FACTORIZATION OF DEGREE k-ALMOST SIMPLE POLYNOMIALS

Abstract

In this work, we introduce the notion of k-almost prime polynomials formed by the product over a Galois field of an arbitrary characteristic p of exactly k irreducible polynomials. The study aims to develop an effective algorithm for factorizing the degree of such composite polynomials providing a minimum of computational complexity. The proposed algorithm is based on the solution of a system of two equations functionally related to the a priori unknown degree of the components of the compound polynomial. One of these equations reflects that the degree of the composite polynomial equals the sum of degrees of elements of this polynomial. The second equation is based on the so-called cycle period of the compound polynomial, which coincides with the least common multiple of the unknown degrees of the components of the compound polynomial. It is found that the expansion of the cycle period contains all the degrees of the factorizable compound polynomial. The computational volume reduction is achieved by switching from the linear scale of the multiplicative group of subtractions generated by the cycle period of the compound polynomial to the logarithmic scale.