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.

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.

It is shown that the generalized Rothaus construction of p-ary bent functions can be extended to a construction of a vectorial bent function with non-weakly regular components, for which in general the duals are not a bent function, i.e., they belong to the class of non-dual bent functions. This complements results on other two constructions of non-weakly regular bent functions, the generalized Maiorana-McFarland construction and the semi-direct sum, for which vectorial versions are presented and the properties of their duals are investigated in the literature. The distribution of the values of the Walsh transform of vectorial bent functions (with non-weakly regular components) is then analysed in detail. Among others, a condition on the values of the Walsh transform of a vectorial bent function from$\mathbb {F}_{p}^{n}$to$\mathbb {F}_{p}^{m}$is presented, which implies that$m \le \lceil n/2\rceil $. This refines a classical result by Nyberg, which states that for an$(n,m)$bent function, n even, with only regular components, m can be at most$n/2$. Some results on the weight distribution of codes obtained from vectorial bent functions with non-weakly regular components complement the article.

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.

For a function $f$ from $\mathbb {F}_{2}^{n}$ to $\mathbb {F}_{2}^{n}$ , the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure of the distance between $f$ and the set of almost perfect nonlinear functions. In some cases, the number of vanishing flats serves as an “intermediate” concept between differential uniformity and differential spectrum, which contains more information than differential uniformity, however less than the differential spectrum. Secondly, the set of vanishing flats forms a combinatorial configuration called partial quadruple system, since it conveys a detailed structural information about $f$ . We initiate this study by considering the number of vanishing flats and the partial quadruple systems associated with monomials and Dembowski-Ostrom polynomials. In addition, we present an application of vanishing flats to the partition of a vector space into disjoint equidimensional affine spaces. We conclude the paper with several further questions and challenges.

A natural generalization of bent functions is a class of functions from F2nto Z(2k) which is known as generalized bent (gbent) functions. The construction and characterization of gbent functions are commonly described in terms of the Walsh transforms of the associated Boolean functions. Using similar approach, we first determine the dual of a gbent function when n is even. Then, depending on the parity of n, it is shown that the Gray image of a gbent function is (k - 1) or (k - 2) plateaued, which generalizes previous results for k = 2,3, and 4. We then completely characterize gbent functions as algebraic objects. More precisely, again depending on the parity of n, a gbent function is a (k - 1)-dimensional affine space of bent functions or semi-bent functions with certain interesting additional properties, which we completely describe. Finally, we also consider a subclass of functions from F2nto Z(2k), called Zq-bent functions (which are necessarily gbent), which essentially gives rise to relative difference sets similarly to standard bent functions. Two examples of this class of functions are provided and it is demonstrated that many gbent functions are not Zq-bent.

In this paper, we introduce generalized hyperbent functions from F2nto ℤ2k, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions f from F2nto ℤ2kconsist of components which are generalized (hyper)bent functions from F2ntoZ2k'for some k' <; k. For even n, most notably we show that the g-hyperbentness of f is equivalent to the hyperbentness of the components of f with some conditions on the Walsh-Hadamard coefficients. For odd n, we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even n, where the associated Boolean functions are bent.

Bent functions can be classified into regular bent functions, weakly regular but not regular bent functions, and non-weakly regular bent functions. Regular and weakly regular bent functions always appear in pairs, since their duals are also bent functions. In general, this does not apply to non-weakly regular bent functions. However, the first known construction of non-weakly regular bent functions by Çeşmelioğlu et al. yields bent functions for which the dual is also bent. In this paper, the first construction of non-weakly regular bent functions for which the dual is not bent is presented. We call such functions non-dual-bent functions. Until now, only sporadic examples found via computer search were known. We then show that with the direct sum of bent functions and with the construction by Çeşmelioğlu et al., one can obtain infinitely many non-dual-bent functions once one example of a non-dual-bent function is known.

