Permutation polynomial

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 $$x \mapsto g(x)$$ is a bijection. In case the ring is a finite field, the Dickson polynomials, which are closely related to the Chebyshev polynomials, provide examples. Over a finite field, every function, so in particular every permutation of the elements of that field, can be written as a polynomial function.

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.

Single variable permutation polynomials over finite fields
Let $F_{q} = GF(q)$ be the finite field of characteristic $p$, that is, the field having $q$ elements where $q = p^{e}$ for some prime $p$. A polynomial $f$ with coefficients in $F_{q}$ (symbolically written as $f ∈ F_{q}[x]$) is a permutation polynomial of $F_{q}$ if the function from $F_{q}$ to itself defined by $$c \mapsto f(c)$$ is a permutation of $F_{q}$.

Due to the finiteness of $F_{q}$, this definition can be expressed in several equivalent ways:
 * the function $$ c \mapsto f(c)$$ is onto (surjective);
 * the function $$c \mapsto f(c)$$ is one-to-one (injective);
 * $f(x) = a$ has a solution in $F_{q}$ for each $a$ in $F_{q}$;
 * $f(x) = a$ has a unique solution in $F_{q}$ for each $a$ in $F_{q}$.

A characterization of which polynomials are permutation polynomials is given by

(Hermite's Criterion) $f ∈ F_{q}[x]$ is a permutation polynomial of $F_{q}$ if and only if the following two conditions hold:
 * 1) $f$ has exactly one root in $F_{q}$;
 * 2) for each integer $t$ with $1 ≤ t ≤ q − 2$ and $$t \not \equiv 0 \!\pmod p$$, the reduction of $f(x)^{t} mod (x^{q} − x)$ has degree $≤ q − 2$.

If $f(x)$ is a permutation polynomial defined over the finite field $GF(q)$, then so is $g(x) = a f(x + b) + c$ for all $a ≠ 0, b$ and $c$ in $GF(q)$. The permutation polynomial $g(x)$ 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 $x^{n&minus;1}$ 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 $$D_n(x,a) = xD_{n-1}(x,a)-a D_{n-2}(x,a), $$ with the initial conditions $$D_0(x,a) = 2$$ and $$D_1(x,a) = x$$. The first few Dickson polynomials are:
 * $F_{q}$ permutes $F_{q}$ if and only if $q$ and $x^{n}$ are coprime (notationally, $GF(q)$).
 * If $n$ is in $q − 1$ and $(n, q − 1) = 1$ then the Dickson polynomial (of the first kind) $GF(q)$ is defined by $$D_n(x,a)=\sum_{j=0}^{\lfloor n/2\rfloor}\frac{n}{n-j} \binom{n-j}{j} (-a)^j x^{n-2j}. $$
 * $$ D_2(x,a) = x^2 - 2a $$
 * $$ D_3(x,a) = x^3 - 3ax$$
 * $$ D_4(x,a) = x^4 - 4ax^2 + 2a^2 $$
 * $$ D_5(x,a) = x^5 - 5ax^3 + 5a^2 x.$$

If $n ≥ 1$ and $D_{n}(x,a)$ then $a ≠ 0$ permutes GF(q) if and only if $n > 1$. If $D_{n}(x, a)$ then $(n, q^{2} − 1) = 1$ and the previous result holds.
 * If $a = 0$ is an extension of $D_{n}(x, 0) = x^{n}$ of degree $a$, then the linearized polynomial $$L(x) = \sum_{s=0}^{r-1} \alpha_s x^{q^s},$$ with $GF(q^{r})$ in $GF(q)$, is a linear operator on $α_{s}$ over $GF(q^{r})$. A linearized polynomial $GF(q^{r})$ permutes $GF(q)$ if and only if 0 is the only root of $L(x)$ in $GF(q^{r})$. This condition can be expressed algebraically as $$ \det\left ( \alpha_{i-j}^{q^j} \right ) \neq 0 \quad (i, j= 0,1,\ldots,r-1).$$

The linearized polynomials that are permutation polynomials over $L(x)$ form a group under the operation of composition modulo $$x^{q^r} - x$$, which is known as the Betti-Mathieu group, isomorphic to the general linear group $GF(q^{r})$.
 * If $GF(q^{r})$ is in the polynomial ring $GL(r, F_{q})$ and $g(x)$ has no nonzero root in $F_{q}[x]$ when $r$ divides $g(x^{s})$, and $GF(q)$ is relatively prime (coprime) to $q − 1$, then $r > 1$ permutes $q − 1$.
 * Only a few other specific classes of permutation polynomials over $x^{r}(g(x^{s}))^{(q - 1)/s}$ have been characterized. Two of these, for example, are: $$ x^{(q + m - 1)/m} + ax $$ where $s$ divides $GF(q)$, and $$ x^r \left(x^d - a\right)^{\left(p^n - 1\right)/d}$$ where $m$ divides $GF(q)$.

