Higher Degree Polynomials
Consider polynomial for the ring Z/pkZ. In the same way as for quadratic polynomials one can see:
Lemma: if and i>0, then polynomial g(x) defines a permutation for the elements of the ring Z/pkZ for k>1.
However contrary to the case of the quadratic polynomials the lemma is not if and only if. This can be seen from the following statement.
Lemma: consider finite field Z/pZ for some prime number p. The cubic polynomial defines a permutation if and only if for all it is true that, i.e. the Legendre symbol 
.
Evaluation of the Legendre symbol can be achieved with the help of quadratic reciprocity law.
So one can see that the analysis of higher degree polynomials to define a permutation is a quite subtle question.
Read more about this topic: Permutation Polynomial
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