8×8 S-boxes over Elliptic Curves and Chaotic maps: The Optimization Technique by Using Symmetric group S256

Abstract

In mathematics, the symmetric group S(n) is essential. Its significance cannot be overstated as it occurs in a wide variety of circumstances. In mathematical journals, there are tens of thousands of pages of research papers that in one way or another involve this group. Recently, many researchers have used different symmetric groups to increase the working abilities of their respective substi tution boxes (S-boxes) according to their designed schemes. Most of the S-boxes, designed over Elliptic curves and chaotic maps have a maximum nonlinearity up to 107. In literature, there are also some well-designed S-boxes having the same basis but their nonlinearities are even below 105 or nearly 100. In our work, we used the elements of the symmetric group S256 to enhance the nonlinearity of such Elliptic curves and chaotic maps-based S-boxes to make them more efficient so that they can be used like the current S-boxes with effective nonlinearity which are being used in various applications of cryptography. In the first chapter, we gave an introductory definition of cryptography and its types, S-box, and its nonlinearity, the purpose of an elliptic curve and chaotic maps-based cryptosystem, and how symmetric groups can affect the nonlinearity of an S-box. In the second chapter, the basic structures related to the field of abstract algebra and cryptog raphy are defined according to the literature. In the third chapter, we described how we used the symmetric group S256 to increase the non linearity of the required S-boxes including the symmetric group elements and their effect on the nonlinearity of S-boxes. The complete description of the increase in the nonlinearity of the S-boxes is also given in the tabular form. Lastly, we emphasized the use of the symmetric groups to enhance the working of the S-boxes in the literature so that they can meet the requirements of the present applications of cryptography