Root-Hadamard transforms and complementary sequences

In this paper we define a new transform on (generalized) Boolean functions, which generalizes the Walsh-Hadamard, nega-Hadamard, $2^k$-Hadamard, consta-Hadamard and all $HN$-transforms. We describe the behavior of what we call the root- Hadamard transform for a generalized Boolean function $f$ in terms of the binary components of $f$. Further, we define a notion of complementarity (in the spirit of the Golay sequences) with respect to this transform and furthermore, we describe the complementarity of a generalized Boolean set with respect to the binary components of the elements of that set.Comment: 19 page

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

Decomposing generalized bent and hyperbent functions

In this paper we introduce generalized hyperbent functions from $F_{2^n}$ to $Z_{2^k}$, and investigate decompositions of generalized (hyper)bent functions. We show that generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^k}$ consist of components which are generalized (hyper)bent functions from $F_{2^n}$ to $Z_{2^{k^\prime}}$ for some $k^\prime < k$. For odd $n$, we show that the Boolean functions associated to a generalized bent function form an affine space of semibent functions. This complements a recent result for even $n$, where the associated Boolean functions are bent.Comment: 24 page

On the Equivalence Between a Minimal Codomain Cardinality Riesz Basis Construction, a System of Hadamard&#x2013;Sylvester Operators, and a Class of Sparse, Binary Optimization Problems

Piecewise, low-order polynomial, Riesz basis families are constructed such that they share the same coefficient functionals of smoother, orthonormal bases in a localized indexing subset. It is shown that a minimal cardinality basis codomain can be realized by inducing sparsity, via l1 regularization, in the distributional derivatives of the basis functions and that the optimal construction can be found numerically by constrained binary optimization over a suitably large dictionary. Furthermore, it is shown that a subset of these solutions are equivalent to a specific, constrained analytical solution, derived via Sylvester-type Hadamard operators

A new class of negabent functions

Negabent functions were introduced as a generalization of bent functions, which have applications in coding theory and cryptography. In this paper, we have extended the notion of negabent functions to the functions defined from $\mathbb{Z}_q^n$ to $\mathbb{Z}_{2q}$ ($2q$-negabent), where $q \geq 2$ is a positive integer and $\mathbb{Z}_q$ is the ring of integers modulo $q$. For this, a new unitary transform (the nega-Hadamard transform) is introduced in the current set up, and some of its properties are discussed. Some results related to $2q$-negabent functions are presented. We present two constructions of $2q$-negabent functions. In the first construction, $2q$-negabent functions on $n$ variables are constructed when $q$ is an even positive integer. In the second construction, $2q$-negabent functions on two variables are constructed for arbitrary positive integer $q \ge 2$. Some examples of $2q$-negabent functions for different values of $q$ and $n$ are also presented