Permutation polynomial explained

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.^{[1] [2]}

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 (symbolically written as) is a permutation polynomial of if the function from to itself defined by

c \mapsto f(c) is a permutation of .

Due to the finiteness of, 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);

• has a solution in for each in ;
• has a unique solution in for each in .

A characterization of which polynomials are permutation polynomials is given by

(Hermite's Criterion) is a permutation polynomial of if and only if the following two conditions hold:

1. has exactly one root in ;
2. for each integer with and

t \not \equiv 0 \pmod p , 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 (provided the characteristic does not divide the degree of the polynomial) 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:

Normalized Permutation Polynomial of
x	any q
x ^{2}	q \equiv 0 \pmod 2
x ^{3}	q \not \equiv 1 \pmod 3
x ^{3} - ax (a not a square)	q \equiv 0 \pmod 3
x ^{4} \pm 3 x	q = 7
x ^{4} + a _{1} x ^{2} + a _{2} x (if its only root in is 0)	q \equiv 0 \pmod 2
x ^{5}	q \not \equiv 1 \pmod 5
x ^{5} - ax (a not a fourth power)	q \equiv 0 \pmod 5
x ^{5} + ax (a ^{2} = 2 )	q = 9
x ^{5} \pm 2 x ^{2}	q = 7
x ^{5} + ax ^{3} \pm x ^{2} + 3 a ^{2} x (a not a square)	q = 7
x ^{5} + ax ^{3} + 5 ^{-1} a ^{2} x (a arbitrary)	q \equiv \pm 2 \pmod 5
x ^{5} + ax ^{3} + 3 a ^{2} x (a not a square)	q = 13
x ^{5} - 2 ax ^{3} + a ^{2x} (a not a square)	q \equiv 0 \pmod 5

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.

• permutes if and only if and are coprime (notationally,).
• If is in and then the Dickson polynomial (of the first kind) is defined by $$D\_n(x,a)=\backslash sum\_^\backslash frac\; \backslash binom\; (-a)^j\; x^.$$

These can also be obtained from the recursion$$D\_n(x,a)\; =\; xD\_(x,a)-a\; D\_(x,a),$$with the initial conditions

D _{0(x,a)} = 2 and D _{1(x,a)} = x .The first few Dickson polynomials are:D _{2(x,a)} = x ^{2} - 2 a D _{3(x,a)} = x ^{3} - 3 ax D _{4(x,a)} = x ^{4} - 4 ax ^{2} + 2 a ^{2} D _{5(x,a)} = x ^{5} - 5 ax ^{3} + 5 a ^{2} x.

If and then permutes GF(q) if and only if . If then and the previous result holds.

• If is an extension of of degree, then the linearized polynomial $$L(x)\; =\; \backslash sum\_^\; \backslash alpha\_s\; x^,$$ with in, is a linear operator on over . A linearized polynomial permutes if and only if 0 is the only root of in . This condition can be expressed algebraically as $$\backslash det\backslash left\; (\backslash alpha\_^\; \backslash right)\; \backslash neq\; 0\; \backslash quad\; (i,\; j=\; 0,1,\backslash ldots,r-1).$$

The linearized polynomials that are permutation polynomials over 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 .

• If is in the polynomial ring and has no nonzero root in when divides, and is relatively prime (coprime) to, then permutes .
• Only a few other specific classes of permutation polynomials over have been characterized. Two of these, for example, are: $$x^\; +\; ax$$ where divides, and $$x^r\; \backslash left(x^d\; -\; a\backslash right)^$$ where divides .

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 (i.e., in) is a permutation polynomial over for infinitely many primes, then it is the composition of linear and Dickson polynomials. (See Schur's conjecture below).

Geometric examples

See main article: Oval (projective plane).

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.^[3]

Permutation polynomials in several variables over finite fields

A polynomial

f \in F _{q[x} _{1,\ldots,x} _{n]} is a permutation polynomial in variables over F _{q} if the equation f(x _{1,\ldots,x} _{n)} = \alpha has exactly q ^{n-1} solutions in F ^{n} _{q} for each \alpha \in 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 ^[4] e.g. page 14 in version 8.8.0).

Simple examples

Consider

g(x) = 2 x ^{2+x} for the ring Z/4Z.One sees: so the polynomial defines the permutation

Consider the same polynomial

g(x) = 2 x ^{2+x} for the other ring Z/8Z.One sees: so the polynomial defines the permutation

Rings Z/p^kZ

Consider

g(x) = ax ^{2+bx+c} for the ring Z/p^kZ.

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/p^kZ) 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 ^{k _{1}} _{1} p ^{k _{2}} _{2} ...p ^{k _{l}} _{l} , where p_t are prime numbers.

