Binomial permutations over finite fields with even characteristic

Ziran Tu; Xiangyong Zeng; Yupeng Jiang; Yan Li

doi:10.1007/s10623-021-00959-4

Binomial permutations over finite fields with even characteristic

Ziran Tu
, Xiangyong Zeng^*
, Yupeng Jiang
, Yan Li

^*Corresponding author for this work

School of Cyber Science and Technology

Hubei University

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

8 Scopus citations

Abstract

In this paper, we study binomials having the form x^r(a+ x³⁽^q^-¹⁾) over the finite field Fq2 with q= 2 ^m, and determine all the r’s and coefficients a’s making them permutations. For even m or odd m with 3 ∤ m, we prove that the characterization is necessary and sufficient. For the case of odd m and 3 ∣ m, we prove that the corresponding sufficient condition is also necessary for almost all r’s. Finally we obtain that the proportion of r’s we cannot prove the necessity is only about 140.

Original language	English
Pages (from-to)	2869-2888
Number of pages	20
Journal	Designs, Codes, and Cryptography
Volume	89
Issue number	12
DOIs	https://doi.org/10.1007/s10623-021-00959-4
State	Published - Dec 2021

Keywords

Finite field
Permutation binomial
Permutation polynomial

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title = "Binomial permutations over finite fields with even characteristic",

abstract = "In this paper, we study binomials having the form xr(a+ x3(q-1)) over the finite field Fq2 with q= 2 m, and determine all the r{\textquoteright}s and coefficients a{\textquoteright}s making them permutations. For even m or odd m with 3 ∤ m, we prove that the characterization is necessary and sufficient. For the case of odd m and 3 ∣ m, we prove that the corresponding sufficient condition is also necessary for almost all r{\textquoteright}s. Finally we obtain that the proportion of r{\textquoteright}s we cannot prove the necessity is only about 140.",

keywords = "Finite field, Permutation binomial, Permutation polynomial",

author = "Ziran Tu and Xiangyong Zeng and Yupeng Jiang and Yan Li",

note = "Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.",

year = "2021",

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doi = "10.1007/s10623-021-00959-4",

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TY - JOUR

T1 - Binomial permutations over finite fields with even characteristic

AU - Tu, Ziran

AU - Zeng, Xiangyong

AU - Jiang, Yupeng

AU - Li, Yan

PY - 2021/12

Y1 - 2021/12

N2 - In this paper, we study binomials having the form xr(a+ x3(q-1)) over the finite field Fq2 with q= 2 m, and determine all the r’s and coefficients a’s making them permutations. For even m or odd m with 3 ∤ m, we prove that the characterization is necessary and sufficient. For the case of odd m and 3 ∣ m, we prove that the corresponding sufficient condition is also necessary for almost all r’s. Finally we obtain that the proportion of r’s we cannot prove the necessity is only about 140.

AB - In this paper, we study binomials having the form xr(a+ x3(q-1)) over the finite field Fq2 with q= 2 m, and determine all the r’s and coefficients a’s making them permutations. For even m or odd m with 3 ∤ m, we prove that the characterization is necessary and sufficient. For the case of odd m and 3 ∣ m, we prove that the corresponding sufficient condition is also necessary for almost all r’s. Finally we obtain that the proportion of r’s we cannot prove the necessity is only about 140.

KW - Finite field

KW - Permutation binomial

KW - Permutation polynomial

UR - https://www.scopus.com/pages/publications/85116775727

U2 - 10.1007/s10623-021-00959-4

DO - 10.1007/s10623-021-00959-4

M3 - 文章

AN - SCOPUS:85116775727

SN - 0925-1022

VL - 89

SP - 2869

EP - 2888

JO - Designs, Codes, and Cryptography

JF - Designs, Codes, and Cryptography

IS - 12

ER -

Binomial permutations over finite fields with even characteristic

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