ALGORITHMS FOR FACTORIZATION AND IRREDUCIBILITY TESTING OF POLYNOMIALS USING ELLIPTIC CURVES

Abstract

In this article irreducibility criteria and factorization algorithms are formulated and proved, that are generalizations of correspondent criteria and algorithms for integers. Generalization of Lenstra’s theorem is of especial interest. It provides correct calculations of multiple points on elliptic curve that is used in Lenstra algorithm. Note that the generalization of this theorem in case of finite field of characteristic 2 is completely different from the classical case and is very non-trivial. Using the criteria obtained, we construct a series of algorithms for factoring and testing of irreducibility of polynomials over finite field. We should note that their performance isn’t their main advantage, but the main advantage is using some abelian group built on elliptic curve. So in case of failure of algorithm we can just chose the other elliptic curve over the corresponding field and repeat the algorithm. These algorithms have no large advantage in speed, like their preimages – analogical algorithms for integers, but they demonstrate new approaches to the solutions of important problem of irreducibility testing and also are very interesting from mathematical point of view.