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On planar functions over $\mathbb{F}_{q^3}$

This paper investigates the planarity of a specific family of polynomials over the finite field $\mathbb{F}_{q^3}$ for odd $q$ , establishing conditions for their planarity and presenting new families of planar trinomials, quadrinomials, and pentanomials using results from the theory of $q$ -polynomials.

Original authors: João Paulo Guardieiro, Adler Marques, Luciane Quoos, Guilherme Tizziotti

Published 2026-05-27

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CC0 1.0

Original authors: João Paulo Guardieiro, Adler Marques, Luciane Quoos, Guilherme Tizziotti

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are working with a very specific, finite universe of numbers. In mathematics, this is called a finite field (denoted as $\mathbb{F}_q$ ). Think of it like a clock that doesn't go from 1 to 12, but from 0 to $q-1$ , and then wraps back around to 0.

The paper you provided is about finding special "rules" (mathematical functions) that operate within a slightly larger version of this universe, specifically one that has $q^3$ numbers. The goal is to find rules that are Planar.

What is a "Planar Function"?

To understand "planar," imagine you have a machine that takes a number, does something to it, and spits out a new number.

The Test: If you take any two different numbers, say $X$ and $X + \epsilon$ (where $\epsilon$ is a small shift), and run them through your machine, you get two results.
The Planar Condition: A function is "planar" if, no matter what shift $\epsilon$ you choose, the difference between the two results is always unique. In other words, if you ask the machine, "What shift $\epsilon$ makes the output difference equal to $Y$ ?", there is exactly one correct answer.

Why does this matter? The paper mentions that these functions are like "perfectly balanced" tools. In the real world (outside the paper), they are used to build unbreakable locks for computers (cryptography) and to fix errors in data transmission (coding theory). The paper focuses on the "odd" versions of these number systems (where $q$ is an odd number).

The Big Puzzle: The Five-Part Recipe

The authors are investigating a specific family of mathematical recipes called pentanomials. Think of a polynomial as a recipe with several ingredients added together. This specific recipe has five ingredients (hence "penta"):
$f(X) = E X^2 + A X^{q+1} + B X^{q^2+1} + C X^{2q} + D X^{2q^2}$

The letters $A, B, C, D,$ and $E$ are the "coefficients" or the amounts of each ingredient. The question the paper asks is: "What specific amounts of these ingredients make the recipe 'planar'?"

If you pick the wrong amounts, the recipe fails the uniqueness test. If you pick the right amounts, it works perfectly.

How They Solved It

The authors didn't just guess. They used a clever mathematical "detector" (based on something called a Dickson matrix, which is like a grid of numbers that checks if a function is unique).

The Detective Work: They calculated a giant formula (the determinant) that tells you if the recipe works.
The Shortcut: Instead of checking every single number in the universe, they realized that if they could break this giant formula down into three smaller, simpler pieces, they could just check if those three pieces were "safe."
The Safety Rule: They found a rule (Lemma 2.2) that says a specific type of piece is "safe" (has no zeros) if the ingredients satisfy a simple equation: $a^3 + b^3 + c^3 - 3abc \neq 0$ .

By forcing their five-ingredient recipe to split into three of these "safe" pieces, they could guarantee the whole thing was planar.

The New Discoveries

Using this method, the authors found several new, specific recipes that work. They are like new "magic spells" for this number universe:

The Three-Ingredient Mix: They found that if you only use three of the five ingredients (specifically $X^2$ , $X^{2q}$ , and $X^{2q^2}$ ), you can make a planar function as long as the amounts $C, D,$ and $E$ satisfy a specific cubic equation ( $C^3 + D^3 + E^3 - 3CDE = 1/2$ ).
The Four-Ingredient Mix: They found a specific combination of four ingredients: $-X^2 + 2X^{q^2+1} + X^{2q} - X^{2q^2}$ . This one works automatically.
The Five-Ingredient Mixes: They discovered two new families of full five-ingredient recipes that work, provided the ingredients aren't zero and satisfy certain relationships (like one ingredient being related to the others by a specific formula).

In Summary

The paper is a guidebook for mathematicians. It says: "If you want to build a perfect, planar function in this specific 3-layered number world, don't just guess. Use these specific recipes we found. If you mix the ingredients exactly as we say, your function will pass the uniqueness test every time."

They didn't invent a new type of math; they found new, reliable ways to use existing math to create these highly useful, perfectly balanced functions.

Technical Summary: Planar Functions over $\mathbb{F}_{q^3}$

Problem Statement
This work investigates the existence and characterization of planar functions over the finite field $\mathbb{F}_{q^3}$ , where $q$ is an odd prime power. A function $f: \mathbb{F}_{q^n} \to \mathbb{F}_{q^n}$ is defined as planar if, for every non-zero $\epsilon \in \mathbb{F}_{q^n}$ , the difference function $F_\epsilon(X) = f(X+\epsilon) - f(X) - f(\epsilon)$ is a permutation polynomial. The authors focus specifically on the family of pentanomials of the form:
$f_{E,A,B,C,D}(X) := EX^2 + AX^{q+1} + BX^{q^2+1} + CX^{2q} + DX^{2q^2} \in \mathbb{F}_q[X].$
The primary challenge addressed is determining the precise algebraic conditions on the coefficients $A, B, C, D, E \in \mathbb{F}_q$ that ensure $f_{E,A,B,C,D}$ satisfies the planarity property over $\mathbb{F}_{q^3}$ .

