Abstract

Substitution boxes or S-boxes play a significant role in encryption and de-cryption of bit level plaintext and cipher-text respectively. Irreducible Poly-nomials (IPs) have been used to construct 4-bit or 8-bit substitution boxes in many cryptographic block ciphers. In Advance Encryption Standard, the ele-ments of 8-bit S-box have been obtained from the Multiplicative Inverse (MI) of elemental polynomials (EPs) of the 1st IP over Galois field GF(28) by adding an additive element. In this paper, a mathematical method and the algorithm of the said method with the discussion of the execution time of the algorithm, to obtain monic IPs over Galois field GF(pq) have been illustrated with example. The method is very similar to polynomial multiplication of two polynomials over Galois field GF(pq) but has a difference in execution. The decimal equivalents of polynomials have been used to identify Basic Polynomials (BPs), EPs, IPs and Reducible polynomials (RPs). The monic RPs have been determined by this method and have been cancelled out to produce monic IPs. The non-monic IPs have been obtained with multiplication of α where α∈ GF(pq) and assume values from 2 to (p − 1) to monic IPs.

Highlights

  • Substitution box or S-box in block ciphers is of utmost importance in Public Key Cryptography from the initial days

  • In Advance Encryption Standard, the elements of 8-bit S-box have been obtained from the Multiplicative Inverse (MI) of elemental polynomials (EPs) of the 1st Irreducible Polynomials (IPs) over Galois field GF(28) by adding an additive element

  • The monic Reducible polynomials (RPs) have been determined by this method and have been cancelled out to produce monic IPs

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Summary

Introduction

Substitution box or S-box in block ciphers is of utmost importance in Public Key Cryptography from the initial days. For 8 bit S-box, the number of elements is 28 or 256, varying from 0 to 255 as used in Advance Encryption Standard or AES [5] [6]. The construction of S-boxes is a major issue in Cryptology from initial days. The search for monic as well as mon-monic IPs has been the untouched stone to break in cryptography

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