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CONSTRUCTING PERMUTATION POLYNOMIALS OVER FINITE FIELDS

Part of: Finite fields and commutative rings (number-theoretic aspects) General field theory

Published online by Cambridge University Press:  07 August 2013

XIAOER QIN  and
SHAOFANG HONG
XIAOER QIN
Affiliation:
Mathematical College, Sichuan University, Chengdu 610064, PR China email qincn328@sina.com College of Mathematics and Computer Science, Yangtze Normal University, Chongqing 408100, PR China email qincn328@sina.com
SHAOFANG HONG*
Affiliation:
Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, PR China email s-f.hong@tom.com, hongsf02@yahoo.com Mathematical College, Sichuan University, Chengdu 610064, PR China email s-f.hong@tom.com, hongsf02@yahoo.com
*
✉sfhong@scu.edu.cn
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Abstract

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In this paper, we construct several new permutation polynomials over finite fields. First, using the linearised polynomials, we construct the permutation polynomial of the form ${ \mathop{\sum }\nolimits}_{i= 1}^{k} ({L}_{i} (x)+ {\gamma }_{i} ){h}_{i} (B(x))$ over ${\mathbf{F} }_{{q}^{m} } $, where ${L}_{i} (x)$ and $B(x)$ are linearised polynomials. This extends a theorem of Coulter, Henderson and Matthews. Consequently, we generalise a result of Marcos by constructing permutation polynomials of the forms $xh({\lambda }_{j} (x))$ and $xh({\mu }_{j} (x))$, where ${\lambda }_{j} (x)$ is the $j$th elementary symmetric polynomial of $x, {x}^{q} , \ldots , {x}^{{q}^{m- 1} } $ and ${\mu }_{j} (x)= {\mathrm{Tr} }_{{\mathbf{F} }_{{q}^{m} } / {\mathbf{F} }_{q} } ({x}^{j} )$. This answers an open problem raised by Zieve in 2010. Finally, by using the linear translator, we construct the permutation polynomial of the form ${L}_{1} (x)+ {L}_{2} (\gamma )h(f(x))$ over ${\mathbf{F} }_{{q}^{m} } $, which extends a result of Kyureghyan.

Keywords

permutation polynomiallinearised polynomiallinear translatorelementary symmetric polynomial

MSC classification

Secondary: 11T06: Polynomials 12E20: Finite fields (field-theoretic aspects)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 89 , Issue 3 , June 2014 , pp. 420 - 430
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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