#### Publication Date

12-2016

#### Date of Final Oral Examination (Defense)

10-28-2016

#### Type of Culminating Activity

Thesis

#### Degree Title

Master of Science in Mathematics

#### Department

Mathematics

#### Major Advisor

Liljana Babinkostova, Ph.D.

#### Advisor

Samuel Coskey, Ph.D.

#### Advisor

Marion Scheepers, Ph.D.

#### Advisor

Jyh-haw Yeh, Ph.D.

#### Abstract

A *latin square* of order-*n* is an *n* x *n* array over a set of *n* symbols such that every symbol appears exactly once in each row and exactly once in each column. Latin squares encode features of algebraic structures. When an algebraic structure passes certain "latin square tests", it is a candidate for use in the construction of cryptographic systems. A *transversal* of a latin square is a list of *n* distinct symbols, one from each row and each column. The question regarding the existence of transversals in latin squares that encode the Cayley tables of finite groups is far from being resolved and is an area of active investigation. It is known that counting the pairs of permutations over a Galois field ��* _{pd}* whose point-wise sum is also a permutation is equivalent to counting the transversals of a latin square that encodes the addition group of ��

*. We survey some recent results and conjectures pertaining to latin squares and transversals. We create software tools that generate latin squares and count their transversals. We confirm previous results that cyclic latin squares of prime order-*

_{pd}*p*possess the

*maximum*transversal counts for 3 ≤

*p*≤ 9. Furthermore, we create a new algorithm that uses these prime order-

*p*cyclic latin squares as "building blocks" to construct

*super-symmetric*latin squares of prime power order-

*p*

*with*

^{d }*d*> 0; using this algorithm we accurately predict that super-symmetric latin squares of order-

*p*

*possess the confirmed maximum transversal counts for 3 ≤*

^{d }*p*≤ 9 and the estimated lower bound on the maximum transversal counts for 9 <

^{d }*p*

*≤ 17. Also, we give some conjectures regarding the number of transversals in a super-symmetric latin square. Lastly, we use the super-symmetric latin square for the additive group of the Galois field (��*

^{d}_{32}, +) to create a simplified version of Grøstl, an iterated hash function, where the compression function is built from two fixed, large, distinct permutations.

#### Recommended Citation

Schmidt, Nathan O., "Latin Squares and Their Applications to Cryptography" (2016). *Boise State University Theses and Dissertations*. 1223.

https://scholarworks.boisestate.edu/td/1223

## Comments

The latest versions of our latin square software tools are open source and are available at: https://sourceforge.net/projects/latin-square-toolbox/.