4, 8, 32, 64 bit Substitution Box generation using Irreducible or Reducible Polynomials over Galois Field GF(p^q) for Smart Applications

Substitution Box or S-Box had been generated using 4-bit Boolean Functions (BFs) for Encryption and Decryption Algorithm of Lucifer and Data Encryption Standard (DES) in late sixties and late seventies respectively. The SBox of Advance Encryption Standard have also been generated using Irreducible Polynomials over Galois field GF(2 8 ) adding an additive constant in early twenty first century. In this paper Substitution Boxes have been generated from Irreducible or Reducible Polynomials over Galois field GF(p q ). Binary Galois fields have been used to generate Substitution Boxes. Since the Galois Field Number or the Number generated from coefficients of a polynomial over a particular Binary Galois field (2 q ) is similar to log 2 q+1 bit BFs. So generation of log 2 q+1 bit S-Boxes is Possible. Now if p = prime or non-prime number then generation of S-Boxes is possible using Galois field GF (p q ). where, q = p-1.