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:
“But I, being man, can kiss
And bed-spread-eagle too;
All flesh shall come to this,
Being less than angel is,
Yet higher far in bliss
As it entwines with you.”
—William Robert Rodgers (1909–1969)
“It is true that genius takes its rise out of the mountains of rectitude; that all beauty and power which men covet are somehow born out of that Alpine district; that any extraordinary degree of beauty in man or woman involves a moral charm.”
—Ralph Waldo Emerson (1803–1882)