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

Abstract: 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 S-box 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 (2q) 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.