Construction of Sudoku, magic rectangles, magic Sudoku, twin Sudoku, and their extension to 3D Sudoku with application in cryptography and data security
Date
2025
Authors
Najafian, Mehrab
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Abstract
Sudoku tables as a special class of Latin squares are studied and a construction for each class of 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, where s is a divisor of d, 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 X, Y , and Z axes, 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 squares 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 the algebraic group Zn are used. Sudoku tables have applications in different fields of study, and some of these applications are investigated. It is 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.
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Keywords
Sudoku Tables, Sudoku Puzzles, Twin Sudoku Tables, Standard Solid Sudoku Cubes, 3D Sudolu Puzzles, Magic Rectangles, Magic Sudoku Tables, Zero-knowledge proof protocols