Mathematical Method to Search for Monic Irreducible Polynomials with Decimal Equivalents of Polynomials over Galois Field GF(pq)

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 α where α ∈GF(p q ) and assumes values from 2 to (p-1).General Terms: Algorithms, Irreducible polynomial.

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

“…• Dey and Ghosh. 4 Here the procedure to multiply of GFNs of two polynomials over the Galois field GF(p q ) is illustrated.…”

“…The first IP over the binary Galois field GF (2 8 ) is used to generate the elements of the substitution box in Advanced Encryption Standard. 7 IPs over the binary Galois field GF(2 4 ) are also used to generate crypto 4-bit S-boxes later. 6 The irreducible polynomials with large values of prime p and extension q can also be used to generate crypto 4-bit, 8-bit, 32 bit as well as 64 bit S-boxes in the same manner.…”

“…Then the obtained digits or coefficients with same degree terms are added and modulated with p to obtain the resultant GFN or the coefficients of the resultant polynomial over the Galois field GF(p q ). 4 At last according to Dey and Ghosh 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 GFNs 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 GFN of the RPs over the Galois field GF(p q ).…”

Intervalo de dobleces

“…Siendo todavía estudiante de secundaria, Gallivan demostró que un único trozo de papel puede ser doblado por la mitad doce veces. El 27 de enero de 2002, después de 8 horas de trabajo duro por parte de tres personas, un papel de seda (1,219.2 m de largo, 0.008382 m de espesor) se dobló 12 veces por la mitad por primera vez en Pomona, California, (Dey & Ghosh, 2017), (Folding --from Wolfram MathWorld, n.d.). ¡Esto demostró que la creencia generalizada de que no es posible doblar un pedazo de papel en la mitad más de ocho veces era falsa!…”