Date

Spring 5-8-2020

Document Type

Honors Project

First Advisor

Helmstutler, Randall

Department Chair or Program Director

Helmstutler, Randall

Degree Name

Bachelor of Science

Major or Concentration

Mathematics

Department or Program

Mathematics

Abstract

We introduce a procedure in which two trusted individuals, Alice and Bob, may share a secret matrix K from the non-abelian general linear group of matrices. In this procedure, the matrix K is concealed from an eavesdropper, Eve, by a sequence of conjugations by elements from a pre-determined abelian subgroup of the general linear group. We demonstrate that the group of invertible circulant matrices is one abelian subgroup that may be able to withstand a brute force attack. To analyze this we need a technique to determine the order of this group, and to do this we make use of a well-known isomorphism between the ring of circulants over a finite field and a quotient of the ring of polynomials over a finite field. After we show empirically that the order of the group of invertible circulants increases exponentially in dimension n with degree q of the field fixed, we will give a universal lower bound on the order of this group to prove this mathematically as well.

Language

English

Included in

Mathematics Commons

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