# 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 $O(\mathsf{M}(n)\log(n))$ field operations, where $\mathsf{M}(n)$ 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

 $A(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\ldots+a_{0},\quad B(x)=b_{n-1}x^{n-1}+\ldots+b% _{0}$

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.

Title fast Euclidean algorithm FastEuclideanAlgorithm 2013-03-22 14:28:52 2013-03-22 14:28:52 mathcam (2727) mathcam (2727) 7 mathcam (2727) Algorithm msc 11A05 Half-GCD Algorithm Lehmer’s Algorithm EuclidsAlgorithm