We show that bent functions $f$ from ${\mathbb F}_{p^m}\times{\mathbb F}_{p^m}$ to ${\mathbb F}_p$, which are constant or affine on the elements of a given spread of ${\mathbb F}_{p^m}\times{\mathbb F}_{p^m}$, either arise from partial spread bent functions, or they are Boolean and a generalization of Dillon's class $H$. For spreads of a presemifield $S$, we show that a bent function of the second class corresponds to an o-polynomial of a presemifield in the Knuth orbit of $S$. In contrast to the finite fields case, we have to consider pairs of (pre)semifields in a Knuth orbit. We give a canonical example of an o-polynomial for commutative presemifields (which also defines a hyperoval on the semifield plane) and show that the corresponding bent functions belong to the completed Maiorana--McFarland class. Using Albert's twisted fields and Kantor's family of presemifields, we explicitly present examples of such bent functions.

The Walsh transform f̂ of a quadratic function f: F(pn) → Fpsatisfies |f̂| ∈ {0,pn+s/2} for an integer 0 ≤ s ≤ n-1, depending on f. In this paper, quadratic functions of the form Fp,n(x) = Trn(Σi=0kaixpi+1) are studied, with the restriction that ai∈ Fp, 0 ≤ i ≤ k. Three methods for enumeration of such functions are presented when the value for s is prescribed. This paper extends earlier enumeration results significantly, for instance, the generating function for the counting function is obtained, when n is odd and relatively prime to p, or when n = 2 m, for odd m and p = 2. The number of bent and semibent functions for various classes of n is also obtained.

In this paper, a technique for constructing$p$-ary bent functions from plateaued functions is presented. This generalizes earlier techniques of constructing bent from near-bent functions. The Fourier spectrum of quadratic monomials is analyzed, and examples of quadratic functions with highest possible absolute values in their Fourier spectrum are given. Applying the construction of bent functions to the latter class of functions yields bent functions attaining upper bounds for the algebraic degree when$p=$3,5. Until now, no construction of bent functions attaining these bounds was known.

The d-ary Sidel'nikov sequence S = s0, s1... of period q-1 for a prime power q=pmis a frequently analyzed sequence in the literature. Recently, it turned out that the linear complexity over Fpof the d-ary Sidel'nikov sequence is considerably smaller than the period if the sequence element s(q-1)/2mod(q-1)is chosen adequately. In this paper this work is continued and tight lower bounds on the linear complexity over Fpof the d-ary Sidel'nikov sequence are given. For certain cases exact values are provided. Finally, results on the k-error linear complexity over Fpof the d-ary Sidel'nikov sequence are presented.

The k-error linear complexity of a periodic binary sequence is defined to be the smallest linear complexity that can be obtained by changing k or fewer bits per period. This contribution focuses on the case of 2n-periodic binary sequences. For k=1,2, the exact formula for the expected k-error linear complexity of a sequence having maximal possible linear complexity 2n, and the exact formula of the expected 1-error linear complexity of a random 2n-periodic binary sequence are provided. For k ges 2, lower and upper bounds on the expected value of the k-error linear complexity of a random 2n-periodic binary sequence are established

Rueppel (1986) conjectured that periodic binary sequences have expected linear complexity close to the period length N. In this paper, we determine the expected value of the linear complexity of N-periodic sequences explicitly and confirm Rueppel's conjecture for arbitrary finite fields. Cryptographically strong sequences should not only have a large linear complexity, but also the change of a few terms should not cause a significant decrease of the linear complexity. This requirement leads to the concept of the k-error linear complexity of N-periodic sequences. We present a method to establish a lower bound on the expected k-error linear complexity of N-periodic sequences based on the knowledge of the counting function /spl Nscr//sub N/,/sub 0/(c), i.e., the number of N-periodic sequences with given linear complexity c. For some cases, we give explicit formulas for that lower bound and we also determine /spl Nscr//sub N,0/(c).

Let p be a prime, r a positive integer, q=p/sup r/, and d a divisor of p(q-1). We derive lower bounds on the linear complexity over the residue class ring Z/sub d/ of a (q-periodic) sequence representing the residues modulo d of the discrete logarithm in F/sub q/. Moreover, we investigate a sequence over F/sub q/ representing the values of a certain polynomial over F/sub q/ introduced by Mullen and White (1986) which can be identified with the discrete logarithm in F/sub q/ via p-adic expansions and representations of the elements of F/sub q/ with respect to some fixed basis.