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Substitution Box or S-Box had been generated using 4-bit Boolean Functions(BFs) for Encryption and Decryption Algorithm of Lucifer and Data EncryptionStandard (DES) in late sixties and late seventies respectively. The S-box ofAdvance Encryption Standard have also been generated using IrreduciblePolynomials over Galois field GF(2^8) adding an additive constant in earlytwenty first century. In this paper Substitution Boxes have been generated fromIrreducible or Reducible Polynomials over Galois field GF(p^q). Binary Galoisfields have been used to generate Substitution Boxes. Since the Galois FieldNumber or the Number generated from coefficients of a polynomial over aparticular Binary Galois field (2q) is similar to log 2 q+1 bit BFs. Sogeneration of log 2 q+1 bit S-boxes is Possible. Now if p = prime or non-primenumber then generation of S-Boxes is possible using Galois field GF (p^q).where, q = p-1.

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