Permutation Polynomial - Higher Degree Polynomials

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.