### Abstract

For any given polynomial f over the finite field F_{q} with degree at most q-1, we associate it with a q × q matrix A(f) = (a_{ik}) consisting of coefficients of its powers (f(x))^{k}= ∫^{q-1} _{i=0} aikx^{i} modulo x^{q}-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)}(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 F_{q} when f is a permutation polynomial over F_{q}.

Original language | English (US) |
---|---|

Title of host publication | Contemporary Developments in Finite Fields and Applications |

Publisher | World Scientific Publishing Co. Pte Ltd |

Pages | 270-281 |

Number of pages | 12 |

ISBN (Electronic) | 9789814719261 |

ISBN (Print) | 9789814719254 |

DOIs | |

State | Published - Aug 1 2016 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)

### Cite this

*Contemporary Developments in Finite Fields and Applications*(pp. 270-281). World Scientific Publishing Co. Pte Ltd. https://doi.org/10.1142/9789814719261_0016

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*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 L.; Muratović-Ribić, Amela; Wang, Qiang.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

TY - CHAP

T1 - On coefficients of powers of polynomials and their compositionsover finite fields

AU - Mullen, Gary L.

AU - Muratović-Ribić, Amela

AU - Wang, Qiang

PY - 2016/8/1

Y1 - 2016/8/1

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85018833107&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85018833107&partnerID=8YFLogxK

U2 - 10.1142/9789814719261_0016

DO - 10.1142/9789814719261_0016

M3 - Chapter

AN - SCOPUS:85018833107

SN - 9789814719254

SP - 270

EP - 281

BT - Contemporary Developments in Finite Fields and Applications

PB - World Scientific Publishing Co. Pte Ltd

ER -