On coefficients of powers of polynomials and their compositionsover finite fields

Gary Lee Mullen, Amela Muratović-Ribić, Qiang Wang

Research output: Chapter in Book/Report/Conference proceedingChapter

Abstract

For any given polynomial f over the finite field Fq with degree at most q-1, we associate it with a q × q matrix A(f) = (aik) consisting of coefficients of its powers (f(x))k= ∫q-1 i=0 aikxi modulo xq-x for k = 0; 1 q-1. This matrix has some interesting properties such as A(gf) = A(f)A(g) where (gf)(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 ≥ 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 ā = -a0, a1, a2, generated by an = f(n)(a0) with initial value a0, in terms of the order of the associated matrix. Finally we show that A(f) is diagonalizable in some extension field of Fq when f is a permutation polynomial over Fq.

Original languageEnglish (US)
Title of host publicationContemporary Developments in Finite Fields and Applications
PublisherWorld Scientific Publishing Co. Pte Ltd
Pages270-281
Number of pages12
ISBN (Electronic)9789814719261
ISBN (Print)9789814719254
DOIs
StatePublished - Aug 1 2016

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Permutation Polynomial
Galois field
Polynomial
Coefficient
Q-matrix
Permutation Matrix
Field extension
Modulo
Cardinality

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

Cite this

Mullen, G. L., Muratović-Ribić, A., & Wang, Q. (2016). On coefficients of powers of polynomials and their compositionsover finite fields. In Contemporary Developments in Finite Fields and Applications (pp. 270-281). World Scientific Publishing Co. Pte Ltd. https://doi.org/10.1142/9789814719261_0016
Mullen, Gary Lee ; Muratović-Ribić, Amela ; Wang, Qiang. / On coefficients of powers of polynomials and their compositionsover finite fields. Contemporary Developments in Finite Fields and Applications. World Scientific Publishing Co. Pte Ltd, 2016. pp. 270-281
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Mullen, GL, Muratović-Ribić, A & Wang, Q 2016, On coefficients of powers of polynomials and their compositionsover finite fields. in Contemporary Developments in Finite Fields and Applications. World Scientific Publishing Co. Pte Ltd, pp. 270-281. https://doi.org/10.1142/9789814719261_0016

On coefficients of powers of polynomials and their compositionsover finite fields. / Mullen, Gary Lee; Muratović-Ribić, Amela; Wang, Qiang.

Contemporary Developments in Finite Fields and Applications. World Scientific Publishing Co. Pte Ltd, 2016. p. 270-281.

Research output: Chapter in Book/Report/Conference proceedingChapter

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Mullen GL, Muratović-Ribić A, Wang Q. On coefficients of powers of polynomials and their compositionsover finite fields. In Contemporary Developments in Finite Fields and Applications. World Scientific Publishing Co. Pte Ltd. 2016. p. 270-281 https://doi.org/10.1142/9789814719261_0016