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

Abstract

Substitution boxes or S-boxes play a significant role in encryption and decryption of bit level plaintext and cipher-text respectively. Irreducible Polynomials (IPs) have been used to construct 4-bit or 8-bit substitution boxes in many cryptographic block ciphers. In Advance Encryption Standard, the ele-

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

“…The algebraic method or the above pseudo code has been tested on GF(3 3 ),GF(7 3 ),GF(11 3 ), GF(101 3 ), GF(3 5 ), GF(7 5 ), GF(3 7 ), GF(7 7 ),. Number of Monic IPs given by this algorithm is same as in hands on calculation by the theorem to count Monic IPs over Galois Field GF(p q ) [8].…”

“…• Dey and Ghosh. 5 In this algorithm the decimal equivalents of each of two monic EPs over the Galois field GF(p q ) at a time with highest degree d and (q-d) where d € {0,… ,(q-1)/2}, have been split into the p-nary coefficients of each term of two said monic EPs over the Galois field GF(p q ). The coefficients of each term in each two Monic EPs or two GFNs are multiplied, added respectively with each other and modulated to obtain the p-nary coefficients of each term of the RPs over the Galois field GF(p q ).…”

“…The DE of BPs over the Galois field GF(p q ) belonging to the list of RPs over the Galois field GF(p q ) have been canceled leaving behind the monic IPs over the Galois field GF(p q ). 5 Now, here a new multiplication algorithm is introduced to multiply two monic EPs over the Galois field GF(p q ). The procedure is same as decimal multiplication but the each digit in product must be modulated with prime modulus p and the quotient is considered as carry to obtain the result.…”