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:
“The higher we rise up, the smaller we appear to those who are unable to fly.”
—Friedrich Nietzsche (1844–1900)
“There is always a degree of ridicule that attends a disappointment, though often very unjustly, if the expectation was reasonably grounded; however, it is certainly most prudent not to communicate, prematurely, one’s hopes or one’s fears.”
—Philip Dormer Stanhope, 4th Earl Chesterfield (1694–1773)