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- W833851724 abstract "For any given polynomial $f$ over the finite field $mathbb{F}_q$ with degree at most $q-1$, we associate it with a $qtimes q$ 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 $A(gcirc f)=A(f)A(g)$ where $(gcirc f)(x) = g(f(x))$ 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 $mathbb{F}_q$." @default.
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- W833851724 date "2016-06-16" @default.
- W833851724 modified "2023-09-26" @default.
- W833851724 title "On coefficients of powers of polynomials and their compositions over finite fields" @default.
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- W833851724 doi "https://doi.org/10.1142/9789814719261_0016" @default.
- W833851724 hasPublicationYear "2016" @default.
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