Abstract Recently, the first constructions of bent partitions of elementary abelian groups that do not induce partial difference sets or Latin square type partial difference set packings (LP-packings) have been presented (Anbar et al.; Wang et al., 2025). Motivated by observations on the differential properties of examples of these bent partitions, the notions of twisted partial difference sets and twisted LP-packings in abelian groups $$\mathcal {G}$$ G are introduced in this article. Basic properties of twisted partial difference sets and twisted LP-packings, as well as properties of the corresponding character values, are investigated. It is shown that the sets arising from all recently introduced bent partitions are twisted partial difference sets, and that these bent partitions induce twisted LP-packings. As a consequence, all nontrivial bent partitions of elementary abelian groups known so far are shown to correspond either to LP-packings or to twisted LP-packings.

Abstract In Pasalic et al. (IEEE Trans Inf Theory 69:2702–2712, 2023), and in Anbar and Meidl (Cryptogr Commun 10:235–249, 2018), two different vectorial negabent and vectorial bent-negabent concepts are introduced, which leads to seemingly contradictory results. One of the main motivations for this article is to clarify the differences and similarities between these two concepts. Moreover, the negabent concept is extended to generalized Boolean functions from $${\mathbb {F}}_2^n$$ F 2 n to the cyclic group $${\mathbb {Z}}_{2^k}$$ Z 2 k . It is shown how to obtain nega- $${\mathbb {Z}}_{2^k}$$ Z 2 k -bent functions from $${\mathbb {Z}}_{2^k}$$ Z 2 k -bent functions, or equivalently, corresponding non-splitting relative difference sets from the splitting relative difference sets. This generalizes the shifting results for Boolean bent and negabent functions. We finally point to constructions of $${\mathbb {Z}}_8$$ Z 8 -bent functions employing permutations with the $$({\mathcal {A}}_m)$$ ( A m ) property, and more generally we show that the inverse permutation gives rise to $${\mathbb {Z}}_{2^k}$$ Z 2 k -bent functions.

We present secondary constructions of vectorial functions respectively partitions of elementary abelian groups, which simultaneously yield vectorial dual-bent functions with certain properties, bent partitions, and under some conditions, Latin square partial difference set packings (LP-packings). First, we analyse constructions via the direct sum of vectorial functions and then present a version of the generalized Maiorana-McFarland construction. Next, we generalize a construction of vectorial dual-bent functions by Wang, Fu, and Wei (2023). Finally, we use a lifting procedure of LP-packings from Jedwab and Li (2021) to construct vectorial dual-bent functions, bent partitions, and LP-packings in elementary abelian groups. With these constructions, a large variety of vectorial bent functions, bent partitions, LP-packings, and related amorphic association schemes can be obtained.

We introduce a new concept, theAPN-defect, which can be thought of as measuring the distance of a given function$G:\mathbb {F}_{2^{n}} \rightarrow \mathbb {F}_{2^{n}}$to the set of almost perfect nonlinear (APN) functions. This concept is motivated by the detailed analysis of the differential behaviour of non-APN functions (of low differential uniformity)Gusing the so-calleddifference squares. Indeed, the insight into some structural qualities of S-boxes provided by this new approach is particularly useful in the light of recent refinements of differential cryptanalysis. We describe the relations between the APN-defect and other current concepts of similar nature. Values of APN-defect for several classes of functions of interest, including Dembowski-Ostrom polynomials are given. This enables one to identify thequasi-APNones, i.e., those with favourable differential behavior. The difference square corresponding to a modification of the inverse function is determined, its APN-defect depending onnis evaluated, the partial quadruple system associated to it is described, and the implications are discussed. In the forthcoming second part of this work we further examine the APN-defect of modifications of the inverse function and address some questions concerning CCZ-equivalence. We also study modifications of classes of functions of low differential uniformity over infinitely many extensions of$\mathbb {F}_{2^{n}}$and present quantitative results on their differential behaviour.

