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

TL;DR

This paper presents a method for generating substitution boxes (S-boxes) of various bit sizes using irreducible or reducible polynomials over Galois fields GF(p^q), expanding cryptographic options beyond traditional binary fields.

Contribution

It introduces a novel approach to generate S-boxes of 4, 8, 32, and 64 bits from Galois fields GF(p^q) using both irreducible and reducible polynomials, generalizing previous methods.

Findings

S-boxes of 4, 8, 32, and 64 bits can be generated from GF(p^q).

Generation is possible for prime and non-prime p values.

The method extends cryptographic S-box design beyond binary fields.

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.