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Irreducible polynomials play an important role till now, in construction of8-bit S-Boxes in ciphers. The 8-bit S-Box of Advanced Encryption Standard is alist of decimal equivalents of Multiplicative Inverses (MI) of all theelemental polynomials of a monic irreducible polynomial over Galois FieldGF(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 decimalequivalents of each monic elemental polynomial (ep), two at a time, are splitinto the p-nary coefficients of each term, of those two monic elementalpolynomials. From those coefficients the p-nary coefficients of the resultantmonic basic polynomials (BP) have been obtained. The decimal equivalents ofresultant basic polynomials with p-nary coefficients are treated as decimalequivalents of the monic reducible polynomials, since monic reduciblepolynomials must have two monic elemental polynomials as its factor. Thedecimal equivalents of polynomials belonging to the list of reduciblepolynomials are cancelled leaving behind the monic irreducible polynomials. Anon-monic irreducible polynomial is computed by multiplying a monic irreduciblepolynomial by alpha where alpha belongs to GF(p^q) and assumes values from 2 to(p-1).

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