Chapter

Open Problems in Mathematics and Computational Science

pp 243-273

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On Semi-bent Functions and Related Plateaued Functions Over the Galois Field \(\mathbb{F}_{2^{n}}\)

  • Sihem MesnagerAffiliated withDepartment of Mathematics, University of Paris VIII, LAGA (Laboratoire Analyse, GĂ©ometrie et Applications), UMR 7539, CNRSUniversity of Paris XIII, Sorbonne Paris Cité Email author 

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Abstract

Plateaued functions were introduced in 1999 by Zheng and Zhang as good candidates for designing cryptographic functions since they possess desirable various cryptographic characteristics. They are defined in terms of the Walsh–Hadamard spectrum. Plateaued functions bring together various nonlinear characteristics and include two important classes of Boolean functions defined in even dimension: the well-known bent functions and the semi-bent functions. Bent functions (including their constructions) have been extensively investigated for more than 35 years. Very recently, the study of semi-bent functions has attracted the attention of several researchers. Much progress in the design of such functions has been made. The chapter is devoted to certain plateaued functions. The focus is particularly on semi-bent functions defined over the Galois field \(\mathbb{F}_{2^{n}}\) (n even). We review what is known in this framework and investigate constructions.