Construction of bent functions of 2 k variables from a basis of (original) (raw)
Constructions of Bent Functions from Two Known Bent Functions
2000
A (1, -1)-matrix will be called a bent type matrix if each row and each column arebent sequences. A similar description can be found in Carlisle M. Adams and StaffordE. Tavares, Generating and counting binary sequences, IEEE Trans. Inform. Theory,vol. 36, no. 5, pp. 1170-1173, 1990, in which the authors use the properties of benttype matrices to construct a class
A new method for secondary constructions of vectorial bent functions
Designs, Codes and Cryptography, 2021
In 2017, Tang et al. have introduced a generic construction for bent functions of the form f (x) = g(x) + h(x), where g is a bent function satisfying some conditions and h is a Boolean function. Recently, Zheng et al. [22] generalized this result to construct large classes of bent vectorial Boolean function from known ones in the form F (x) = G(x)+h(X), where G is a bent vectorial and h a Boolean function. In this paper we further generalize this construction to obtain vectorial bent functions of the form F (x) = G(x)+H(X), where H is also a vectorial Boolean function. This allows us to construct new infinite families of vectorial bent functions, EA-inequivalent to G, which was used in the construction. Most notably, specifying H(x) = h(T r n 1 (u 1 x),. .. , T r n 1 (u t x)), the function h : F t 2 → F 2 t can be chosen arbitrary which gives a relatively large class of different functions for a fixed function G. We also propose a method of constructing vectorial (n, n)-functions having maximal number of bent components.
Notes on Bent Functions in Polynomial Forms
The existence and construction of bent functions are two of the most widely studied problems in Boolean functions. For monomial functions f (x) = T r n 1 (ax s), these problems were examined extensively and it was shown that the bentness of the monomial functions is complete for n ≤ 20. However, in the binomial function case, i.e. f (x) = T r n 1 (ax s 1) + T r k 1 (bx s 2), this characterization is not complete and there are still open problems. In this paper, we give a summary of the literature on the bentness of binomial functions and show that there exist no bent functions of the form T r n 1 (ax r(2 m −1)) + T r m 1 (bx s(2 m +1)) where n = 2m, gcd(r, 2 m + 1) = 1, gcd(s, 2 m − 1) = 1. Also, we give a bent function example of the form f a,b (x) = T r n 1 (ax 2 m −1) + T r 2 1 (bx 2 n −1 3) for n = 4, although, it is stated in [9] that there is no such bent function of this form for any value of a and b.
Construction methods for generalized bent functions
Discrete Applied Mathematics, 2018
Generalized bent (gbent) functions is a class of functions f : Z n 2 → Z q , where q ≥ 2 is a positive integer, that generalizes a concept of classical bent functions through their codomain extension. A lot of research has recently been devoted towards derivation of the necessary and sufficient conditions when f is represented as a collection of Boolean functions. Nevertheless, apart from the necessary conditions that these component functions are bent when n is even (respectively semi-bent when n is odd), no general construction method has been proposed yet for n odd case. In this article, based on the use of the well-known Maiorana-McFarland (MM) class of functions, we give an explicit construction method of gbent functions, for any even q > 2 when n is even and for any q of the form q = 2 r (for r > 1) when n is odd. Thus, a long-term open problem of providing a general construction method of gbent functions, for odd n, has been solved. The method for odd n employs a large class of disjoint spectra semi-bent functions with certain additional properties which may be useful in other cryptographic applications.
