On coefficients of powers of polynomials and their compositions over finite fields

TL;DR

This paper introduces a matrix representation for polynomials over finite fields, revealing properties related to polynomial composition, permutation polynomials, and pseudorandom sequence periods, with implications for algebraic and cryptographic applications.

Contribution

It defines a matrix $A(f)$ for polynomials over finite fields, establishing its properties and connections to polynomial composition, permutation polynomials, and sequence periodicity, providing new algebraic insights.

Findings

The matrix $A(f)$ encodes polynomial powers and compositions.

Rank of $A(f)$ equals the value set size of $f$.

For permutation polynomials, $A(f)$ is invertible and diagonalizable.

Abstract

For any given polynomial $f$ over the finite field $\mathbb{F}_q$ with degree at most $q-1$, we associate it with a $q\times q$ matrix $A(f)=(a_{ik})$ consisting of coefficients of its powers $(f(x))^k=\sum_{i=0}^{q-1}a_{ik} x^i$ modulo $x^q -x$ for $k=0,1,\ldots,q-1$. This matrix has some interesting properties such as $A(g\circ f)=A(f)A(g)$ where $(g\circ f)(x) = g(f(x))$ is the composition of the polynomial $g$ with the polynomial $f$. In particular, $A(f^{(k)})=(A(f))^k$ for any $k$-th composition $f^{(k)}$ of $f$ with $k \geq 0$. As a consequence, we prove that the rank of $A(f)$ gives the cardinality of the value set of $f$. Moreover, if $f$ is a permutation polynomial then the matrix associated with its inverse $A(f^{(-1)})=A(f)^{-1}=PA(f)P$ where $P$ is an antidiagonal permutation matrix. As an application, we study the period of a nonlinear congruential pseduorandom sequence $\bar{a} = \{a_0, a_1, a_2, ... \}$ generated by $a_n = f^{(n)}(a_0)$ with initial value $a_0$, in terms of the order of the associated matrix. Finally we show that $A(f)$ is diagonalizable in some extension field of $\mathbb{F}_q$ when $f$ is a permutation polynomial over $\mathbb{F}_q$.