A Weil Sum Approach to Permutation Polynomials over Quadratic Extensions of Finite Fields

Abstract: In this article, we introduce several classes of permutation polynomials over $\mathbb{F}<em>{q<sup>2}$. More precisely, we characterize permutation polynomials of the forms $x^q + b x² + c x + d$ and $x^{q+1} + b x^q + c x + d$ over $\mathbb{F}</em>{q<sup>2}$. To this end, we determine the exact number of zeros of these polynomials using existing results on certain special Weil sums. We also present the compositional inverses of the permutation polynomials obtained in this paper.

• The paper rigorously characterizes permutation polynomials over quadratic extensions by establishing necessary and sufficient conditions using Weil sum evaluations.
• It applies explicit root-counting via canonical character sums to distinguish polynomial behavior in both odd and even finite field characteristics, leading to concrete inversion formulas.
• The study identifies strong negative results and posits an open conjecture, thereby motivating further research in algebraic theory and cryptographic applications.

Weil Sum Methods for Characterizing Permutation Polynomials in Quadratic Finite Field Extensions

Background and Motivation

Permutation polynomials (PPs) over finite fields, which induce bijections on the underlying field, are central in numerous areas including error-correction coding, cryptographic primitives, and combinatorial constructions. For quadratic extensions $\mathbb{F}_{q^2}$, the explicit characterization and construction of PPs is more challenging than over prime fields or extensions of modest degree due to the richer algebraic structure and subtler interplay between additive and multiplicative properties. This paper investigates PPs of forms $x^q + b x^2 + c x + d$ and $x^{q+1} + b x^q + c x + d$, departing from classical families (e.g., Wan-Lidl or affine $q$-polynomials), exploring new structural properties via Weil sums—a class of character sums that have proved effective in analyzing zero distributions in polynomials over finite fields.

Approach: Weil Sums and Zero Enumeration

The key analytical tool is the computation of the number of zeros of the target polynomials for each $t \in \mathbb{F}_{q^2}$, using Weil sums associated with canonical additive characters. Specifically, for $f(x)$ over $\mathbb{F}_{q^2}$, the number $M$ of zeros is expressed as

$$M = \frac{1}{q^2} \sum_{x \in \mathbb{F}_{q^2} }\sum_{y \in \mathbb{F}_{q^2} } \chi(y f(x)),$$

where $\chi$ is the canonical additive character. This reduction allows the exploitation of deep results on explicit Weil sum evaluations—in both even and odd characteristics—to precisely count roots and thus delineate permutation behavior.

Main Structural Results

PPs of the Form $x^q + b x^2 + c x + d$0 in $x^q + b x^2 + c x + d$1

Odd Characteristic ($x^q + b x^2 + c x + d$2 Odd):

• The necessary and sufficient conditions are established: $x^q + b x^2 + c x + d$3 is a PP if and only if $x^q + b x^2 + c x + d$4 and $x^q + b x^2 + c x + d$5, where $x^q + b x^2 + c x + d$6 denotes the $x^q + b x^2 + c x + d$7-th roots of unity. For $x^q + b x^2 + c x + d$8, detailed character sum analysis demonstrates the nonexistence of PPs in this form; for every choice of $x^q + b x^2 + c x + d$9, the associated sum never yields the required unicity of zeros.

Even Characteristic ($x^{q+1} + b x^q + c x + d$0):

• Two families of PPs arise:
  • If $x^{q+1} + b x^q + c x + d$1 and $x^{q+1} + b x^q + c x + d$2,
  • If $x^{q+1} + b x^q + c x + d$3 and $x^{q+1} + b x^q + c x + d$4.
  • Here, the symmetry and additive structure in characteristic $x^{q+1} + b x^q + c x + d$5 allow more flexible construction, and the rigorous use of explicit root-counting via character sum evaluations provides strength and precision in the classification.

PPs of the Form $x^{q+1} + b x^q + c x + d$6

Strong negative results (contradictory to some classical intuitions) are demonstrated:

• For $x^{q+1} + b x^q + c x + d$7 even, such polynomials never permute $x^{q+1} + b x^q + c x + d$8 (Proposition~\ref{p3.8}), corroborated by independent character sum analysis and connection to prior work (2606.14529).
• For $x^{q+1} + b x^q + c x + d$9 odd and $q$0, non-permutation behavior is explicitly proved; for $q$1, computational evidence (via SageMath enumeration) suggests NPP in all cases, motivating a conjecture that no such PPs exist—leaving a significant open theoretical problem.

Explicit Inverses

The paper leverages structural results on $q$2-linearized polynomials and Dickson matrices to provide explicit compositional inverses for the classified PPs:

• For $q$3, $q$4 (odd $q$5):

$q$6

• For $q$7, two cases (even $q$8):
  • $q$9: $t \in \mathbb{F}_{q^2}$0,
  • $t \in \mathbb{F}_{q^2}$1: $t \in \mathbb{F}_{q^2}$2, with explicit algebraic constructions (see detailed expressions in the original paper).

Numerical and Structural Implications

The results delineate precise boundaries for permutation behavior across different structural regimes. The explicit root enumeration, especially cases with $t \in \mathbb{F}_{q^2}$3 even, demonstrates surprisingly rich possibilities and, in some regimes, contradicts intuition—showing existence only in specific parameter choices. Strong negative claims (for $t \in \mathbb{F}_{q^2}$4) provide clarity, and the unresolved $t \in \mathbb{F}_{q^2}$5 odd, $t \in \mathbb{F}_{q^2}$6 case defines a substantive territory for further investigation.

Theoretical and Practical Implications

From a theoretical standpoint, the paper's methods and classifications push the boundary of what can be achieved through character sum techniques, taking advantage of nuanced algebraic and number-theoretic arguments. Practically, the explicit constructions and inverses benefit cryptographic algorithm design and coding theory, where permutation polynomials with concrete inverses are essential. The critical open conjecture regarding nonexistence for the $t \in \mathbb{F}_{q^2}$7 family in odd characteristic is likely to motivate extensive further work both in algebraic geometry of finite fields and computational explorations.

Outlook and Future Directions

The extension of the Weil sum root-counting strategy to higher-degree polynomials or more general forms, possibly incorporating additional automorphism or symmetry constraints, appears promising. Addressing the open conjecture for odd $t \in \mathbb{F}_{q^2}$8, $t \in \mathbb{F}_{q^2}$9 could yield new insights into the limitations of character sum methods and perhaps invite alternative approaches (e.g., algebraic geometry or group theoretic techniques). The explicit inverse constructions also suggest directions for constructing new classes of cryptographically strong permutations.

Conclusion

This work rigorously characterizes permutation and non-permutation behavior of specific polynomial families over quadratic extensions of finite fields, utilizing Weil sums and detailed character sum computations. The strong numerical results provide clarity in both even and odd characteristics, and the explicit compositional inverses extend the practical utility of such classification. The theoretical implications, especially the formulated conjecture, shape a future research agenda aimed at unlocking deeper structural understanding of permutation polynomials in higher field extensions.