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
Famous quotes containing the words higher and/or degree:
“There is a higher law affecting our relation to pines as well as to men. A pine cut down, a dead pine, is no more a pine than a dead human carcass is a man.”
—Henry David Thoreau (1817–1862)
“I cannot see how a man of any large degree of humorous perception can ever be religious—except he purposely shut the eyes of his mind & keep them shut by force.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)