What are some applications of binary finite fields in CS? I was looking at details on finite fields. Finite binary fields, e.g. $\mathbb{F_2}$, are used in CS in some places such as circuit theory [1]. 


  What are some key applications of finite fields in CS?


I am also looking for uses of $\mathbb{F_{2}^n}$ which Mathworld shows can be represented as binary vectors.



[1] Noga Alon and Gil Cohen. On Rigid Matrices and Subspace Polynomials. Electronic Colloquium on Computational Complexity, Report No. 133 (2012).
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.