Construction of bent functions of 2 k variables from a basis of (original) (raw)
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SeMA Journal, 2015
Given a bent function f (x) of n variables, its max-weight and min-weight functions are introduced as the Boolean functions f + (x) and f − (x) whose supports are the sets {a ∈ F n 2 | w(f ⊕l a) = 2 n−1 + 2 n 2 −1 } and {a ∈ F n 2 | w(f ⊕ l a) = 2 n−1 − 2 n 2 −1 } respectively, where w(f ⊕ l a) denotes the Hamming weight of the Boolean function f (x) ⊕ l a (x) and l a (x) is the linear function defined by a ∈ F n 2. f + (x) and f − (x) are proved to be bent functions. Furthermore, combining the 4 minterms of 2 variables with the max-weight or min-weight functions of a 4-tuple (f 0 (x), f 1 (x), f 2 (x), f 3 (x)) of bent functions of n variables such that f 0 (x) ⊕ f 1 (x) ⊕ f 2 (x) ⊕ f 3 (x) = 1, a bent function of n + 2 variables is obtained. A family of 4-tuples of bent functions satisfying the above condition is introduced, and finally, the number of bent functions we can construct using the method introduced in this paper are obtained. Also, our construction is compared with other constructions of bent functions.
An Analysis of the 𝒞 Class of Bent Functions
Fundamenta Informaticae, 2016
Two (so-called C, D) classes of permutation-based bent Boolean functions were introduced by Carlet [4] two decades ago, but without specifying some explicit construction methods for their construction (apart from the subclass D 0). In this article, we look in more detail at the C class, and derive some existence and nonexistence results concerning the bent functions in the C class for many of the known classes of permutations over F 2 n. Most importantly, the existence results induce generic methods of constructing bent functions in class C which possibly do not belong to the completed Maiorana-McFarland class. The question whether the specific permutations and related subspaces we identify in this article indeed give bent functions outside the completed Maiorana-McFarland class remains open.
2010 Information Theory and Applications Workshop (ITA), 2010
Bent functions were first introduced by Rothaus in 1976 as an interesting combinatorial object with the important property of having the maximum distance to all affine functions. Bent functions have many applications to coding theory, cryptography and sequence designs. For many years the focus was on the construction of binary bent functions. There are several known examples of binary monomial and binomial bent functions. In 1985, Kumar, Scholtz and Welch generalized bent functions to the case of an arbitrary finite field. In the recent years, new results on nonbinary bent functions have appeared. This paper gives an updated overview of some of the recent results and open problems on generalized bent functions. This includes some recent constructions of weakly regular monomial and binomial bent functions and examples of non-weakly regular bent functions.
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.