An

irreducible polynomial is polynomial that cannot be written as a product of nontrivial polynomials over the same field.

and an

irreducible polynomial. These

irreducible polynomials are, in the indicated order (k = -1,0,1),

Since in cryptography, an S-box is the salient component used to produce confusion in the data, it is worth studying that the confusion creating ability is associated with the choice of the

irreducible polynomial used to form the background Galois field.

This analysis is based on the study of the mathematical model of polynomial representation, in which it is stated: If p(x) is the

irreducible polynomial, then the multiplication of two elements of the field, represented as the polynomials A(x) and B(x) is the algebraic product of the two polynomials, and the modulus operation of the polynomial p(x), also known as modular reduction, is that shown in equation 1.

Mix Columns: This is a substitution that makes use of arithmetic over GF (28), with the

irreducible polynomial m(x) = x8 + x4 + x3 + x +1.

Let [alpha] be a root of an

irreducible polynomial g(x) of degree n over GF(2).

Therefore we use locator set [mathematical expression not reproducible], where [mathematical expression not reproducible] is formal derivative of denominator [mathematical expression not reproducible] and [mathematical expression not reproducible] is an

irreducible polynomial on [mathematical expression not reproducible].

As a rule linear automaton follows the structure of an

irreducible polynomial p(x) of degree m= deg p(x).

Known results on orthogonal systems and permutation polynomials vectors over finite fields are extended to modular algebras of the form [L.sub.v] = K[X] (p[(X).sup.v]), where K is a finite field, p(X) [member of] K[X] is an

irreducible polynomial, v = 1 2, and to the algebra of formal power series L[[Z]], where [L.sub.1] = K[X] (p(X)) = L.

Let x be a root of the polynomial, then the

irreducible polynomial G is represented as a following equation.

Among the topics are experimental computation with oscillatory integrals, expressions for harmonic number exponential generating functions, a new algorithm, for the recursion of hyper-geometric multi-sums with improved universal denominator, an algorithmic approach to the Mellin transform method, the distance to an

irreducible polynomial, towards an automation of the circle method, and experimentation at the frontiers of reality in Schubert calculus.

(1) Let n = 5k, for k odd, and in GF([2.sup.5]), let [x.sup.5] + [x.sup.2] + 1 = 0, be the

irreducible polynomial. If e = 1 + x + [x.sup.3], b = [x.sup.2], c = [x.sup.3] then the semifield (S, +, *) of order [2.sup.5k] admits a subfield isomorphic to GF(4).