Recently, the concept of (normal) bent partitions, which are partitions of elementary abelian groups having similar properties to spreads, has been introduced by Anbar and Meidl. A large number of bent partitions, so-called generalized semifield spreads, can be obtained from semifields with certain properties. A strongly related concept, namely Latin square partial difference set packings (LP-packings) in finite abelian groups, has also been introduced recently by Jedwab and Li. The examples for LP-packings in an elementary abelian group are obtained from spreads. LP-packings yield bent partitions (not only for elementary abelian groups). In this paper, we first point out that conversely, generalized semifield spreads yield LP-packings. As a result, there is a huge amount of LP-packings in elementary abelian groups, other than spreads. With some examples from ternary bent functions, we then show that normal bent partitions and LP-packings are not the same concept. Finally, we extend the lifting procedure from spreads to LP-packings in nonelementary abelian groups to a lifting procedure from some generalized spreads to LP-packings in nonelementary abelian groups and in larger elementary abelian groups. This potentially yields bent partitions other than generalized semifield spreads.

The recently introduced concept of a bent partition of a$2m$-dimensional vector space$\mathbb {V}_{2m}^{(p)}$over a prime field$\mathbb {F}_{p}$exhibits similar properties as a partition from a spread. In particular, it gives rise to a large family of bent functions obtained in the same manner as spread bent functions. We show that the first non-spread construction of bent partitions introduced by Pirsic and the third author ($p=2$), respectively, the first and the third author ($p$odd), gives rise to a large variety of different bent partitions. Especially, we show that the sets of bent functions obtained with any two such bent partitions do not intersect. We then show that every union of sets from one of these bent partitions always forms a partial difference set. This generalizes some known results on partial difference sets from spreads. Some general results on partial difference sets from bent partitions of$\mathbb {V}_{2m}^{(2)}$are given in the last section.

A function$F: \mathbb {F}_{2}^{n}\rightarrow \mathbb {F} _{2}^{n}$,$n=2m$, can have at most$2^{n}-2^{m}$bent component functions. Trivial examples are vectorial bent functions from$\mathbb {F}_{2}^{n}$to$\mathbb {F}_{2}^{m}$, seen as functions on$\mathbb {F}_{2}^{n}$. The first nontrivial example is given in univariate form as$x^{2^{r}} {\rm Tr^{n}_{m}}(x), 1\le r < m$(Pott et al. 2018), a few more examples of similar shape are given by Mesnager et al. 2019, and finally it has been shown that the quadratic function$F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$, has$2^{n}-2^{m}$bent components if and only if$\Lambda $is a linearized permutation polynomial of$\mathbb {F}_{2^{m}}[x]$(Anbar et al. 2021). In the first part of this article, an upper bound for the nonlinearity of plateaued functions with$2^{n}-2^{m}$bent components is shown, which is attained by the example$x^{2^{r}} {\rm Tr^{n}_{m}}(x)$. We then analyse in detail nonlinearity and differential spectrum of the class of functions$F(x) = x^{2^{r}} {\rm Tr^{n}_{m}}(\Lambda (x))$, which, as will be seen, requires the study of the functions$x^{2^{r}}\Lambda (x)$. In the last part we demonstrate that this class belongs to a larger class of functions with$2^{n}-2^{m}$Maiorana-McFarland bent components, which also contains nonquadratic and non-plateaued functions.

Pott et al. (2018) showed that F(x) = x2r Trn m(x), n = 2m, r ≥ 1, is a nontrivial example of a vectorial function with the maximal possible number 2n -2m of bent components. Mesnager et al. (2019) generalized this result by showing conditions on Λ(x) = x+ ∑σ j=1 αjx2tj, αj ∈ 2 F2m, under which F(x) = x2r Trn m(Λ(x)) has the maximal possible number of bent components. We simplify these conditions and further analyse this class of functions. For all related vectorial bent functions F(x) = Trn m(γF(x)), γ ∈ 2 F2n F2m, which as we will point out belong to the Maiorana-McFarland class, we describe the collection of the solution spaces for the linear equations DaF(x) = F(x) + F(x + a) + F(a) = 0, which forms a spread of F2n. Analysing these spreads, we can infer neat conditions for functions H(x) = (F(x);G(x)) from F2n to F2m × F2m to exhibit small differential uniformity (for instance for Λ(x) = x and r = 0 this fact is used in the construction of Carlet’s, Pott-Zhou’s, Taniguchi’s APN-function). For some classes of H(x) we determine differential uniformity and with a method based on Bezout’s theorem nonlineariy.