Reversed Dickson polynomials over finite fields

Xiang dong Hou; Gary L. Mullen; James A. Sellers; Joseph L. Yucas

doi:10.1016/j.ffa.2009.06.004

Reversed Dickson polynomials over finite fields

Xiang dong Hou, Gary L. Mullen, James A. Sellers, Joseph L. Yucas

Research output: Contribution to journal › Article › peer-review

52 Scopus citations

Abstract

Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials D_n (x, a) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on F_q with q < 200, with only one exception, all belong to the RDPP families established in this paper.

Original language	English (US)
Pages (from-to)	748-773
Number of pages	26
Journal	Finite Fields and their Applications
Volume	15
Issue number	6
DOIs	https://doi.org/10.1016/j.ffa.2009.06.004
State	Published - Dec 2009
Externally published	Yes

Keywords

Almost perfect nonlinear function
Dickson polynomial
Finite field
Reversed Dickson polynomial

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10.1016/j.ffa.2009.06.004

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title = "Reversed Dickson polynomials over finite fields",

abstract = "Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials Dn (x, a) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on Fq with q < 200, with only one exception, all belong to the RDPP families established in this paper.",

keywords = "Almost perfect nonlinear function, Dickson polynomial, Finite field, Reversed Dickson polynomial",

author = "Hou, {Xiang dong} and Mullen, {Gary L.} and Sellers, {James A.} and Yucas, {Joseph L.}",

year = "2009",

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doi = "10.1016/j.ffa.2009.06.004",

language = "English (US)",

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journal = "Finite Fields and their Applications",

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TY - JOUR

T1 - Reversed Dickson polynomials over finite fields

AU - Hou, Xiang dong

AU - Mullen, Gary L.

AU - Sellers, James A.

AU - Yucas, Joseph L.

PY - 2009/12

Y1 - 2009/12

N2 - Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials Dn (x, a) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on Fq with q < 200, with only one exception, all belong to the RDPP families established in this paper.

AB - Reversed Dickson polynomials over finite fields are obtained from Dickson polynomials Dn (x, a) over finite fields by reversing the roles of the indeterminate x and the parameter a. We study reversed Dickson polynomials with emphasis on their permutational properties over finite fields. We show that reversed Dickson permutation polynomials (RDPPs) are closely related to almost perfect nonlinear (APN) functions. We find several families of nontrivial RDPPs over finite fields; some of them arise from known APN functions and others are new. Among RDPPs on Fq with q < 200, with only one exception, all belong to the RDPP families established in this paper.

KW - Almost perfect nonlinear function

KW - Dickson polynomial

KW - Finite field

KW - Reversed Dickson polynomial

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EP - 773

JO - Finite Fields and their Applications

JF - Finite Fields and their Applications

IS - 6

ER -

Reversed Dickson polynomials over finite fields

Abstract

Keywords

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