Fast Hensel's lifting implementation using partial fraction decomposition

When factoring polynomials defined over Z[x], one computes the factors modulo m from the factors modulo p, where p is a suitable prime number and m a power of p, greater than rn, a bound on the absolute value of the coefficient of any factor of the polynomial. This is done using the Hensel lemma. We present a new way to perform this lifting process based on the idea of partial fraction decomposition (denoted by p.f.d. in the following). Section 3 contains a presentation of the classical Hensel’s method, Section 4 presents Kung and Tong’s algorithm for p.f.d. and describes how Viry [6] suggests to use the p.f.d. for the lifting process. Section 5 describes the new algorithm based on Viry’s idea and Kung and Tong’s algorithm. An analysis of complexity is given and the advantages of the new algorithm are discussed. Computing times help us to compare the new algorithm to the two different ways of performing the Hensel’s lifting, the serial one and the parallel one. These measurements show that the new algorithm is more efficient.