51³Ô¹Ï

Topic

Construction of Sudoku, Magic Rectangles, Magic Sudoku, Twin Sudoku, and Their Extension to 3D Sudoku With Application in Cryptography and Data Security

Department of Electrical and Computer Engineering

Date & location

• Friday, July 18, 2025

• 11:00 A.M.

• Virtual Defence

Reviewers

Supervisory Committee

• Dr. T. Aaron Gulliver, Department of Electrical and Computer Engineering, 51³Ô¹Ï (Co-Supervisor)

• Dr. Morteza Esmaeili, Department of Electrical and Computer Engineering, UVic (Co-Supervisor)

• Dr. Venkatesh Srinivasan, Department of Computer Science, UVic (Outside Member) 

External Examiner

• Dr. John Lorch, Department of Mathematical Sciences, Ball State University 

Chair of Oral Examination

• Dr. Charles Curry, School of Earth and Ocean Sciences, UVic

   

Abstract

Sudoku tables as a special class of Latin squares are studied and a construction for each class of the Sudoku tables is introduced. We provide a definition of a Sudoku table of size m as an m ×m table with blocks of size s × d. Twin Sudoku tables of size m with (s,d)-subtables are defined as Sudoku tables with (s × d)-subtables and (d×s)-subtables, simultaneously. Solid Sudoku cubes of size m, which are cubes with subcubes of size s×s×d, where s ≤ d and s·d = m, can be divided into subcubes in any face of the Sudoku cube. They have the property that each layer of the cube from each face which is a table of size m is a twin Sudoku table. Standard solid Sudoku cubes and 3D Sudoku puzzles are defined as solid cubes in which the subcubes of size m contain m different numbers and they can be divided into subcubes of size x ×y×z along the axes X, Y, and Z, respectively. Magic rectangles and nested magic rectangles are constructed. In addition, orthogonal balanced magic Sudoku tables as a class of Sudoku tables in which their blocks are magic are created and a method to construct nested Latin square is presented. New classes of Sudoku puzzles are obtained from the construction of the Sudoku tables and Sudoku cubes. Since a Sudoku puzzle can be obtained from a Sudoku table by randomly removing some entries of the Sudoku table, creating a new class of Sudoku tables leads to a new class of Sudoku puzzles. To construct these new classes of Sudoku tables cyclotomic cosets of an algebraic group Zn are used. Sudoku tables have applications in different fields of study and some of these applications will be investigated. It will be shown that they can be used in cryptography and zero knowledge proof protocols. Sudoku puzzles are in the class of NP-complete problems so zero knowledge proof protocols can be based on this class of problems. Sudoku tables like Latin squares can be used to generate locally repairable codes which have application in distributed storage systems. It is shown how they can be used to construct locally repairable codes and their properties are investigated.