#### Brandon Grossardt, M.S.

he/him

**Education:** Bachelor of Science in mathematics (May 2001)

Master of Science in statistics from Kansas State University

**McNair Project:** Cryptanalysis: Investigating the Cyclic Nature of Block-Matrix Ciphers (2000)

**Mentor:** Charles Moore, Ph.D.

We begin by discussing a specific matrix encryption method and investigating its cyclic properties. Specifically, we will examine how an initial text always resurfaces after multiple encryptions. The encryption algorithm is given, followed by a more rigorous mathematical approach. The mathematical method is explained in detail for the encryption process. Some preliminary information, including an elementary explanation of circulant matrices and a review of requisite number theoretical concepts is given. Preliminary proofs are given, leading to a final proof that gives a function relating the length and complexity of a message to the number of encryptions necessary to return the initial text. Several implications of the functional relation are discussed, includign results involving the Euler phi-function. We continue by introducing and briefly investigating the cyclic properties of multi-reference ciphers. In closing, a brief introduction to circulant-matrix ciphers as well as permutation-matrix ciphers is given. An analysis of the cyclic tendencies of these similar ciphers is given. A final generalization encompassing all of the cipher methods is discussed, and a proof assuring the return of the initial text is provided.