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Four decades of research on bent functions. (English) Zbl 1378.94028
As indicated in the thesis of J. Dillon [Elementary Hadamard difference sets. College Park: University of Maryland (PhD Dissertation) (1974)], the first paper in English on bent functions has been written by O. S. Rothaus in 1966, but its final version was published ten years later [J. Comb. Theory, Ser. A 20, 300–305 (1976; Zbl 0336.12012)]. These two pioneer documents on bent functions had a huge impact on research in many directions, many ideas and concepts which developed as research topics were present in these two documents.
In this excellent survey paper, the authors (as they also indicate themselves) revisit the results from Dillon’s thesis and Rothaus’ paper, which gives them an itinerary for presenting the current state of the art in bent functions and to provide a survey on these functions with historical viewpoint. The paper can serve as introduction aid to the topic of bent functions, and it is also very beneficial for researchers who are familiar with the topic as a reference work covering several aspects on bent and related functions, and containing a comprehensive reference list.
After a short introduction, the basic concepts for Boolean and \(p\)-ary functions are recalled in the preliminaries. Bent functions are defined in various ways. In Section 3, the concept of the dual of a bent function is presented, Section 4 discusses equivalence of bent functions, and the problem of classification and enumeration of bent functions. Normality, a property many classes of bent functions possess, is dealt with in Section 5. The primary constructions of bent functions, i.e. the Maiorana-McFarland class, the partial spread class and Dillon’s class \(\mathcal{H}\), are presented in Section 6, some main secondary constructions of (Boolean) bent functions are recalled in Section 8. Sections 7 and 10 deal with representations of bent functions. Whereas Section 7 recalls a general result on a representation of bent functions via geometric properties, in Section 10 some classes of bent functions in univariate and bivariate representation are collected. Section 9 is on \(\mathbb{Z}\)-bent functions. Section 11 collects results on various subclasses and super-classes of bent functions, like hyper-bent functions, rotation symmetric respectively idempotent bent functions, homogeneous bent functions, and partially bent and plateaued functions. Vectorial bent functions and aspects on applications for S-Boxes are discussed in Section 12. In Section 13 the authors point to some generalized bent concepts like \(\mathbb{Z}_4\)-valued bent functions. In the final Section 14 some results on \(p\)-ary bent functions in univariate form are presented.

MSC:
94A60 Cryptography
06E30 Boolean functions
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