On generalized bent functions (original) (raw)
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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 ā G F(2) n is 2 n ā 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 ā 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).
An Analysis of the š 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.
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 ā¤ 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 ā1)) + T r m 1 (bx s(2 m +1)) where n = 2m, gcd(r, 2 m + 1) = 1, gcd(s, 2 m ā 1) = 1. Also, we give a bent function example of the form f a,b (x) = T r n 1 (ax 2 m ā1) + T r 2 1 (bx 2 n ā1 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.
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 ā (x) whose supports are the sets {a ā F n 2 | w(f āl a) = 2 nā1 + 2 n 2 ā1 } and {a ā F n 2 | w(f ā l a) = 2 nā1 ā 2 n 2 ā1 } respectively, where w(f ā l a) denotes the Hamming weight of the Boolean function f (x) ā l a (x) and l a (x) is the linear function defined by a ā F n 2. f + (x) and f ā (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) ā f 1 (x) ā f 2 (x) ā 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.
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
Construction methods for generalized bent functions
Discrete Applied Mathematics, 2018
Generalized bent (gbent) functions is a class of functions f : Z n 2 ā Z q , where q ā„ 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.