Paper Title
Construct Mathematical S-Box For Symmetric Key Cryptosystems

Abstract
Advanced Encryption Standard (AES) is a block cipher system which widely used in cryptographic application. Substitution box (S-box), nonlinear substitution operation, is the main factor of the AESstrength.The reason of construction S-boxes for symmetric key cryptosystems is to yield good properties such as high nonlinearity andlow autocorrelation. The main objective of this research is suggesting mathematical approach to generate random S-boxessuitable for symmetric key cryptosystems. The fact that S-boxes are randomly key-dependent and unknown is the main strength of the new approach, since both linear and differential cryptanalysis requires known S-boxes. In this paper, briefly analyze of the AES algorithm, substation-boxes, linear and differential cryptanalysis are introduced. Moreover, a randomly key-dependent S-box, inverse S-box generation algorithm andthe significant properties of proposed S-boxes are described. Preliminary results show that our proposed algorithm has good cryptographic strength, fast, low complexity, and resistant against linear and differential cryptanalysis. Keywords - Cryptography, Symmetric key,AES and RijndaelS-box.Cryptanalysis.