In mathematics, a permutation polynomial (for a given ring) is a polynomial that acts as a permutation of the elements of the ring, i.e. the map
Contents
- Quadratic permutation polynomials QPP
- Simple examples
- Rings ZpkZ
- Rings ZnZ
- Higher degree polynomials
- Finite fields
- Geometric examples
- Schurs conjecture
- Computational complexity
- References
In the case of finite rings Z/nZ, such polynomials have also been studied and applied in the interleaver component of error detection and correction algorithms.
Quadratic permutation polynomials (QPP)
For the finite ring Z/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 Evolution mobile telecommunication standard (see 3GPP technical specification 36.212 e.g. page 14 in version 8.8.0).
Simple examples
Consider
Let us consider the same polynomial
Rings Z/pkZ
Consider
Lemma: for k=1 (i.e. Z/pZ) such polynomial defines a permutation only in the case a=0 and b not equal to zero. So the polynomial is not quadratic, but linear.
Lemma: for k>1, p>2( Z/pkZ) such polynomial defines a permutation if and only if
Rings Z/nZ
Consider
Lemma: any polynomial
As a corollary one can construct plenty quadratic permutation polynomials using the following simple construction. Consider
Consider
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
Higher degree polynomials
A polynomial g(x) for the ring Z/pkZ is a permutation polynomial if and only if it permutes the finite field Z/pZ and
Lemma: consider the finite field Z/pZ for some prime number p. The cubic polynomial
Evaluation of the Legendre symbol can be achieved with the help of quadratic reciprocity law.
Finite fields
There are many open questions concerning permutation polynomials defined over finite fields (see Lidl & Mullen (1988) and Lidl & Mullen (1993)).
If f(x) is a permutation polynomial defined over the finite field GF(q), where q = pe for some prime p, then so is g(x) = a f(x + b) + c for all a ≠ 0, b and c in GF(q). We say that g(x) is in normalized form if a, b and c are chosen so that g(x) is monic, g(0) = 0 and (provided the characteristic p does not divide the degree n of the polynomial) the coefficient of xn-1 is 0.
The following list, while not exhaustive, contains almost all of the known major classes of permutation polynomials over finite fields.
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, PG(2,q) with q 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 GF(q).
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 ideals P, 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.
Computational complexity
The problem of testing whether a given polynomial over a finite field is a permutation polynomial can be solved in polynomial time.