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:
“If all power is in the people, if there is no higher law than their will, and if by counting their votes, their will may be ascertained—then the people may entrust all their power to anyone, and the power of the pretender and the usurper is then legitimate. It is not to be challenged since it came originally from the sovereign people.”
—Walter Lippmann (1889–1974)
“To the degree that respect for professors ... has risen in our society, respect for writers has fallen. Today the professorial intellect has achieved its highest public standing since the world began, while writers have come to be called “men of letters,” by which is meant people who are prevented by some obscure infirmity from becoming competent journalists.”
—Robert Musil (1880–1942)