Is there any popular robust implementation for a finite field in java? (An implementation of prime fields ,i.e. $GF(p)$ might also be good.
You can check Rings Java library (http://ringsalgebra.io). It has quite efficient support for Galois fields $GF(p, k)$ (both for small and for arbitrary large $p$).
You could look at Givaro c++ library for finite fields (I used it, and it is quite okay, and source code is accessible), and try to adapt it to java, because for java I am afraid you have nothing analogue. (Last time I looked for this, I found only some buyable library, and for $p=2$ only.)