A class of cubic polynomial semi-bent functions over F 2 n

Construction methods for generalized bent functions

In this paper, we introduce a class of cubic rotation symmetric (RotS) functions and prove that it can yield bent and semi-bent functions. To the best of our knowledge, this is the second primary construction of an infinite class of nonquadratic RotS bent functions which could be found and the first class of nonquadratic RotS semi-bent functions. We also study a class of idempotents (giving RotS functions through the choice of a normal basis of \(GF(2^n)\) over \(GF(2)\)). We derive a characterization of the bent functions among these idempotents and we relate their precise determination to a problem studied in the framework of APN functions. Incidentally, the proofs of bentness given here are useful for a paper studying a construction of idempotents from RotS functions, entitled “A secondary construction and a transformation on rotation symmetric functions, and their action on bent and semi-bent functions” by the same authors, to appear in the journal JCT series A.

Plateaued functions have been introduced by Zheng and Zhang in 1999 as good candidates for designing cryptographic functions since they possess many desirable cryptographic characteristics. Plateaued functions bringtogether various nonlinear characteristics and include twoimportant classes of Boolean functions defined in even dimension: the well-known bent functions ($0$-plateaued functions) and the semi-bent functions ($2$-plateaued functions). Bent functions have been extensively investigated since 1976. Very recently, the study of semi-bent functions has attracted a lot of attention in symmetric cryptography. Many intensive progresses in the design of such functions have been made especially in recent years.The paper is devoted to the construction of semi-bent functions on the finite field $\mathbb{F}_{2^n}$ ($n=2m$) in the line of a recent work of S. Mesnager [IEEE Transactions on Information Theory, Vol 57, No 11, 2011]. We extend Mesnager's results and present a new construction of infinite classes of binary semi-bent functions in polynomial trace. The extension is achieved by inserting mappings $h$ on $\mathbb{F}_{2^n}$ which can be expressed as $h(0) = 0$ and $h(uy) = h_1(u)h_2(y)$ with $u$ ranging over the circle $U$ of unity of $\mathbb{F}_{2^n}$, $y \in \mathbb{F}_{2^m}^{*}$ and $uy \in \mathbb{F}_{2^n}^{*}$, where $h_1$ is a isomorphism on $U$ and $h_2$ is an arbitrary mapping on $\mathbb{F}_{2^m}^{*}$. We then characterize the semi-bentness property of the extended family in terms of classical binary exponential sums and binary polynomials.

Semi-bent functions have very high nonlinearity and hence they have many applications in symmetric-key cryptography, binary sequence design for communications, and combinatorics. In this paper, we focus on studying the additive autocorrelation of semi-bent functions. We provide a lower bound on the maximum additive autocorrelation absolute value of semi-bent functions with three-level additive autocorrelation. Semi-bent functions with three-level additive autocorrelation achieving this bound with equality are said to have perfect three-level additive autocorrelation. We present two classes of balanced semi-bent functions with optimal algebraic degree and perfect three-level additive autocorrelation. The functions in the first class are constructed from the generalized Maiorana-McFarland class. The functions in the second one are derived from the concatenation of two bent functions, and we obtain a necessary and sufficient condition on bent functions such that the constructed semi-bent functions have perfect three-level additive autocorrelation.

Equivalence for negabent functions and their relative difference sets

This article is the successor of Dempwolff and Neumann (Des. Codes Cryptogr. 57:373---381, 2010). We now consider semibent functions with a linear structure. Semibent functions of partial spread type with a linear structure seem to be rare. We distinguish four classes of such semibent functions. For three classes we exhibit some examples.

A natural generalization of bent functions is a class of functions from ${\mathbb F}_{2}^{n}$ to ${\mathbb Z}_{2^{k}}$ 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 ${\mathbb F}_{2}^{n}$ to ${\mathbb Z}_{2^{k}}$ , called ${\mathbb Z}_{q}$ -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 ${\mathbb Z}_{q}$ -bent.

A class of balanced semi-bent functions with an even number of variables is proposed. It is shown that they include one subclass of semi-bent functions with maximum algebraic degrees. Furthermore, an example of semi-bent functions in a small field is given by using the zeros of some Kloosterman sums. Based on the result given by S. Kim et al., an example of infinite families of semi-bent functions is also obtained.

Semi-bent functions are a kind of Boolean functions with high nonlinearity. They have important applications in cryptography and communications. In this paper, two classes of semi-bent functions with Niho exponents are proposed. It is shown that all semi-bent functions of the first class attain the maximum algebraic degree, and there exists one subclass of semi-bent functions with maximum algebraic degree in the second class. Furthermore, two examples of semi-bent functions in a small field are given by using the zeros of some Kloosterman sums. Based on the result given by Kim et al., two examples of infinite families of semi-bent functions are also obtained. These results provide more available Boolean functions with high nonlinearity and high algebraic degrees for designing the filter generators of stream ciphers.

