New quadratic bent functions in polynomial forms with coefficients in extension fields

  • Dongmei Huang
  • Chunming TangEmail author
  • Yanfeng Qi
  • Maozhi Xu
Original Paper


In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form \(f(x)=\sum _{i=1}^{{n}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^i})+ \mathrm {Tr}_1^{n/2}(c_{n/2}x^{1+2^{n/2}})\), where n is even, \(c_i\in \mathrm {GF}(2^n)\) for \(1\le i \le {n}/{2}-1\) and \(c_{n/2}\in \mathrm {GF}(2^{n/2})\). The bentness of these functions can be connected with linearized permutation polynomials. Hence, methods for constructing quadratic bent functions are given. Further, we consider a subclass of quadratic Boolean functions of the form \(f(x)=\sum _{i=1}^{{m}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^{ei}})+ \mathrm {Tr}_1^{n/2}(c_{m/2}x^{1+2^{n/2}})\), where \(n=em\), m is even, and \(c_i\in \mathrm {GF}(2^e)\). The bentness of these functions is characterized and some methods for deriving new quadratic bent functions are given. Finally, when m and e satisfy some conditions, we determine the number of these quadratic bent functions.


Bent function Boolean function Linearized permutation polynomial Cyclotomic polynomial Semi-bent function 

Mathematics Subject Classification

06E75 94A60 



The authors are very grateful to the anonymous reviewers and Prof. Teo Mora for their valuable comments and suggestions. This work is supported by the National Natural Science Foundation of China (Grant Nos. 11871058, 11531002, 11701129, 61672059). C. Tang also acknowledges support from 14E013, CXTD2014-4 and the Meritocracy Research Funds of China West Normal University. Y. Qi also acknowledges support from Zhejiang provincial Natural Science Foundation of China (LQ17A010008, LQ16A010005).


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Dongmei Huang
    • 1
  • Chunming Tang
    • 1
    Email author
  • Yanfeng Qi
    • 2
  • Maozhi Xu
    • 3
  1. 1.School of Mathematics and InformationChina West Normal UniversityNanchongChina
  2. 2.School of ScienceHangzhou Dianzi UniversityHangzhouChina
  3. 3.LMAM, School of Mathematical SciencesPeking UniversityBeijingChina

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