for really big numbers there are the generalisations of karatsuba and toom-cook for higher numbers of splits, and beyond that there are the FFT multiplication methods which i believe are the fastest general case methods for very large numbers.

also the laddering scheme used for the 'exponentiation' can be in different forms, the left-to-right or right-to-left form or even a Montgomery ladder...

i seemed to recall there is a 'fastest known' method for generating fibonacci or lucas numbers... but google is not helping me.

pretty sure i had seen it in this book: https://www.amazon.co.uk/Prime-Numbers-Computational-Carl-Po... i'll have to check when i have access to it next. :)