "locally" planar graphs are acyclically 7-colourable, an analog of Thomassen's result that "locally" planar graphs are 5-colourable. We say that a graph G is critical for (acyclic) k-colouring if G is not (acyclically) k-colourable, but all proper subgraphs of G are. In 1997, Thomassen proved that for every k >= 5 and every surface S, there are only finitely many graphs that embed in S that are critical for k-colouring. Here we prove the analogous result that for each k >= 12 and each surface S, there are finitely many graphs embeddable on S that are critical for acyclic k-colouring. This result implies that there exists a linear time algorithm that, given a surface S and large enough k, decides whether a graph embedded in S is acyclically k-colourable.
2018-09-04T00:00:00ZDensity and Structure of Homomorphism-Critical GraphsSmith-Roberge, Evelynehttp://hdl.handle.net/10012/136432018-08-23T02:30:17Z2018-08-22T00:00:00ZDensity and Structure of Homomorphism-Critical Graphs
Smith-Roberge, Evelyne
Let $H$ be a graph. A graph $G$ is $H$-critical if every proper subgraph of $G$ admits a homomorphism to $H$, but $G$ itself does not. In 1981, Jaeger made the following conjecture concerning odd-cycle critical graphs: every planar graph of girth at least $4t$ admits a homomorphism to $C_{2t+1}$ (or equivalently, has a $\tfrac{2t+1}{t}$-circular colouring). The best known result for the $t=3$ case states that every planar graph of girth at least 18 has a homomorphism to $C_7$. We improve upon this result, showing that every planar graph of girth at least 16 admits a homomorphism to $C_7$. This is obtained from a more general result regarding the density of $C_7$-critical graphs. Our main result is that if $G$ is a $C_7$-critical graph with $G \not \in \{C_3, C_5\}$, then $e(G) \geq \tfrac{17v(G)-2}{15}$. Additionally, we prove several structural lemmas concerning graphs that are $H$-critical, when $H$ is a vertex-transitive non-bipartite graph.
2018-08-22T00:00:00Z