**Finite
fields, Permutation Polynomials.**

**Computational
aspects with**

**applications****
to public key cryptography**

We will start from finite fields, their construction and their interpolation properties. This will lead to permutation polynomials. First we will review a few basic properties and examples. Next, we will explain a possible use of permutation polynomials in cryptography. This will require recalling the classical Diffie Hellmann key exchange protocol. Finally we will propose the standard enumeration problems for permutation polynomials and some results about them.