What are some applications of binary finite fields in CS?
Finite fields come up in many places. Here are just a few examples:


The Razborov-Smolensky polynomial method.
Fourier analysis, as used for example in the proof of the PCP theorem, or fast integer multiplication.
List decoding - codes like Reed-Muller are algebraic codes.
Algebraization, the method used to prove IP=PSPACE.
Elliptic curves over finite fields are used in cryptography.