A new class of $ p $-ary regular bent functions

Chunming Tang; Maozhi Xu; Yanfeng Qi; null Mingshuo Zhou

doi:10.3934/amc.2020042

Article Contents

2021, Volume 15, Issue 1: 55-64. Doi: 10.3934/amc.2020042

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$ C_{2t}\times C_2 $

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A new class of $ p $-ary regular bent functions

1.
School of Mathematics and Information, China West Normal University, Sichuan Nanchong, 637002, China
2.
School of Mathematical Sciences, Peking University, Beijing, 100871, China
3.
School of Science, Hangzhou Dianzi University, Hangzhou, Zhejiang, 310018, China
4.
Center of Applied Mathematics, School of Mathematics, Tianjin University, Tianjin, 300072, China

^* Corresponding author: Chunming Tang
^* Corresponding author: Chunming Tang

Received: March 2019

Revised: July 2019

Early access: November 2019

Published: February 2021

This work is supported by the National Natural Science Foundation of China (Grant No. 11871058, 61672059, 11701129, 11531002). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi and M. Zhou also acknowledge support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005)

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Abstract

Bent functions have many important applications in cryptography and coding theory. This paper considers a class of $ p $-ary functions with the Dillon exponent of the form

$ f(x) = \sum\limits_{i = 0}^{q-1}(Tr^n_1(a_1x^{(r i+s)(q-1)})+Tr^n_1(a_2x^{(r i+s)(q-1)+\frac{q^2-1}{2}}))+bx^{\frac{q^2-1}{2}}, $

where $ n = 2m $, $ q = p^m $, $ p $ is an odd prime, $ a_1,a_2\in \mathbb{F}_{p^n} $, and $ b\in \mathbb{F}_p $. With the help of Kloosterman sums, we present an explicit characterization of these $ p $-ary regular bent functions for the case $ gcd(s-r,\frac{q+1}{2}) = 1 $ and $ gcd(r,q+1) = 1 $ or $ 2 $. Our results generalize results of Li et al. [IEEE Trans. Inf. Theory 59 (2013) 1818-1831].

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Mathematics Subject Classification: Primary: 06E75, 94A60, 11T23.

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References

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