Constructing vectorial bent functions via second-order derivatives

Let $n$ be an even positive integer, and $m<n$ be one of its positive divisors. In this paper, inspired by a nice work of Tang et al. on constructing large classes of bent functions from known bent functions [27, IEEE TIT, 63(10): 6149-6157, 2017], we consider the construction of vectorial bent and vectorial plateaued $(n,m)$-functions of the form $H(x)=G(x)+g(x)$, where $G(x)$ is a vectorial bent $(n,m)$-function, and $g(x)$ is a Boolean function over $\mathbb{F}_{2^{n}}$. We find an efficient generic method to construct vectorial bent and vectorial plateaued functions of this form by establishing a link between the condition on the second-order derivatives and the key condition given by [27]. This allows us to provide (at least) three new infinite families of vectorial bent functions with high algebraic degrees. New vectorial plateaued $(n,m+t)$-functions are also obtained ($t\geq 0$ depending on $n$ can be taken as a very large number), two classes of which have the maximal number of bent components

Graphs of Vectorial Plateaued Functions as Difference Sets

A function $F:\mathbb{F}_{p^n}\rightarrow \mathbb{F}_{p^m},$ is a vectorial $s$-plateaued function if for each component function $F_{b}(\mu)=Tr_n(\alpha F(x)), b\in \mathbb{F}_{p^m}^*$ and $\mu \in \mathbb{F}_{p^n}$, the Walsh transform value $|\widehat{F_{b}}(\mu)|$ is either $0$ or $p^{\frac{n+s}{2}}$. In this paper, we explore the relation between (vectorial) $s$-plateaued functions and partial geometric difference sets. Moreover, we establish the link between three-valued cross-correlation of $p$-ary sequences and vectorial $s$-plateaued functions. Using this link, we provide a partition of $\mathbb{F}_{3^n}$ into partial geometric difference sets. Conversely, using a partition of $\mathbb{F}_{3^n}$ into partial geometric difference sets, we constructed ternary plateaued functions $f:\mathbb{F}_{3^n}\rightarrow \mathbb{F}_3$. We also give a characterization of $p$-ary plateaued functions in terms of special matrices which enables us to give the link between such functions and second-order derivatives using a different approach.Comment: regular research pape

Full characterization of generalized bent functions as (semi)-bent spaces, their dual, and the Gray image

In difference to many recent articles that deal with generalized bent (gbent) functions $f:\mathbb{Z}_2^n \rightarrow \mathbb{Z}_q$ for certain small valued $q\in \{4,8,16 \}$, we give a complete description of these functions for both $n$ even and odd and for any $q=2^k$ in terms of both the necessary and sufficient conditions their component functions need to satisfy. This enables us to completely characterize gbent functions as algebraic objects, namely as affine spaces of bent or semi-bent functions with interesting additional properties, which we in detail describe. We also specify the dual and the Gray image of gbent functions for $q=2^k$. We discuss the subclass of gbent functions which corresponds to relative difference sets, which we call $\mathbb{Z}_q$-bent functions, and point out that they correspond to a class of vectorial bent functions. The property of being $\mathbb{Z}_q$-bent is much stronger than the standard concept of a gbent function. We analyse two examples of this class of functions.Comment: 20 page

Large Sets of Orthogonal Sequences Suitable for Applications in CDMA Systems

In this paper, we employ the so-called semi-bent functions to achieve significant improvements over currently known methods regarding the number of orthogonal sequences per cell that can be assigned to a regular tessellation of hexagonal cells, typical for certain code-division multiple-access (CDMA) systems. Our initial design method generates a large family of orthogonal sets of sequences derived from vectorial semi-bent functions. A modification of the original approach is proposed to avoid a hard combinatorial problem of allocating several such orthogonal sets to a single cell of a regular hexagonal network, while preserving the orthogonality to adjacent cells. This modification increases the number of users per cell by starting from shorter codewords and then extending the length of these codewords to the desired length. The specification and assignment of these orthogonal sets to a regular tessellation of hexagonal cells have been solved regardless of the parity and size of $m$ (where $2^m$ is the length of the codewords). In particular, when the re-use distance is $D=4$ the number of users per cell is $2^{m-2}$ for almost all $m$, which is twice as many as can be obtained by the best known methods.Comment: 24 pages, 4 figures, 5 tables. IEEE Transactions on Information Theory,vol.62, no.6, 201

