A New Algebraic Method to Search Irreducible Polynomials Using Decimal Equivalents of Polynomials over Galois Field GF(p^q)

TL;DR

This paper introduces a novel algebraic method for identifying monic irreducible polynomials over Galois fields GF(p^q) by using decimal equivalents and p-nary coefficients, aiding cryptographic constructions.

Contribution

The paper presents a new algebraic approach to efficiently search for monic irreducible polynomials over GF(p^q) using decimal and p-nary representations, improving existing methods.

Findings

Successfully identifies monic irreducible polynomials over GF(p^q)

Provides a systematic cancellation method for reducible polynomials

Enables computation of non-monic irreducible polynomials by multiplication

Abstract

Irreducible polynomials play an important role till now, in construction of 8-bit S-Boxes in ciphers. The 8-bit S-Box of Advanced Encryption Standard is a list of decimal equivalents of Multiplicative Inverses (MI) of all the elemental polynomials of a monic irreducible polynomial over Galois Field GF(2^8) [1]. In this paper a new method to search monic Irreducible Polynomials (IPs) over Galois fields GF(p^q) has been introduced. Here the decimal equivalents of each monic elemental polynomial (ep), two at a time, are split into the p-nary coefficients of each term, of those two monic elemental polynomials. From those coefficients the p-nary coefficients of the resultant monic basic polynomials (BP) have been obtained. The decimal equivalents of resultant basic polynomials with p-nary coefficients are treated as decimal equivalents of the monic reducible polynomials, since monic reducible polynomials must have two monic elemental polynomials as its factor. The decimal equivalents of polynomials belonging to the list of reducible polynomials are cancelled leaving behind the monic irreducible polynomials. A non-monic irreducible polynomial is computed by multiplying a monic irreducible polynomial by alpha where alpha belongs to GF(p^q) and assumes values from 2 to (p-1).