Generic Construction of Bent Functions and Bent Idempotents With Any Possible Algebraic Degrees
Generic Construction of Bent Functions and Bent Idempotents With Any Possible Algebraic Degrees
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DOI：
10.1109/tit.2017.2717966
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发表时间：
2017-06
影响因子：
2.5
通讯作者：
Chunming Tang;Zhengchun Zhou;Yanfeng Qi;Xiaosong Zhang;Cuiling Fan;T. Helleseth
中科院分区：
文献类型：
--
作者：
Chunming Tang;Zhengchun Zhou;Yanfeng Qi;Xiaosong Zhang;Cuiling Fan;T. Helleseth
As a class of optimal combinatorial objects, bent functions have important applications in cryptography, sequence design, and coding theory. Bent idempotents are a subclass of bent functions and of great interest, since they can be stored in less space and allow faster computation of the Walsh-Hadamard transform. The objective of this paper is to present a generic construction of bent functions from known ones. It includes the previous constructions of bent functions by Mesnager and Xu <italic>et al.</italic> as special cases, and produces new bent functions, which cannot be produced by earlier ones. In particular, it also generates infinite families of bent idempotents over <inline-formula> <tex-math notation="LaTeX">${F}_{2^{2m}}$ </tex-math></inline-formula> of any algebraic degree between 2 and <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>. This together with a recent construction by Su and Tang gives a positive answer to an open problem on bent idempotents proposed by Carlet. In addition, an infinite family of anti-self-dual bent functions is obtained in which the sum of any three distinct functions is again an anti-self-dual bent function in this family. This solves an open problem recently proposed by Mesnager.
As a class of optimal combinatorial objects, bent functions have important applications in cryptography, sequence design, and coding theory. Bent idempotents are a subclass of bent functions and of great interest, since they can be stored in less space and allow faster computation of the Walsh-Hadamard transform. The objective of this paper is to present a generic construction of bent functions from known ones. It includes the previous constructions of bent functions by Mesnager and Xu <italic>et al.</italic> as special cases, and produces new bent functions, which cannot be produced by earlier ones. In particular, it also generates infinite families of bent idempotents over <inline-formula> <tex-math notation="LaTeX">${F}_{2^{2m}}$ </tex-math></inline-formula> of any algebraic degree between 2 and <inline-formula> <tex-math notation="LaTeX">$m$ </tex-math></inline-formula>. This together with a recent construction by Su and Tang gives a positive answer to an open problem on bent idempotents proposed by Carlet. In addition, an infinite family of anti-self-dual bent functions is obtained in which the sum of any three distinct functions is again an anti-self-dual bent function in this family. This solves an open problem recently proposed by Mesnager.