Single variable permutation polynomials over finite fields
Let be the finite field of characteristic, that is, the field having elements where for some prime. A polynomial with coefficients in is a permutation polynomial of if the function from to itself defined by is a permutation of. Due to the finiteness of, this definition can be expressed in several equivalent ways: A characterization of which polynomials are permutation polynomials is given by is a permutation polynomial of if and only if the following two conditions hold:
has exactly one root in ;
for each integer with and, the reduction of has degree.
If is a permutation polynomial defined over the finite field, then so is for all and in. The permutation polynomial is in normalized form if and are chosen so that is monic, and the coefficient of is 0. There are many open questions concerning permutation polynomials defined over finite fields.
Small degree
Hermite's criterion is computationally intensive and can be difficult to use in making theoretical conclusions. However, Dickson was able to use it to find all permutation polynomials of degree at most five over all finite fields. These results are: A list of all monic permutation polynomials of degree six in normalized form can be found in.
Some classes of permutation polynomials
Beyond the above examples, the following list, while not exhaustive, contains almost all of the known major classes of permutation polynomials over finite fields.
These can also be obtained from the recursion with the initial conditions and. The first few Dickson polynomials are: If and then permutes GF if and only if. If then and the previous result holds.
The linearized polynomials that are permutation polynomials over form a group under the operation of composition modulo, which is known as the Betti-Mathieu group, isomorphic to the general linear group.
If is in the polynomial ring and has no nonzero root in when divides, and is relatively prime to, then permutes.
Only a few other specific classes of permutation polynomials over have been characterized. Two of these, for example, are:
Exceptional polynomials
An exceptional polynomial over is a polynomial in which is a permutation polynomial on for infinitely many. A permutation polynomial over of degree at most is exceptional over. Every permutation of is induced by an exceptional polynomial. If a polynomial with integer coefficients is a permutation polynomial over for infinitely many primes, then it is the composition of linear and Dickson polynomials..
Geometric examples
In finite geometry coordinate descriptions of certain point sets can provide examples of permutation polynomials of higher degree. In particular, the points forming an oval in a finite projective plane, with a power of 2, can be coordinatized in such a way that the relationship between the coordinates is given by an o-polynomial, which is a special type of permutation polynomial over the finite field.
Computational complexity
The problem of testing whether a given polynomial over a finite field is a permutation polynomial can be solved in polynomial time.
Permutation polynomials in several variables over finite fields
A polynomial is a permutation polynomial in variables over if the equation has exactly solutions in for each.
Quadratic permutation polynomials (QPP) over finite rings
For the finite ringZ/nZ one can construct quadratic permutation polynomials. Actually it is possible if and only if n is divisible by p2 for some prime number p. The construction is surprisingly simple, nevertheless it can produce permutations with certain good properties. That is why it has been used in the interleaver component of turbo codes in 3GPP Long Term Evolutionmobile telecommunication standard.
Simple examples
Consider for the ring Z/4Z. One sees:, so the polynomial defines the permutation Consider the same polynomial for the other ring Z/8Z. One sees:, so the polynomial defines the permutation
Rings Z/''pk''Z
Consider for the ring Z/pkZ. Lemma: for k=1 such polynomial defines a permutation only in the case a=0 and bnot equal to zero. So the polynomial is not quadratic, but linear. Lemma: for k>1, p>2 such polynomial defines a permutation if and only if and.
Rings Z/''n''Z
Consider, where pt are prime numbers. Lemma: any polynomial defines a permutation for the ring Z/nZ if and only if all the polynomials defines the permutations for all rings, where are remainders of modulo . As a corollary one can construct plenty quadratic permutation polynomials using the following simple construction. Consider, assume that k1 >1. Consider, such that, but ; assume that,i>1. And assume that for all i=1...l. . Then such polynomial defines a permutation. To see this we observe that for all primes pi,i>1, the reduction of this quadratic polynomial modulo pi is actually linear polynomial and hence is permutation by trivial reason. For the first prime number we should use the lemma discussed previously to see that it defines the permutation. For example, consider Z/12Z and polynomial. It defines a permutation
A polynomial g for the ring Z/pkZ is a permutation polynomial if and only if it permutes the finite field Z/pZ and for all x in Z/pkZ, where g′ is the formal derivative of g.
Schur's conjecture
Let K be an algebraic number field with R the ring of integers. The term "Schur's conjecture" refers to the assertion that, if a polynomial f defined over K is a permutation polynomial on R/P for infinitely many prime idealsP, then f is the composition of Dickson polynomials, degree-one polynomials, and polynomials of the form xk. In fact, Schur did not make any conjecture in this direction. The notion that he did is due to Fried, who gave a flawed proof of a false version of the result. Correct proofs have been given by Turnwald and Müller.
