[Paper Review] Permutation polynomials of the form x+c*Tr(x^k)

Gohar Kyureghyan, Michael E. Zieve|arXiv (Cornell University)|Mar 3, 2016

Coding theory and cryptography1 references3 citations

TL;DR

This paper classifies permutation polynomials of the form $X + \gamma \operatorname{Tr}_{q^n/q}(X^k)$ over finite fields $\mathbb{F}_{q^n}$, proving they permute the field under specific conditions on $q$, $n$, $k$, and $\gamma$. It establishes nine distinct families using diverse methods—ranging from trace function analysis and root-of-unity permutation criteria to Gröbner bases and planar function techniques—offering a comprehensive framework for such binomial permutation polynomials.

ABSTRACT

Let F_{q^n} be the field of order q^n, and let Tr be the trace map from F_{q^n} to its q-element subfield. We exhibit nine sequences of polynomials of the form f(x):=x+c*Tr(x^k), with c in F_{q^n}, such that for each polynomial the function F_{q^n}-->F_{q^n} given by c-->f(c) is a permutation of F_{q^n}. We also computed all permutation polynomials of this form over finite fields of size less than 5000, and found that our examples comprise all examples with n>1 except for some simple cases where the polynomial induces a homomorphism of the additive group of F_{q^n}, along with a few sporadic examples. One intriguing feature is that our proofs of the different sequences use various different methods, including a new variant of Dobbertin's method among others.

Motivation & Objective

• To systematically identify and classify all permutation polynomials of the form $X + \gamma \operatorname{Tr}_{q^n/q}(X^k)$ over finite fields $\mathbb{F}_{q^n}$, excluding trivial cases.
• To understand the structural and arithmetic conditions under which such binomials permute $\mathbb{F}_{q^n}$, especially when $\gamma \in \mathbb{F}_{q^n}^*$ and $k$ is not a power of $p$.
• To unify disparate techniques—such as trace function manipulation, root-of-unity permutation criteria, and Gröbner basis computation—into a coherent framework for proving permutation behavior.
• To demonstrate that the observed permutation polynomials are not random but arise from deep algebraic and number-theoretic structures.
• To show that the vast majority of such permutation polynomials are captured by the nine families listed in Theorem 1.1, with only a few exceptional cases outside these families.

Proposed method

• Uses Proposition 2.2 to reduce permutation of $\mathbb{F}_{q^n}$ to permutation of the set $\mu_{(q^n-1)/(q-1)}$ of $(q^n-1)/(q-1)$-th roots of unity.
• Applies a reformulation of the trace condition via rational functions and power maps to simplify verification in cases (a)–(d) and (g).
• Employs a variant of Dobbertin’s method based on Gröbner basis computation in multivariate polynomial rings to handle cases (e) and (f), where other methods fail.
• Utilizes a square-nonsquare decomposition strategy in case (h), showing that the polynomial permutes both sets separately by leveraging affine equivalence to planar functions.
• Applies an additive analogue of Proposition 2.2 (Proposition 2.1) in case (i), reducing the permutation condition to checking that associated polynomials induce functions equivalent to $X + c$ over $\mathbb{F}_q$.
• Employs a key identity $L(H(x^2)) = L(x)^2$ in case (h), linking the behavior on squares and nonsquares to the squaring map and enabling direct bijectivity verification.

Experimental results

Research questions

• RQ1 Under what conditions on $q$, $n$, $k$, and $\gamma$ does the polynomial $X + \gamma \operatorname{Tr}_{q^n/q}(X^k)$ permute $\mathbb{F}_{q^n}$?
• RQ2 Why do such permutation polynomials exist in abundance despite the low density of permutation functions among all functions $\mathbb{F}_{q^n} \to \mathbb{F}_{q^n}$?
• RQ3 What structural or number-theoretic reasons explain the existence of infinite families of such permutation binomials?
• RQ4 How can different algebraic techniques—such as root-of-unity permutation, Gröbner bases, and planar function methods—be effectively combined to prove permutation behavior?
• RQ5 To what extent do the constructed families in Theorem 1.1 account for all nontrivial permutation polynomials of this form?

Key findings

• The polynomial $X + \gamma \operatorname{Tr}_{q^n/q}(X^k)$ permutes $\mathbb{F}_{q^n}$ when $n=2$, $q \equiv \pm 1 \pmod{6}$, $\gamma = -1/3$, and $k = 2q - 1$, as shown via rational function permutation on $\mu_{q+1}$.
• For $n=2$, $q \equiv 5 \pmod{6}$, $\gamma^3 = -1/27$, and $k = 2q - 1$, the polynomial permutes $\mathbb{F}_{q^2}$, with the proof relying on cubic rational function behavior on roots of unity.
• When $n=2$, $q \equiv 1 \pmod{3}$, $\gamma = 1$, and $k = (q^2 + q + 1)/3$, the polynomial permutes $\mathbb{F}_{q^2}$ due to the power map $X^N$ permuting $\mu_{q+1}$.
• For $n=2$, $q \equiv 1 \pmod{4}$, $(2\gamma)^{(q+1)/2} = 1$, and $k = (q+1)^2/4$, the polynomial permutes $\mathbb{F}_{q^2}$ by analyzing its action separately on squares and nonsquares in $\mu_{q+1}$.
• In the case $n=3$, $q$ odd, $\gamma = 1$, $k = (q^2 + 1)/2$, the polynomial permutes $\mathbb{F}_{q^3}$ via a double application of Proposition 2.2 and reduction modulo $X^{q^2+q+1} - 1$.
• For $n=2\ell r$, $\gamma^{q^{2\ell}-1} = -1$, and $k = q^\ell + 1$, the polynomial permutes $\mathbb{F}_{q^n}$ by showing that associated additive polynomials induce $X + c$ functions over $\mathbb{F}_q$, as guaranteed by Proposition 2.1.