Mathematics Commons^™

The Digraph Of The Square Mapping On Elliptic Curves, Katrina Glaeser

Mathematical Sciences Technical Reports (MSTR)

Consider a subgroup of an elliptic curve generated by a point P of order n. It is possible to match any point Q to an integer k (mod n) such that Q = kP using a brute force method. By observing patterns in the digraph of the squaring map on the integers modulo n it is possible to perform this matching. These techniques can be applied to solving the Elliptic Curve Discrete Log Problem given a complete graph of the square mapping k P -> k^2 P for the elliptic curve points.

Operations Research Methods For Optimization In Radiation Oncology, M Ehrgott, Allen Holder

Mathematical Sciences Technical Reports (MSTR)

Operations Research has a successful tradition of applying mathematical analysis to a wide range of applications, with one of the burgeoning areas of growth being in medical physics. The original application was in the optimal design of the influence map for a radiotherapy treatment, a problem that has continued to receive attention. However, operations research has been applied to other clinical problems like patient scheduling, vault design, and image alignment. The overriding theme of this article is to present how techniques in operations research apply to clinical problems, which we accomplish in three parts. First, we present the perspective from …

Flattening A Cone, Sean A. Broughton

Mathematical Sciences Technical Reports (MSTR)

We want to manufacture a cut-off slanted cone from a flat sheet of metal. If the cone were a normal right cone we know that we would simply cut out a sector of a circle and roll it up. However the cone is slanted. We want to know what the flattened shape looks like so that we can cut it out and roll it up to closely approximate correct final shape. We also want to minimize the amount of wasted metal after the shape is cut out.

The problem, and it generalizations may be solved analytically but the analytical solution …

Solving The P-Median Problem With Insights From Discrete Vector Quantization, Gino J. Lim, Allen Holder, Josh Reese

Mathematical Sciences Technical Reports (MSTR)

The goals of this paper are twofold. First, we formally equate the p-median problem from facility location to the optimal design of a vector quantizer. Second, we use the equivalence to show that the Maranzana Algorithm can be interpreted as a projected Lloyd Algorithm, a fact that improves complexity. Numerical results verify significant improvements in run-time.

A Decomposition Of The Pure Parsimony Problem, Allen Holder, Thomas M. Langley

Mathematical Sciences Technical Reports (MSTR)

We partially order a collection of genotypes so that we can represent the problem of inferring the least number of haplotypes in terms of substructures we call g-lattices. This representation allows us to prove that if the genotypes partition into chains with certain structure, then the NP-Hard problem can be solved efficiently. Even without the specified structure, the decomposition shows how to separate the underlying integer programming model into smaller models.

A Clustering Approach For Optimizing Beam Angles In Imrt Planning, Gino J. Lim, Allen Holder, Josh Reese

Mathematical Sciences Technical Reports (MSTR)

In this paper we introduce a p-median problem based clustering heuristic for selecting efficient beam angles for intensity-modulated radiation therapy. The essence of the method described here is the clustering of beam angles according to probability that an angle will be observed in the final solution and similarities among different angles and the selection of a representative angle from each of the p resulting cluster cells. We conduct experiments using several combinations of modeling parameters to find the conditions where the heuristic best performs. We found a combination of such parameters that outperformed all other parameters on three of the …

Statistical Investigation Of Structure In The Discrete Logarithm, Andrew Hoffman

Mathematical Sciences Technical Reports (MSTR)

The absence of an efficient algorithm to solve the Discrete Logarithm Problem is often exploited in cryptography. While exponentiation with a modulus is extremely fast with a modern computer, the inverse is decidedly not. At the present time, the best algorithms assume that the inverse mapping is completely random. Yet there is at least some structure, and to uncover additional structure that may be useful in constructing or refining algorithms, statistical methods are employed to compare modular exponential mappings to random mappings. More concretely, structure will be defined by representing the mappings as functional graphs and using parameters from graph …

Discrete Logarithm Over Composite Moduli, Marcus L. Mace

Mathematical Sciences Technical Reports (MSTR)

In an age of digital information, security is of utmost importance. Many encryption schemes, such as the Diffie-Hellman Key Agreement and RSA Cryptosystem, use a function which maps x to y by a modular power map with generator g. The inverse of this function - trying to find x from y - is called the discrete logarithm problem. In most cases, n is a prime number. In some cases, however, n may be a composite number. In particular, we will look at when n = p^b for a prime p. We will show different techniques of obtaining graphs of this …

Structural Properties Of Power Digraphs Modulo N, Joseph Kramer-Miller

Mathematical Sciences Technical Reports (MSTR)

We define G(n, k) to be a directed graph whose set of vertices is {0, 1, ..., n−1} and whose set of edges is defined by a modular relation. We say that G(n, k) is symmetric of order m if we can partition G(n, k) into subgraphs, each containing m components, such that all the components in a subgraph are isomorphic. We develop necessary and sufficient conditions for G(n, k) to contain symmetry when n is odd and square-free. Additionally, we use group theory to describe the structural properties of the subgraph of G(n, k) containing only those vertices relatively …