The Mathematics Behind Sudoku and How to Create Magic Squares

Sudoku puzzles date back to the 1800s in France and were introduced to America in the late 1970s. Since then, the puzzle has become a worldwide phenomenon. This thesis will be of expositional nature including the works of books and mathematical papers such as [2], [14], and [15], among others. The pages ahead will contain answers to some common questions about Sudoku such as, what is the minimum number of starting clues that will produce a unique solution? On the other side of the spectrum, what is the maximum number of starting clues that won’t produce a unique solution? Taking Sudoku one step farther, this paper will talk about Magic Squares and the algorithm used in making them. Even more interesting is that the al- gorithm provides the makings of a multimagic square, where every entry in a magic square is squared, with the rows, columns, and diagonals still adding to the same number [15]. See the Appendix for the computer code in the program MATLAB that creates multimagic squares

Synthetic steganography: Methods for generating and detecting covert channels in generated media

Issues of privacy in communication are becoming increasingly important. For many people and businesses, the use of strong cryptographic protocols is sufficient to protect their communications. However, the overt use of strong cryptography may be prohibited or individual entities may be prohibited from communicating directly. In these cases, a secure alternative to the overt use of strong cryptography is required. One promising alternative is to hide the use of cryptography by transforming ciphertext into innocuous-seeming messages to be transmitted in the clear. ^ In this dissertation, we consider the problem of synthetic steganography: generating and detecting covert channels in generated media. We start by demonstrating how to generate synthetic time series data that not only mimic an authentic source of the data, but also hide data at any of several different locations in the reversible generation process. We then design a steganographic context-sensitive tiling system capable of hiding secret data in a variety of procedurally-generated multimedia objects. Next, we show how to securely hide data in the structure of a Huffman tree without affecting the length of the codes. Next, we present a method for hiding data in Sudoku puzzles, both in the solved board and the clue configuration. Finally, we present a general framework for exploiting steganographic capacity in structured interactions like online multiplayer games, network protocols, auctions, and negotiations. Recognizing that structured interactions represent a vast field of novel media for steganography, we also design and implement an open-source extensible software testbed for analyzing steganographic interactions and use it to measure the steganographic capacity of several classic games. ^ We analyze the steganographic capacity and security of each method that we present and show that existing steganalysis techniques cannot accurately detect the usage of the covert channels. We develop targeted steganalysis techniques which improve detection accuracy and then use the insights gained from those methods to improve the security of the steganographic systems. We find that secure synthetic steganography, and accurate steganalysis thereof, depends on having access to an accurate model of the cover media

Colorings and Sudoku Puzzles

Map colorings refer to assigning colors to different regions of a map. In particular, a typical application is to assign colors so that no two adjacent regions are the same color. Map colorings are easily converted to graph coloring problems: regions correspond to vertices and edges between two vertices exist for adjacent regions. We extend these notions to Shidoku, 4x4 Sudoku puzzles, and standard 9x9 Sudoku puzzles by demanding unique entries in rows, columns, and regions. Motivated by our study of ring and field theory, we expand upon the standard division algorithm to study Gr\ obner bases in multivariate polynomial rings. We utilize Gr\ obner bases of an ideal of a multivariate polynomial ring over a finite field to solve coloring, Shidoku, and Sudoku problems. In the last section, we note Gr\ obner bases are also well-suited to hypergraph coloring problems