Obtaining Algebraic expression of an S-box Based on Inversion Mapping Using Finite Field Theory

Recently proposed block ciphers like AES, Square, Shark use S-boxes that are based on inversion mapping over a Unite field F = CF(p"). Because of the simple algebraic structure of S-boxes generated in this way, these ciphers usually use a bitwise affine transformation after the inversion mapping. In this study, we show how we obtain algebraic expression of an S-box based on inversion mapping using finite field theory and trace function combined with theoretical preliminaries related with this theory.