Methodology
The authors employ the theory of $q$ -polynomials (linearized polynomials) to analyze the planarity condition. The methodology proceeds through the following logical steps:

Reduction to Permutation Polynomials: By definition, $f$ is planar if and only if $F_\epsilon(X)$ is a permutation polynomial for all $\epsilon \in \mathbb{F}_{q^3}^*$ . The authors explicitly compute $F_\epsilon(X)$ for the target pentanomial family, showing it results in a $q$ -polynomial of the form $L(X) = c_0 X + c_1 X^q + c_2 X^{q^2}$ .
Dickson Matrix Criterion: Utilizing Theorem 1.1 and Lemma 1.2, the authors establish that a $q$ -polynomial is a permutation polynomial if and only if its associated Dickson matrix is non-singular (i.e., has a non-zero determinant).
Determinant Analysis: The determinant $\Delta_{F_\epsilon}$ of the Dickson matrix for $F_\epsilon(X)$ is calculated as a polynomial in $\epsilon$ . Proposition 2.1 provides a general, albeit complex, characterization: the pentanomial is planar if and only if this determinant is non-zero for all $\epsilon \in \mathbb{F}_{q^3}^*$ .
Factorization Strategy: Recognizing that verifying the non-vanishing of the general determinant is algebraically difficult, the authors develop a factorization approach (Theorem 2.5). They seek conditions on the coefficients such that $\Delta_{F_\epsilon}$ factors into a product of three simpler $q$ -polynomials, $P_1(\epsilon)P_2(\epsilon)P_3(\epsilon)$ .
Root-Free Conditions: The authors leverage Lemma 2.2, which states that a $q$ -polynomial $aX^{q^2} + bX^q + cX$ has no roots in $\mathbb{F}_{q^3}^*$ if and only if $a^3 + b^3 + c^3 - 3abc \neq 0$ . By constructing specific systems of equations (Theorem 2.5 and Corollary 2.6) that force the determinant to factor into such root-free polynomials, they derive sufficient conditions for planarity.
Specific Matrix Families: To verify the non-vanishing of determinants for specific cases, the authors analyze two families of $3 \times 3$ matrices (Propositions 2.3 and 2.4), proving their determinants are non-zero for all non-zero $\epsilon$ .

Key Contributions and Results
The paper provides a systematic framework for constructing planar pentanomials over $\mathbb{F}_{q^3}$ and presents five new explicit families of planar functions:

General Characterization (Theorem 2.5): A system of equations is established that, when satisfied by the coefficients $(E, A, B, C, D)$ , guarantees the determinant of the associated Dickson matrix factors into three $q$ -polynomials with no roots in $\mathbb{F}_{q^3}^*$ .
Planar Trinomials (Theorem 3.1): A characterization for trinomials of the form $EX^2 + CX^{2q} + DX^{2q^2}$ . The function is planar if and only if $C^3 + D^3 + E^3 - 3CDE = 1/2$ .
Specific Quadrinomial (Theorem 3.2): The polynomial $f(X) = -X^2 + 2X^{q^2+1} + X^{2q} - X^{2q^2}$ is proven to be planar over $\mathbb{F}_{q^3}$ .
Two-Parameter Pentanomial Family (Theorem 3.3): A family of pentanomials $f(X) = EX^2 + 2(E-D)X^{q+1} + 2DX^{q^2+1} + (E-D)X^{2q} + DX^{2q^2}$ is shown to be planar provided $E(3D^2 - 3DE + E^2) \neq 0$ .
Two New Pentanomials (Theorem 3.4): Two specific pentanomials are identified as planar:
- $X^2 - X^{q+1} - X^{q^2+1} + X^{2q} + X^{2q^2}$
- $X^2 + X^{q+1} + X^{q^2+1} + \frac{1}{2}X^{2q} + \frac{1}{2}X^{2q^2}$

Significance and Claims
The paper claims to provide new characterizations for the planarity property of polynomials over $\mathbb{F}_{q^3}$ and to present new families of planar trinomials, quadrinomials, and pentanomials. The work extends the existing literature on planar functions, which has applications in cryptography (specifically S-box design), coding theory, and signal sets. The authors note that while the existence of planar mappings has been extensively studied, constructing them or characterizing the conditions for their existence remains a significant problem in finite field theory. This work addresses that problem by offering a constructive method based on the factorization of Dickson matrix determinants, yielding concrete examples of planar functions that were previously uncharacterized in this specific form.

What is a "Planar Function"?

The Big Puzzle: The Five-Part Recipe

How They Solved It

The New Discoveries

In Summary

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