I was playing with an algorithm which at one step, calculated $f(x) = x^{-1} \mod p$ for $0 < x < p$. I used Knuth's algorithm for calculating the Extended GCD, but I was wondering if there was a trick for making it run in constant time, within a word?

The trick I found online used Montgomery Multiplication, and therefore wouldn't fit within a 64-bit word.

My objective is to write a 64-bit cipher which (is probably insecure, definitely slow, and) uses inversion for confusion and diffusion.