Constructing new superclasses of bent functions from known ones
Cryptography and Communications, 2022
Some recent research articles [23, 24] addressed an explicit specification of indicators that specify bent functions in the so-called C and D classes, derived from the Maiorana-McFarland (M) class by C. Carlet in 1994 [5]. Many of these bent functions that belong to C or D are provably outside the completed M class. Nevertheless, these modifications are performed on affine subspaces, whereas modifying bent functions on suitable subsets may provide us with further classes of bent functions. In this article, we exactly specify new families of bent functions obtained by adding together indicators typical for the C and D class, thus essentially modifying bent functions in M on suitable subsets instead of subspaces. It is shown that the modification of certain bent functions in M gives rise to new bent functions which are provably outside the completed M class. Moreover, we consider the so-called 4-bent concatenation (using four different bent functions on the same variable space) of the (non)modified bent functions in M and show that we can generate new bent functions in this way which do not belong to the completed M class either. This result is obtained by specifying explicitly the duals of four constituent bent functions used in the concatenation. The question whether these bent functions are also excluded from the completed versions of PS, C or D remains open and is considered difficult due to the lack of membership indicators for these classes.
Algebra, 2014
We present a method to iteratively construct new bent functions of n+2 variables from a bent function of n variables and its cyclic shift permutations using minterms of n variables and minterms of 2 variables. In addition, we provide the number of bent functions of n+2 variables that we can obtain by applying the method here presented, and finally we compare this method with a previous one introduced by us in 2008 and with the Rothaus and Maiorana-McFarland constructions.
On the representation of bent functions by bent rectangles
2005
We propose a representation of boolean bent functions by bent rectangles, that is, by special matrices with restrictions on rows and columns. Using this representation, we exhibit new classes of bent functions, give an algorithm to construct bent functions, improve a lower bound for the number of bent functions.
On the maximum number of bent components of vectorial functions
IEEE Transactions on Information Theory, 2017
In this paper, we show that the maximum number of bent component functions of a vectorial function F : G F(2) n → G F(2) n is 2 n − 2 n/2. We also show that it is very easy to construct such functions. However, it is a much more challenging task to find such functions in polynomial form F ∈ G F(2 n)[x], where F has only a few terms. The only known power functions having such a large number of bent components are x d , where d = 2 n/2 + 1. In this paper, we show that the binomials F i (x) = x 2 i (x + x 2 n/2) also have such a large number of bent components, and these binomials are inequivalent to the monomials x 2 n/2 +1 if 0 < i < n/2. In addition, the functions F i have differential properties much better than x 2 n/2 +1. We also determine the complete Walsh spectrum of our functions when n/2 is odd and gcd(i, n/2) = 1. Index Terms-Cryptography, Boolean functions, bent functions, vectorial bent functions, trace functions, equivalence of functions. I. INTRODUCTION B ENT functions are extremal combinatorial objects with several areas of application, such as coding theory, maximum length sequences, cryptography, the theory of difference sets to name a few. The term bent Boolean function was introduced by Rothaus [42]; another early thorough investigation of bent functions is [19]. For a recent survey article, see [12] and the two books [35], [43], see also [17] for a more general discussion of functions on finite fields. Among other equivalent characterizations of bent functions, the one that is most often used is a characterization of bent functions as a class of Boolean functions having so-called flat Walsh-Hadamard spectra. It means that for any bent function over G F(2) n , its Hamming distance to any affine function in n variables is constant including the distance to the all-zero function (or all-one function).
On the Iterative Construction of Bent Functions1
In this paper we present two methods to construct iteratively bent functions of n + 2 variables from bent functions of n variables. Our methods use bent functions expressed as sum of minterms.
A note of generalized bent functions
Journal of Pure and Applied Algebra, 1996
Kumar et al. (1985) introduced the concept of generalized bent functionsj': z: + Z, where q is a positive integer > 1, and gave constructions for such functions for every possible value of L/ and II other than II odd and y = 2 (mod 4). Furthermore, they have shown the non-existence in the remaining case under certain sufficient conditions. The main purpose of this paper is to understand the extent of the set of parameters for which no generalized bent functions exist. In particular. the non-existence of Bent functions on L:,,. with p = 7 (mod 8) and r 2 1 is examined. The result obtained generalizes recent works of Bi (1991) and Pei (1993).