fast Euclidean algorithm

Given two polynomials of degree $n$ with coefficients from a field $K$, the straightforward Eucliean Algorithm uses $O(n^2)$ field operations to compute their greatest common divisor. The Fast Euclidean Algorithm computes the same GCD in $O(𝖬(n)\log (n))$ field operations, where $𝖬(n)$ is the time to multiply two $n$-degree polynomials; with FFT multiplication the GCD can thus be computed in time $O(n{\log }^2(n)\log (\log (n)))$. The algorithm can also be used to compute any particular pair of coefficients from the Extended Euclidean Algorithm, although computing every pair of coefficients would involve $O(n^2)$ outputs and so the efficiency is not as helpful when all are needed.

The algorithm can be made to work over $\mathbb{Z}$ but it is very tricky. A newer version that is easier to understand was published by Damien Stehlé and Paul Zimmerman, “A Binary Recursive Gcd Algorithm.”

Here we sketch the algorithm over $K[x]$. The basic idea is that the quotients $q_i$ computed by the Euclidean Algorithm can usually be computed by looking at only the first few coefficients of the polynomial; for example, if

$$A(x)=a_n{x}^n+a_{n-1}{x}^{n-1}+\mathrm{\dots }+a_0,B(x)=b_{n-1}{x}^{n-1}+\mathrm{\dots }+b_0$$

then

$$quo(A(x),B(x))=\frac{a_n}{b_{n-1}}x+\frac{b_{n-1}{a}_{n-1}-a_n{b}_{n-2}}{b_{n-1}^2}$$

With more detailed analysis, we can show that in fact a divide-and-conquer approach can be used to calculate the GCD. First, we remove the terms whose degree is $n/2$ or less from both polynomials $A$ and $B$. Then, we recursively compute their GCD and Euclidean coefficients. We then apply the Euclidean coefficients to $A$ and $B$, and recursively complete the Euclidean Algorithm.

The full algorithm, and a comprehensive runtime analysis is given in “Modern Computer Algebra” by von zur Gathen and Gerhard.