An Algorithm to Find the Irreducible Polynomials Over Galois Field GF(pm)

Abstract

Irreducible Polynomials over GF(p m) and the multiplicative inverses under it are important in cryptography. Presently the method of deriving irreducible polynomials of a particular prime modulus is very primitive and time consuming. In this paper, in order to find all irreducible polynomials, be it monic or non-monic, of all prime moduli p with all its order m, a fast deterministic computer algorithm based on an algebraic method producing a (m×m) matrix is proposed. The maximum number of terms in each column of the matrix is 2 j where j is the column index.

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

“…If for any division the residue is 1 then the two monic EPs over the Galois field GF(p q ) are multiplicative inverses (MIs) of each other. 3 In the contribution of Dey and Ghosh the procedure to multiply of GFNs of two polynomials over the Galois field GF(p q ) is illustrated. The each digit of a GFN or the coefficients of each degree term of the polynomial over the Galois field GF(p q ) are multiplied to all digits of other GFN consecutively.…”

Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field <I>GF</I>(<I>p<sup>q</sup></I>)

“…The residue of division of BPs by IPs must be 1 and this reported to literature a bit later [47]. The IPs with several coefficients of different categories had been illustrated in literature a bit later [48].…”

A review of cryptographic properties of S-boxes with generation and analysis of crypto secure S-boxes.

“…Polynomials over Finite field or Galois field GF(p q ) have been of utmost importance in Public Key Cryptography [BS96]. The polynomials over Galois field GF(p q ) with degree q have been termed as Basic Polynomials or BPs over Galois field GF(p q ) and Polynomials with degree <q have been termed as Elemental Polynomials or EPs over Galois field GF(p q ) [SJ15]. The EPs over Galois field GF(p q ) with only constant terms have been termed as Constant Polynomials or CPs over Galois field GF(p q ).…”

“…The EPs over Galois field GF(p q ) with only constant terms have been termed as Constant Polynomials or CPs over Galois field GF(p q ). The BPs over Finite field or Galois field GF(p q ) that cannot be factored into at least two non-constant EPs have been termed as Irreducible polynomials or IPs over Finite field or Galois field GF(p q ) and the rest have been termed as Reducible polynomials or RPs over Finite field or Galois field GF(p q ) [SJ15]. The polynomials over Galois field GF(p q ) with coefficient of the highest degree term as 1 have been termed as monic polynomials over Galois field GF(p q ) and rest have been termed as nonmonic Polynomials over Galois field GF(p q ) [SJ15].…”

“…An algorithm to search for primitive polynomials had also been notified at the same time [WJ14]. The residue of division of BPs by IPs must be 1 and this reported to literature a bit later [SJ15]. The IPs with several coefficients of different categories had been illustrated in literature a bit later [HJ16].…”