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Some classes of permutation pentanomials

Zhiguo Ding, Michael E. Zieve

TL;DR

The paper constructs two expansive families of permutation polynomials over $\mathbb{F}_{q^2}$ of the form $f(X)=X^r B(X^{q-1})$, with $B(X)$ a short polynomial whose coefficients lie in $\{1,-1\}$, for primes $p\neq 3$ and $q=p^k$. Using a noncomputational, cubic-map framework built from $\rho$ and $\eta$ on $\mu_{q+1}$ and a Z-Redei style reduction, the authors derive explicit necessary and sufficient gcd conditions on the exponents $r,Q,R,S$ to guarantee that $f$ permutes $\mathbb{F}_{q^2}$; they obtain two large classes with at most five-term $B_z(X)$ polynomials. In characteristic $2$ these results subsume 76 prior results and conjectures, while for odd $p>2$ the polynomials appear to be new and address open problems in the literature. A key finding is that, in the special case $r=Q+R+S$, the permutation polynomials decompose into simple, linear-equivalent compositions, connecting the finite-field permutation behavior to basic monomial and bi-variable maps. Overall, the paper unifies and extends the landscape of low-term permutation polynomials over $\mathbb{F}_{q^2}$ with broad applicability to theory and applications in finite-field permutations.

Abstract

For each prime p other than 3, and each power q=p^k, we present two large classes of permutation polynomials over F_{q^2} of the form X^r B(X^{q-1}) which have at most five terms, where B(X) is a polynomial with coefficients in {1,-1}. The special case p=2 of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case p>2, no instances of our permutation polynomials have appeared in the literature, and the construction of such polynomials had been posed as an open problem. Our proofs are short and involve no computations, in contrast to the proofs of many of the special cases of our results which were published previously.

Some classes of permutation pentanomials

TL;DR

The paper constructs two expansive families of permutation polynomials over of the form , with a short polynomial whose coefficients lie in , for primes and . Using a noncomputational, cubic-map framework built from and on and a Z-Redei style reduction, the authors derive explicit necessary and sufficient gcd conditions on the exponents to guarantee that permutes ; they obtain two large classes with at most five-term polynomials. In characteristic these results subsume 76 prior results and conjectures, while for odd the polynomials appear to be new and address open problems in the literature. A key finding is that, in the special case , the permutation polynomials decompose into simple, linear-equivalent compositions, connecting the finite-field permutation behavior to basic monomial and bi-variable maps. Overall, the paper unifies and extends the landscape of low-term permutation polynomials over with broad applicability to theory and applications in finite-field permutations.

Abstract

For each prime p other than 3, and each power q=p^k, we present two large classes of permutation polynomials over F_{q^2} of the form X^r B(X^{q-1}) which have at most five terms, where B(X) is a polynomial with coefficients in {1,-1}. The special case p=2 of our results comprises a vast generalization of 76 recent results and conjectures in the literature. In case p>2, no instances of our permutation polynomials have appeared in the literature, and the construction of such polynomials had been posed as an open problem. Our proofs are short and involve no computations, in contrast to the proofs of many of the special cases of our results which were published previously.
Paper Structure (5 sections, 7 theorems, 12 equations, 6 tables)

This paper contains 5 sections, 7 theorems, 12 equations, 6 tables.

Table of Contents

  1. Introduction
  2. Background results
  3. Proof of Theorem \ref{['1']}
  4. Proof of Theorem \ref{['2']}
  5. Proof of Theorem \ref{['3']}

Key Result

Theorem 1.1

Let $p$ be prime with $p\ne 3$, let $q=p^k$ where $k$ is a positive integer, and pick $Q,R,S\in\{p^i:i\ge 0\}$. Pick an order-$3$ element $\omega\in\mathbb{F}_{q^2}^*$, and let $r$ be a positive integer such that $r\equiv Q+R+S\pmod{q+1}$. Write and $C_2(X)\colonequals X^{Q+R+S} C_1(1/X)$. Let $\beta=\omega^{Q+R}-\omega^{S+1}$ if $Q+R+S\equiv 1\pmod 3$, and $\beta=\omega^{Q+R+S}-\omega$ otherwis

Theorems & Definitions (13)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Lemma 2.2
  • Lemma 2.3
  • ...and 3 more