On fast division algorithm for polynomials using Newton iteration

Abstract

The classical division algorithm for polynomials requires O(n 2) operations for inputs of size n. Using reversal technique and Newton iteration, it can be improved to O(M(n)), where M is a multiplication time. But the method requires that the degree of the modulo, x l , should be the power of 2. If l is not a power of 2 and f(0)∈=∈1, Gathen and Gerhard suggest to compute the inverse, f -∈1, modulo x ⌈l/2r⌉, x ⌈l/2r-1⌉,..., x ⌈l/2⌉, x l, separately. But they did not specify the iterative step. In this paper, we show that the original Newton iteration formula can be directly used to compute f -1 mod x l without any additional cost, when l is not a power of 2. © 2012 Springer-Verlag.