Plateaued functions were introduced in 1999 by Zheng and Zhang as good candidates for designing cryptographic functions since they possess desirable various cryptographic characteristics. They are defined in terms of the Walsh–Hadamard spectrum. Plateaued functions bring together various nonlinear characteristics and include two important classes of Boolean functions defined in even dimension: the well-known bent functions and the semi-bent functions. Bent functions (including their constructions) have been extensively investigated for more than 35 years. Very recently, the study of semi-bent functions has attracted the attention of several researchers. Much progress in the design of such functions has been made. The chapter is devoted to certain plateaued functions. The focus is particularly on semi-bent functions defined over the Galois field \(\mathbb{F}_{2^{n}}\) (n even). We review what is known in this framework and investigate constructions.

Plateaued functions are significant in cryptography as they possess various desirable cryptographic properties. Two important classes of plateaued functions are those of bent functions and semi-bent functions, due to their combinatorial and algebraic properties. Constructions of bent functions have been extensively investigated. However only few constructions of semi-bent functions have been proposed in the literature. In general, finding new constructions of bent and semi-bent functions is not a simple task. The paper is devoted to the construction of semi-bent functions with even number of variables. We show that bent functions give rise to primary and secondary-like constructions of semi-bent functions.

The so-called plateaued functions in n variables (or r-plateaued functions) have been introduced in 1999 by Zheng and Zhang in [61] for 0 < r < n. They were firstly studied by these authors in [62, 63] and further by Carlet and Prouff in [16] as good candidates for designing cryptographic functions. The Walsh–Hadamard spectrum is a very important tool do define and design plateaued functions. An n-variable Boolean function is said to be r-plateaued if the values of its Walsh transform belong to the set \(\{0,\pm 2^{\frac{n+r} {2} }\}\) for some fixed r, 0 ≤ r ≤ n. Consequently, plateaued functions have low Hadamard transform, which provides protection against fast correlation attacks [42] and linear cryptanalysis [41]. It has been shown in [61] that plateaued functions are significant in cryptography as they possess various desirable characteristics such as high nonlinearity, resiliency, low additive autocorrelation, high algebraic degree and satisfy propagation criteria. Plateaued functions bring together various nonlinear characteristics. They include three significant classes of Boolean functions: the well-known bent functions, the near-bent functions and the semi-bent functions. More precisely, bent functions are exactly 0-plateaued functions, near-bent (also called semi-bent in odd dimension) are 1-plateaued functions and semi-bent functions are 2-plateaued functions. 0-plateaued functions and 2-plateaued functions on \(\mathbb{F}_{2^{n}}\) exist when n is even, while 1-plateaued functions on \(\mathbb{F}_{2^{n}}\) exist when n is odd.

Semi-bent functions with even number of variables are a class of important Boolean functions whose Hadamard transform takes three values. Semi-bent functions have been extensively studied due to their applications in cryptography and coding theory. In this paper we are interested in the property of semi-bentness of Boolean functions defined on the Galois field <${\mathbb F}_2^n$ (n even) with multiple trace terms obtained via Niho functions and two Dillon-like functions (the first one has been studied by the author and the second one has been studied very recently by Wang et al. using an approach introduced by the author). We subsequently give a connection between the property of semi-bentness and the number of rational points on some associated hyperelliptic curves. We use the hyperelliptic curve formalism to reduce the computational complexity in order to provide an efficient test of semi-bentness leading to substantial practical gain thanks to the current implementation of point counting over hyperelliptic curves.

A class of semi-bent functions with an even number of variables is constructed by using the values of Kloosterman sums. These semi-bent functions are Boolean functions with four trace terms. Moreover, it is shown that the algebraic degrees of the new semi-bent functions attain the maximum values.

In this paper, we construct two classes of q-ary balanced functions which have good global avalanche characteristics (GAC) measured in terms of sum-of-squares-modulus indicator (SSMI), modulus indicator(MI), and propagation criterion (PC). We show that the SSMI, MI, and PC of q-ary functions are invariant under affine transformations. Also, we give a construction of q-ary s-plateaued functions and obtain their SSMI. We provide a relationship between the autocorrelation spectrum of a cubic Boolean function and the dimension of the kernel of the bilinear form associated with the derivative of the function. Using this result, we identify several classes of cubic semi-bent Boolean functions which have good bounds on their SSMI and MI, and hence show good behaviour with respect to the GAC.