The GMP development page
http://gmplib.org/devel/
has some code for fast GCD/XGCD of huge operands. It might not appear for a while in official releases, but they claim it's stable, and I'm interested in patching it into SAGE. I tried to do this and got stuck on configuration problems that a linux/GNU guru could help with.

Component: basic arithmetic

Issue created by migration from https://trac.sagemath.org/ticket/424