[Paper Review] A condition for scattered linearized polynomials involving Dickson matrices

Corrado Zanella|arXiv (Cornell University)|Jan 1, 2019

Coding theory and cryptography24 references27 citations

TL;DR

This paper introduces a new condition involving Dickson matrices to determine when linearized polynomials over finite fields are scattered, a key property in projective geometry and coding theory. The authors apply this criterion to two binomials: they prove that the Lunardon-Polverino binomial $x^{q^s} + \delta x^{q^n - q^s}$ is never scattered if the norm $N_{q^n/q}(\delta) = 1$, and establish a necessary and sufficient condition for $x^{q^s} + b x^{q^{2s}}$ to be scattered via a plane algebraic curve. The result resolves the scarcity of such polynomials, showing $x^q + b x^{q^2}$ is never scattered for $n \geq 5$.

ABSTRACT

A linearized polynomial over $\mathbb F_{q^n}$ is called scattered when for any $t,x\in\mathbb F_{q^n}$, the condition $xf(t)-tf(x)=0$ holds if and only if $x$ and $t$ are $\mathbb F_q$-linearly dependent. General conditions for linearized polynomials over $\mathbb F_{q^n}$ to be scattered can be deduced from the recent results in [4,7,15,19]. Some of them are based on the Dickson matrix associated with a linearized polynomial. Here a new condition involving Dickson matrices is stated. This condition is then applied to the Lunardon-Polverino binomial $x^{q^s}+\delta x^{q^{n-s}}$, allowing to prove that for any $n$ and $s$, if $\mathbb N_{q^n/q}(\delta)=1$, then the binomial is not scattered. Also, a necessary and sufficient condition for $x^{q^s}+bx^{q^{2s}}$ to be scattered is shown which is stated in terms of a special plane algebraic curve.

Motivation & Objective

• To develop a new algebraic condition based on Dickson matrices to determine when linearized polynomials over $\mathbb{F}_{q^n}$ are scattered.
• To resolve the open question of whether the Lunardon-Polverino binomial $x^{q^s} + \delta x^{q^n - q^s}$ can be scattered when $N_{q^n/q}(\delta) = 1$.
• To provide a necessary and sufficient condition for the binomial $x^{q^s} + b x^{q^{2s}}$ to be scattered, expressed in terms of a plane algebraic curve.
• To demonstrate that $x^q + b x^{q^2}$ is never scattered for $n \geq 5$, confirming the rarity of such polynomials.

Proposed method

• Introduce a generalized $\sigma$-matrix of Dickson associated with a linearized polynomial $f(x) = \sum_{i=1}^{n-1} a_i x^{\sigma^i}$, where $\sigma = q^s$, $\gcd(s,n) = 1$, and $\sigma$ is a power of $q$.
• Define a new polynomial $g(t) = -f(x)t + f(xt)$ and associate it with the $\sigma$-matrix $M_{\sigma,g}$, which captures the kernel structure of $xf(t) - t f(x)$.
• Establish that $f(x)$ is scattered if and only if, for all $x \in \mathbb{F}_{q^n}^*$, the $(n-1) \times (n-1)$ minors of $M_{\sigma,g}$ are nonsingular.
• Use Laplace expansion and properties of trace and norm to reduce the scatteredness condition to a single equation involving $\sum_{i=0}^{n-1} z^{(\sigma^i - 1)/(\sigma - 1)} = 0$ for $z = \delta x^{q^n - 2q^s}$.
• Apply the condition to the Lunardon-Polverino binomial by substituting $z = \delta x^{q^n - 2q^s}$ and analyzing the norm condition $N_{q^n/q}(\delta) = 1$, showing it implies existence of a solution to the sum equation.
• For $x^{q^s} + b x^{q^{2s}}$, derive a necessary and sufficient condition involving the existence of a point $(x_0, y_0) \in \mathbb{F}_{q^n}^{*2}$ on the curve $b^{-1}X^{q-1} + Y^{q^s - 1} + 1 = 0$ with $\operatorname{Tr}_{q^n/q}(y_0) = 0$.

Experimental results

Research questions

• RQ1 Is the Lunardon-Polverino binomial $x^{q^s} + \delta x^{q^n - q^s}$ scattered when $N_{q^n/q}(\delta) = 1$?
• RQ2 What is a necessary and sufficient condition for the binomial $x^{q^s} + b x^{q^{2s}}$ to be scattered, given $\gcd(s,n) = 1$?
• RQ3 Can the polynomial $x^q + b x^{q^2}$ be scattered for $n \geq 5$?
• RQ4 How does the structure of the Dickson matrix relate to the scatteredness of linearized polynomials?

Key findings

• The Lunardon-Polverino binomial $x^{q^s} + \delta x^{q^n - q^s}$ is not scattered if $N_{q^n/q}(\delta) = 1$, resolving a long-standing question.
• For $n = 5$, the polynomial $x^q + b x^{q^2}$ is non-scattered for all $b \in \mathbb{F}_{q^5}^*$, confirming its absence in this case.
• The binomial $x^{q^s} + b x^{q^{2s}}$ is scattered if and only if the algebraic curve $b^{-1}X^{q-1} + Y^{q^s - 1} + 1 = 0$ has no point $(x_0, y_0)$ with $x_0, y_0 \in \mathbb{F}_{q^n}^*$ and $\operatorname{Tr}_{q^n/q}(y_0) = 0$, providing a geometric characterization.
• The condition $N_{q^n/q}(\delta) = 1$ is necessary for the Lunardon-Polverino binomial to fail scatteredness, generalizing earlier results for $n=4$, $s=1$, and odd $n$.
• The new condition based on $(n-1)$-order minors of the $\sigma$-Dickson matrix provides a single equation test for non-scatteredness, offering a more efficient alternative to checking two determinant equations.
• For $n \geq 5$, no polynomial of the form $x^q + b x^{q^2}$ with $b \neq 0$ is scattered, as shown by combining the curve condition with recent results on the existence of solutions to the trace equation.