A new algorithm to find monic irreducible polynomials over extended Galois field GF (p q ) using positional arithmetic

Abstract

Search for monic irreducible polynomials (IPs) over extended Galois field GF(p q ) for a large value of the prime moduli p and a large extension to the Galois Field q is a well‐needed solution in the field of cryptography. In this article, a new algorithm to obtain monic IPs over extended Galois field GF(p q ) for the large values of p and q is introduced. Here in this paper the positional arithmetic is used to multiply all possible two monic elemental polynomials (EPs) with their Galois field number (GFN) to generate all the monic reducible polynomials (RPs). All the monic RPs are canceled out from the list of monic basic polynomials (BPs) leaving behind all the monic IPs. Time complexity analysis of the said algorithm is also executed that ensures the algorithm to be less time consuming.