Non-commutative Massey-Omura Encryption with Symmetric Groups

Author

Shannon HaleyFollow

Date of Award

Spring 4-26-2018

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 two non-commutative variations on the original Massey-Omura encryption system using conjugations in the symmetric group S_n. Patented in 1986, the original system was based on the cyclic group F^* of units in a finite field F. In place of the abelian group F^*, we will work in the non-abelian group S_nusing disjoint permutations as well as maximal abelian subgroups in order to potentially create a more secure system. Introducing the non-abelian group S_n presents the need to create a keyspace of commuting permutations and abelian subgroups of sufficient size. We analyze the security of our modified systems by examining the bit-level security of each and susceptibility to standard message attacks. Additionally, we find that the keycount for the first system grows factorially with n. We show that the keycount for the second variation grows exponentially with n while improving on the first modification by allowing any number of users to participate in communication.

Recommended Citation

Haley, Shannon, "Non-commutative Massey-Omura Encryption with Symmetric Groups" (2018). Student Research Submissions. 254.
https://scholar.umw.edu/student_research/254

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