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 $x^{\lceil l/2^r\rceil}, x^{\lceil l/2^{r-1}\rceil}, \cdots, x^{\lceil l/2\rceil}, x^ l$, 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.