Academic Work

Variations and Attacks on the RSA Algorithm

Abstract: Stallings and Brown of Computer Security: Principles and Practice write, “One of the first public-key schemes was developed in 1977 by Ron Rivest, Adi Shamir, and Len Adleman at MIT and first published in 1978. The RSA scheme has since reigned supreme as the most widely accepted and implemented approach to public-key encryption”([7] p.58). This paper offers insight into the work that has been developed around the RSA algorithm since its creation. Namely, the algorithm, attacks on the algorithm, and updates to the algorithm are discussed.

Combinatorial Species and Graph Enumeration

Abstract: In enumerative combinatorics, it is often a goal to enumerate both labeled and unlabeled structures of a given type. The theory of combinatorial species is a novel toolset which provides a rigorous foundation for dealing with the distinction between labeled and unlabeled structures. The cycle index series of a species encodes the labeled and unlabeled enumerative data of that species. Moreover, by using species operations, we are able to solve for the cycle index series of one species in terms of other, known cycle indices of other species. Section 3 is an exposition of species theory and Section 4 is an enumeration of point-determining bipartite graphs using this toolset. In Section 5, we extend a result about point-determining graphs to a similar result for point-determining Φ-graphs, where Φ is a class of graphs with certain properties. Finally, Appendix A is an expository on species computation using the software Sage [9] and Appendix B uses Sage to calculate the cycle index series of point-determining bipartite graphs.

q-Binomial Coefficients and the q-Binomial Theorem

Abstract: The binomial theorem and binomial coefficient are found in several areas in mathematics–probability, combinatorics, statistics to name a few. Lesser studied are the q-analogs of the theorem and coefficient. This paper offers an introduction to the q-binomial theorem and the q-binomial coefficient. We discuss their definitions, their extensions, and some examples. The paper ends with an application to the q,t-Catalan Numbers.

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