A note on APN permutations in even dimension

APN permutations in even dimension are vectorial Boolean functions that play a special role in the design of block ciphers. We study their properties, providing some general results and some applications to the low-dimension cases. In particular, we prove that none of their components can be quadratic. For an APN vectorial Boolean function (in even dimension) with all cubic components we prove the existence of a component having a large number of balanced derivatives. Using these restrictions, we obtain the first theoretical proof of the non-existence of APN permutations in dimension 4. Moreover, we derive some contraints on APN permutations in dimension 6

More characterizations of generalized bent function in odd characteristic, their dual and the gray image

In this paper, we further investigate properties of generalized bent Boolean functions from $\Z_{p}^n$ to $\Z_{p^k}$, where $p$ is an odd prime and $k$ is a positive integer. For various kinds of representations, sufficient and necessary conditions for bent-ness of such functions are given in terms of their various kinds of component functions. Furthermore, a subclass of gbent functions corresponding to relative difference sets, which we call $\Z_{p^k}$-bent functions, are studied. It turns out that $\Z_{p^k}$-bent functions correspond to a class of vectorial bent functions, and the property of being $\Z_{p^k}$-bent is much stronger then the standard bent-ness. The dual and the generalized Gray image of gbent function are also discussed. In addition, as a further generalization, we also define and give characterizations of gbent functions from $\Z_{p^l}^n$ to $\Z_{p^k}$ for a positive integer $l$ with $l<k$

Characterizations of o-polynomials by the Walsh transform

The notion of o-polynomial comes from finite projective geometry. In 2011 and later, it has been shown that those objects play an important role in symmetric cryptography and coding theory to design bent Boolean functions, bent vectorial Boolean functions, semi-bent functions and to construct good linear codes. In this note, we characterize o-polynomials by the Walsh transform of the associated vectorial functions

APN trinomials and hexanomials

In this paper we give a new family of APN trinomials of the form $X^{2^k+1} + (\mathsf{tr}^{n}_{m}(X))^{2^k+1}$ on $\mathbb{F}_{2^n}$ where $\mathsf{gcd}(k,n)=1$ and $n = 2m = 4t$, and prove its important properties. The family satisfies for all $n = 4t$ an interesting property of the Kim function which is, up to equivalence, the only known APN function equivalent to a permutation on $\mathbb{F}_{2^{2m}}$. As another contribution of the paper, we consider a family of hexanomials $g_{C,k}$ which was shown to be differentially $2^{\mathsf{gcd}(m,k)}$-uniform by Budaghyan and Carlet (2008) when a quadrinomial $P_{C,k}$ has no roots in a specific subgroup. In this paper, for all $(m,k)$ pairs, we characterize, construct and count all $C \in \mathbb{F}_{2^n}$ satisfying the condition. Bracken, Tan and Tan (2014) and Qu, Tan and Li (2014) constructed some elements $C$ satisfying the condition when $m \equiv 2 \textrm{or} 4 \pmod{6}$ and $m \equiv 0 \pmod{6}$ respectively, both requiring $\mathsf{gcd}(m,k) = 1$. Bluher (2013) proved that such $C$ exists if and only if $k \ne m$ without characterizing, constructing or counting those $C$. To prove the results, we effectively use a Trace-$0$/Trace-$1$ (relative to the subfield $\mathbb{F}_{2^m}$) decomposition of $\mathbb{F}_{2^n}$.Comment: 19 pages; submitted Feb 12, 2014; updated Jul 29, 201

Practical Bijective S-box Design

We construct 8 x 8 bijective cryptographically strong S-boxes. Our construction is based on using non-bijective power functions over the finite filed

There are infinitely many bent functions for which the dual is not 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-weaky regular bent functions. However, the first known construction of non-weakly regular bent functions by Ce\c{s}melio\u{g}lu et {\it al.}, 2012, 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 Ce\c{s}melio\u{g}lu et {\it al.} one can obtain infinitely many non-dual-bent functions once one example of a non-dual-bent function is known