Reserarch On Application Of Latin Square In 2D-OOC And Image Encryption

Many problems in combinatorial mathematics are closely related with the fun math games, such as the Kirkman School Girl problem, Konigsberg Seven Bridges problem, Fibonacci Series, Magic Square problem etc. Through several decades or even centuries of efforts on these problems, many results were achieved and gradually applied in real life.Latin square, an important member in combinatorial mathematics can be traced back to the’36 Officer Problem’. The study was firstly proposed by Leonhard Euler. Many results were got from 1779 for this problem during 200 years of research history. In recent decades, the relevant theoretical research and application of Latin square had become a hot spot, especially in this two fields, communication code and encryption, such as error correcting codes design, constant weight codes construction, frequency hopping network, optical orthogonal code construction, image encryption, message authentication code construction, etc.Applications of Latin square in 2D-OOC and image encryption were studied in this paper based on orthogonal Latin square and complete Latin square. Mainly, the study was focused on the following several aspects.(1) Four different types of Latin square construction were proposed. Such as construction of odd order orthogonal Latin square, construction of Latin square with the property that the main diagonal of this Latin square is 0 and side diagonal is n-1, construction of completed Latin square. Based on the idea of’low order orthogonal Latin square can be used to constructed high order orthogonal Latin square’, a tensor product method was proposed to construct high order symmetric Latin square from low order symmetric Latin square.(2) Three novel 2-D optical orthogonal codes, such as CLS/OCS、OOC/MOLS. MOLS/OCS were constructed by these methods that using complete Latin square and orthogonal Latin square as time spreading sequence or wavelength hopping sequence and using one coincidence sequence and 1-D optical orthogonal codes as wavelength hopping or time spreading sequence. Detailed steps and construction were given in this paper. The performance of these codes was compared with other 2-D OOC, such as OCFHC/OCS, OOC/PC etc. These new codes have better BER performance.(3) An encryption/decryption algorithm which can be used in real digital image was designed based on complete Latin square. A double image encryption/decryption algorithm based on complete Latin square was designed by taking advantage of the gray values transformation and pixel transform. Simulation experiments were performed by using different size of image. The related performance was measured and the security performance was analysed. The same ideal effect was achieved when it was used in the encryption of color image. The relationships between the encryption/ decryption time and image size, number of iterations were determined according to this algorithm.