Exceptional polynomials
An exceptional polynomial over $q − 1$ is a polynomial in $p^{n} − 1$ which is a permutation polynomial on $GF(q)$ for infinitely many $d$.

A permutation polynomial over $F_{q}[x]$ of degree at most $GF(q^{m})$ is exceptional over $GF(q)$.

Every permutation of $q^{1/4}$ is induced by an exceptional polynomial.

If a polynomial with integer coefficients (i.e., in $GF(q)$) is a permutation polynomial over $GF(q)$ for infinitely many primes $m$, then it is the composition of linear and Dickson polynomials. (See Schur's conjecture below).

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, $ℤ[x]$ with $GF(p)$ 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 $PG(2,q)$.

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 $$f \in \mathbb{F}_q[x_1,\ldots,x_n]$$ is a permutation polynomial in $p$ variables over $$\mathbb{F}_q$$ if the equation $$f(x_1,\ldots,x_n) = \alpha$$ has exactly $$q^{n-1}$$ solutions in $$\mathbb{F}_q^n$$ for each $$\alpha \in \mathbb{F}_q$$.

Quadratic permutation polynomials (QPP) over finite rings
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 $$ g(x) = 2x^2+x $$ for the ring Z/4Z. One sees: $ g(0) = 0$; $ g(1) = 3$; $ g(2) = 2$; $ g(3) = 1 $, so the polynomial defines the permutation $$\begin{pmatrix} 0 &1 & 2 & 3 \\ 0 &3 & 2 & 1 \end{pmatrix} .$$

Consider the same polynomial $$ g(x) = 2x^2+x $$ for the other ring Z/8Z. One sees: $ g(0) = 0$; $ g(1) = 3$; $ g(2) = 2$; $ g(3) = 5$; $ g(4) = 4$; $ g(5) = 7$; $ g(6) = 6$; $ g(7) = 1$, so the polynomial defines the permutation $$\begin{pmatrix} 0 &1 & 2 & 3 & 4 & 5 & 6 & 7 \\ 0 &3 & 2 & 5 & 4 & 7 & 6 & 1 \end{pmatrix} .$$

Rings Z/pkZ
Consider $$ g(x) = ax^2+bx+c $$ for the ring Z/pkZ.

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 $$a \equiv 0 \pmod p$$ and $$b \not \equiv 0 \pmod p$$.

Rings Z/nZ
Consider $$n=p_1^{k_1}p_2^{k_2}...p_l^{k_l}$$, where pt are prime numbers.

Lemma: any polynomial $ g(x) = a_0+ \sum_{0 < i \leq M} a_i x^i $ defines a permutation for the ring Z/nZ if and only if all the polynomials $ g_{p_t}(x) = a_{0,p_t}+ \sum_{0 < i \leq M} a_{i,p_t} x^i $  defines the permutations for all rings $$Z/p_t^{k_t}Z$$, where $$a_{j,p_t}$$ are remainders of $$a_{j}$$ modulo $$p_t^{k_t}$$.

As a corollary one can construct plenty quadratic permutation polynomials using the following simple construction. Consider $$n = p_1^{k_1} p_2^{k_2} \dots p_l^{k_l}$$, assume that k1 >1.

Consider $$ax^2+bx$$, such that $$ a= 0 \bmod p_1$$, but $$ a\ne 0 \bmod p_1^{k_1}$$; assume that $$ a = 0 \bmod p_i^{k_i}$$, i > 1. And assume that $$b\ne 0 \bmod p_i$$ for all $q$. (For example, one can take $$ a=p_1 p_2^{k_2}...p_l^{k_l} $$ and $$b=1$$). 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 $GF(q)$ and polynomial $$6x^2+x$$. It defines a permutation

Higher degree polynomials over finite rings
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 $$g'(x) \ne 0 \bmod p$$ for all x in Z/pkZ, where g&prime;(x) is the formal derivative of g(x).

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.