Note on fast division algorithm for polynomials using Newton iteration

TL;DR

This paper improves polynomial division algorithms by demonstrating that Newton iteration can directly compute polynomial inverses modulo x^l for non-power-of-two degrees without extra cost, enhancing efficiency.

Contribution

It shows that the original Newton iteration formula can be used directly for non-power-of-two degrees, simplifying and speeding up polynomial division algorithms.

Findings

Newton iteration applies directly to non-power-of-two degrees

Polynomial inverse computation is more efficient without extra steps

Improves classical polynomial division complexity from quadratic to near-linear

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 $x^{\lceil l/2^r\rceil}, x^{\lceil l/2^{r-1}\rceil},..., x^{\lceil l/2\rceil}, x^l$, separately. But they did not specify the iterative step. In this note, 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.