Construction methods for generalized bent functions (original) (raw)
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Bent and generalized bent Boolean functions
Designs, Codes and Cryptography, 2012
In this paper, we investigate the properties of generalized bent functions defined on Z n 2 with values in Z q , where q ≥ 2 is any positive integer. We characterize the class of generalized bent functions symmetric with respect to two variables, provide analogues of Maiorana-McFarland type bent functions and Dillon's functions in the generalized set up. A class of bent functions called generalized spreads is introduced and we show that it contains all Dillon type generalized bent functions and Maiorana-McFarland type generalized bent functions. Thus, unification of two different types of generalized bent functions is achieved. The crosscorrelation spectrum of generalized Dillon type bent functions is also characterized. We further characterize generalized bent Boolean functions defined on Z n 2 with values in Z 4 and Z 8. Moreover, we propose several constructions of such generalized bent functions for both n even and n odd. Keywords Generalized Boolean functions • Generalized bent functions • Walsh-Hadamard transform Mathematics Subject Classification (2000) 94A60 • 94C10 • 06E30 Communicated by C. Carlet.
On bent and semi-bent quadratic Boolean functions
IEEE Transactions on Information Theory, 2005
The maximum length sequences, also called m-sequences, have received a lot of attention since the late sixties. In terms of LFSR synthesis they are usually generated by certain power polynomials over finite field and in addition characterized by a low cross correlation and high nonlinearity. We say that such sequence is generated by a semi-bent function. Some new families of such function, represented
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.
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.
Generalized bent functions - sufficient conditions and related constructions
Advances in Mathematics of Communications, 2017
The necessary and sufficient conditions for a class of functions f : Z n 2 → Z q , where q ≥ 2 is an even positive integer, have been recently identified for q = 4 and q = 8. In this article we give an alternative characterization of the generalized Walsh-Hadamard transform in terms of the Walsh spectra of the component Boolean functions of f , which then allows us to derive sufficient conditions that f is generalized bent for any even q. The case when q is not a power of two, which has not been addressed previously, is treated separately and a suitable representation in terms of the component functions is employed. Consequently, the derived results lead to generic construction methods of this class of functions. The main remaining task, which is not answered in this article, is whether the sufficient conditions are also necessary. There are some indications that this might be true which is also formally confirmed for generalized bent functions that belong to the class of generalized Maiorana-McFarland functions (GMMF), but still we were unable to completely specify (in terms of necessity) gbent conditions.
Construction of Multiple-Valued Bent Functions Using Subsets of Coefficients in GF and RMF Domains
2021
Multiple-valued bent functions are functions with highest nonlinearity which makes them interesting for multiple-valued cryptography. Since the general structure of bent functions is still unknown, methods for construction of bent functions are often based on some deterministic criteria. For practical applications, it is often necessary to be able to construct a bent function that does not belong to any specific class of functions. Thus, the criteria for constructions are combined with exhaustive search over all possible functions which can be very CPU time consuming. A solution is to restrict the search space by some conditions that should be satisfied by the produced bent functions. In this paper, we proposed the construction method based on spectral subsets of multiple-valued bent functions satisfying certain appropriately formulated restrictions in Galois field (GF) and Reed-Muller-Fourier (RMF) domains. Experimental results show that the proposed method efficiently constructs ternary and quaternary bent functions by using these restrictions.
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.
Algebraic Construction of Semi Bent Function Via Known Power Function
Turkish World Mathematical Society Journal of Applied and Engineering Mathematics, 2021
The study of semi bent functions (2-plateaued Boolean function) has attracted the attention of many researchers due to their cryptographic and combinatorial properties. In this paper, we have given the algebraic construction of semi bent functions defined over the finite field F2n (n even) using the notion of trace function and Gold power exponent. Algebraically constructed semi bent functions have some special cryptographical properties such as high nonlinearity, algebraic immunity, and low correlation immunity as expected to use them effectively in cryptosystems. We have illustrated the existence of these properties with suitable examples.