#### Date of Award

Spring 5-8-2020

#### Document Type

Honors Project

#### Degree Name

Bachelor of Science

#### Department

Mathematics

#### Department Chair or Program Director

Helmstutler, Randall

#### First Advisor

Helmstutler, Randall

#### Major or Concentration

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.

#### Recommended Citation

Frederick, Hannah B., "Conjugation by Circulant Matrices in Non-commutative Cryptography" (2020). *Student Research Submissions*. 321.

https://scholar.umw.edu/student_research/321