Factorization of the Degree of Semisimple Polynomials Over the Galois Fields of Arbitrary Characteristics

Abstract

To semisimple polynomials over a Galois field of arbitrary characteristics we mean polynomials formed by the product of two coprime irreducible polynomials with a priori unknown degrees. The main task of this study is to develop an efficient algorithm for factorizing the degree of semisimple polynomials. The efficient factorization algorithms are those that provide a minimum of computational complexity. The proposed algorithm is reduced to solving a system of two equations for the unknown degrees of the factors of a semisimple polynomial. The right-hand sides of the system of equations are as follows: one of them is the degree n of a semisimple polynomial, known a priori, and the second, the cycle period C of the polynomial, is calculated using the so-called fiducial grid. At each rung of the ladder, the simplest recurrent modular computations are carried out, after which the cycle period C of the semisimple polynomial is determined, which is equal to the least common multiple of the degrees of the factors of the polynomial. Reducing the amount of calculations is achieved by switching from a linear scale when determining the cycle period C to a logarithmic one. The proposed factorization algorithms are invariant to the characteristic of the field generated by irreducible polynomials. Various options for the relationship between the parameters n and C are considered.