Several new infinite families of bent functions via second order derivatives
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Several new infinite families of bent functions via second order derivatives Lijing Zheng1,2 · Jie Peng3
· Haibin Kan1 · Yanjun Li3
Received: 21 May 2019 / Accepted: 29 April 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Inspired by a recent work of Tang et al. on constructing bent functions [14, IEEE TIT, 63(1): 6149-6157, 2017], we introduce a property (Pτ ) of any Boolean function that its second order derivatives vanish at any direction (ui , uj ) for some τ -subset {u1 , . . . , uτ } of F2n , and then establish a link between this property and the construction of Tang et al. (IEEE Trans. Inf. Theory 63(10), 6149–6157 2017). It enables us to find more bent functions efficiently. We construct (at least) five new infinite families of bent functions from some known functions: the Gold’s bent functions and some quadratic non-monomial bent functions, Leander’s monomial bent functions, Canteaut-Charpin-Kyureghyan’s monomial bent functions, and the Maiorana-McFarland class of bent functions, respectively. Our result generalizes some recent works on bent functions. We also provide the corresponding dual functions in all our constructions except the quadratic non-monomial one. It also turns out that we can get new bent functions outside the Maiorana-McFarland completed class. Keywords Bent functions · Second order derivative · Algebraic degree · Dual · Walsh spectrum Mathematics Subject Classification 2010 94A60 · 11T71 · 14G50
Jie Peng
[email protected] Lijing Zheng [email protected] Haibin Kan [email protected] Yanjun Li [email protected] 1
School of Computer Sciences, Fudan University, Shanghai, 200433, China
2
School of Mathematics and Physics, University of South China, Hengyang, Hunan, 421001, China
3
Mathematics and Science College, Shanghai Normal University, Guilin Road #100, Shanghai, 200234, China
Cryptography and Communications
1 Introduction In this paper, we often identify the finite field F2n with the n-dimensional vector space over F2 , say Fn2 . Any mapping f from F2n into F2 is called a Boolean function. Bent functions were introduced by Rothaus [12] in 1976. A bent function is a Boolean function in even number of variables which is maximally nonlinear in the sense that its Hamming distance to all the affine Boolean functions is optimal. Over the last four decades, bent functions have attracted a lot of research interest because of their relations to combinatorics, coding theory and applications in cryptography. A survey on bent functions can be found in [4] as well as the book [11]. Bent functions always occur in pairs. In fact, given a bent function g(x), one can always define its dual function g ∗ (x) which is again a bent function. However, computing the dual of a bent function is not an easy task in general. Recently, Mesnager [9] obtained a strong version of [3, Theorem 3], and presented several infinite families of bent functions with their dual functions being computed explicitly. Tang et al. generalized a main result of [9] by pr
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