How can I evaluate the congruency of an AKS primality test? Despite the fact primality test is a mathematical issue, it plays a part on the security of many cryptosystems such as RSA. I was trying to understand how it works until I came to the following congruency: 

〖(X+a)〗^p≡X^p+a (mod X^r-1,p)


The above reduces evaluating the initial congruency 〖(X+a)〗^p≡X^p+a (mod p) to have less coefficient. How do we evaluate the above one?
This is never "evaluated" as such. The above equation must hold for the number to be prime for certain $a$, or else it's composite.

See the algorithm on Wikipedia to exactly see for which $a$ the equation is tested.