Systematic Constructions of Bent-Negabent Functions, 2-Rotation Symmetric Bent-Negabent Functions and Their Duals

Bent-negabent functions have many important properties for their application in cryptography since they have the flat absolute spectrum under the both Walsh-Hadamard transform and nega-Hadamard transform. In this paper, we present four new systematic constructions of bent-negabent functions on $4k, 8k, 4k+2$ and $8k+2$ variables, respectively, by modifying the truth tables of two classes of quadratic bent-negabent functions with simple form. The algebraic normal forms and duals of these constructed functions are also determined. We further identify necessary and sufficient conditions for those bent-negabent functions which have the maximum algebraic degree. At last, by modifying the truth tables of a class of quadratic 2-rotation symmetric bent-negabent functions, we present a construction of 2-rotation symmetric bent-negabent functions with any possible algebraic degrees. Considering that there are probably no bent-negabent functions in the rotation symmetric class, it is the first significant attempt to construct bent-negabent functions in the generalized rotation symmetric class

Bent functions stemming from Maiorana-McFarland class being provably outside its completed version

In early nineties Carlet [1] introduced two new classes of bent functions, both derived from the Maiorana-McFarland ($\mathcal{M}$) class, and named them \cC and \cD class, respectively. Apart from a subclass of \cD, denoted by \cD_0 by Carlet, which is provably outside two main (completed) primary classes of bent functions, little is known about their efficient constructions. More importantly, both classes may easily remain in the underlying $\mathcal{M}$ class which has already been remarked in [21]. Assuming the possibility of specifying a bent function $f$ that belongs to one of these two classes (apart from \cD_0), the most important issue is then to determine whether $f$ is still contained in the known primary classes or lies outside their completed versions. In this article, we further elaborate on the analysis of the set of sufficient conditions given in \cite{OutsideMM} concerning the specification of bent functions in \cC and \cD which are provably outside \cM. It is shown that these conditions, related to bent functions in class \cD, can be relaxed so that even those permutations whose component functions admit linear structures still can be used in the design. It is also shown that monomial permutations of the form $x^{2^r+1}$ have inverses which are never quadratic for $n >4$, which gives rise to an infinite class of bent functions in \cC but outside \cM. Similarly, using a relaxed set of sufficient conditions for bent functions in \cD and outside \cM, one explicit infinite class of such bent functions is identified. We also extend the inclusion property of certain subclasses of bent functions in \cC and \cD, as addressed initially in [1,21], that are ultimately within the completed $\mathcal{M}$ class. Most notably, we specify {\em another generic and explicit subclass} of \cD, which we call \cD_2^\star, whose members are bent functions provably outside the completed $\mathcal{M}$ class

Bent functions, SDP designs and their automorphism groups

PhD ThesisIn a 1976 paper Rothaus coined the term “bent” to describe a function f from a vector space V (n, 2) to F2 with the property that the Fourier coefficients of (−1)f have unit magnitude. Such a function has the maximum possible distance from the set of linear functions, hence the name, and has useful correlation properties. These lead to various applications to coding theory and cryptography, some of which are described. A standard notion of the equivalence of two bent functions is discussed and related to the coding theory setting. Two constructions mentioned by Rothaus and generalised by Maiorana are described. A further generalisation of one of these, involving sets of bent functions on direct summands of the original vector space, is described and proved. Various methods including computer searches are used to find appropriate sets of bent functions and hence many new equivalence classes of bent functions of 8 variables. Equivalence class invariants are used to show that most of these classes cannot be constructed by the earlier methods. Some bounds on numbers of bent functions are discussed. A 2-design is said to have the symmetric difference property (SDP) if the symmetric difference of any three blocks is either a block or the complement of a block — such a design is very close to being a 3-design. All SDP designs are induced by bent functions, and conversely. Work on the automorphism groups of various SDP designs involving computer algebra is described. An SDP design on 256 points with trivial automorphism group is noted. Some connections with strongly-regular graphs are discussed. An infinite class of pseudo-geometric strongly-regular graphs induced by bent functions is noted, and bent functions which are their own Fourier transform duals are investigated. Finally, some open problems and ideas for future work are described

Analysis of Affinely Equivalent Boolean Functions

By walsh transform, autocorrelation function, decomposition, derivation and modification of truth table, some new invariants are obtained. Based on invariant theory, we get two results: first a general algorithm which can be used to judge if two boolean functions are affinely equivalent and to obtain the affine equivalence relationship if they are equivalent. For example, all 8-variable homogenous bent functions of degree 3 are classified into 2 classes; second, the classification of the Reed-Muller code $R(4,6)/R(1,6),R(3,7)/R(1,7),$ which can be used to almost enumeration of 8-variable bent functions