I'm looking for what the title says. Euclidean algorithm works and is fast for just a pair of numbers, but I don't see any obvious generalizations. A quick googling didn't turn up anything too ...

Niels Möller, On Schönhage's Algorithm and Subquadratic Integer GCD Computation, Mathematics of Computation, Vol. 77, No. 261 (Jan., 2008), pp. 589-607 ...

The researchers used a cloud-hosted Microsoft Azure virtual machine and a scalable GCD algorithm for shared factors to analyze a large collection of 175 million RSA certificates.