Lemma: any polynomial $g(x)\; =\; a\_0+\; \backslash sum\_\; a\_i\; x^i$ defines a permutation for the ring Z/nZ if and only if all the polynomials $g\_(x)\; =\; a\_+\; \backslash sum\_\; a\_\; x^i$ defines the permutations for all rings

Z/p ^{k _{t}} _{t} Z , where a _{j,p _{t}} are remainders of a _{j} modulo p ^{k _{t}} _{t} .

As a corollary one can construct plenty quadratic permutation polynomials using the following simple construction. Consider

n = p ^{k _{1}} _{1} p ^{k _{2}} _{2} ... p ^{k _{l}} _{l} , assume that k₁ >1.

Consider

ax ^{2+bx} , such that a = 0 \bmod p _{1} , but a\ne 0 \bmod p ^{k _{1}} _{1} ; assume that a = 0 \bmod p ^{k _{i}} _{i} , i > 1. And assume that b\ne 0 \bmod p _{i} for all .(For example, one can take a = p _{1} p ^{k _{2}} _{2} ...p ^{k _{l}} _{l} and b =1 ).Then such polynomial defines a permutation.

To see this we observe that for all primes p_i, i > 1, the reduction of this quadratic polynomial modulo p_i 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 and polynomial

6 x ^{2+x} .It defines a permutationZ/pZ and

g'(x) \ne 0 \bmod p for all x in Z/p^kZ, where g′(x) is the formal derivative of g(x).^[5]

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 x^k. In fact, Schur did not make any conjecture in this direction. The notion that he did is due to Fried,^[6] who gave a flawed proof of a false version of the result. Correct proofs have been given by Turnwald^[7] and Müller.^[8]

References

• Book: Dickson, L. E.. Linear Groups with an Exposition of the Galois Field Theory. 1958. 1901 . Dover. New York.
• Lidl. Rudolf. Mullen. Gary L.. When Does a Polynomial over a Finite Field Permute the Elements of the Field?. The American Mathematical Monthly. March 1988. 95. 3. 243–246. 10.2307/2323626.
• Lidl. Rudolf. Mullen. Gary L.. When Does a Polynomial over a Finite Field Permute the Elements of the Field?, II. The American Mathematical Monthly. January 1993. 100. 1. 71–74. 10.2307/2324822.
• Book: 0866.11069 . Lidl . Rudolf . Niederreiter . Harald . Harald Niederreiter . Finite fields . 2nd . Encyclopedia of Mathematics and Its Applications . 20 . . 1997 . 0-521-39231-4 . registration . Chapter 7.
• Book: Gary L.. Mullen. Daniel. Panario. Handbook of Finite Fields. 2013. CRC Press. 978-1-4398-7378-6. Chapter 8.
• C.J.. Shallue. I.M.. Wanless. Permutation polynomials and orthomorphism polynomials of degree six. Finite Fields and Their Applications. 20. March 2013. 84–92. 10.1016/j.ffa.2012.12.003. free.

Notes and References

1. Permutation Polynomial Interleavers: An Algebraic-Geometric Perspective. 2006 . Oscar . Takeshita . cs/0601048 . 10.1109/TIT.2007.896870 . 53 . IEEE Transactions on Information Theory . 2116–2132.
2. A New Construction for LDPC Codes using Permutation Polynomials over Integer Rings. 2005 . Oscar . Takeshita . cs/0506091 .
3. 2005 . . Neeraj . Kayal . Recognizing permutation functions in polynomial time . Electronic Colloquium on Computational Complexity . For earlier research on this problem, see: Ma . Keju . von zur Gathen . Joachim . Joachim von zur Gathen . 10.1007/BF01277957 . 1 . Computational Complexity . 1319494 . 76–97 . The computational complexity of recognizing permutation functions . 5 . 1995. Shparlinski . I. E. . 10.1007/BF01202000 . 2 . Computational Complexity . 1190826 . 129–132 . A deterministic test for permutation polynomials . 2 . 1992.
4. http://www.3gpp.org/ftp/Specs/html-info/36212.htm 3GPP TS 36.212
5. Sun . Jing . Takeshita . Oscar . 2005 . Interleaver for Turbo Codes Using Permutation Polynomials Over Integer Rings . IEEE Transactions on Information Theory . 51 . 1. 102 .
6. Fried . M. . On a conjecture of Schur . Michigan Math. J. . 1970 . 41–55 .
7. Turnwald . G. . On Schur's conjecture . J. Austral. Math. Soc. . 1995 . 312–357 .
8. Müller . P. . A Weil-bound free proof of Schur's conjecture . Finite Fields and Their Applications . 1997 . 25–32 .