Odd Number Of Variables Research Articles

Constructions of 2-resilient rotation symmetric Boolean functions with odd number of variables

In this paper, a new method for constructing 2-resilient rotation symmetric Boolean functions with odd number of variables is presented. Based on an equivalent characterization of this class of functions and the relation between the orbit matrices and their complements, a system of equations about 2-tuples distribution matrix is established. Then the constructions of some 2-resilient rotation symmetric Boolean functions with odd number of variables can be converted into the solutions of the system of equations. We also give a sufficient condition for the constructed functions to have the maximum algebraic degree. Moreover, we provide a lower bound on the number of nonlinear 2-resilient rotation symmetric Boolean functions with odd number of variables. Particularly, the lower bound is 152 for seven number of variables.

The Kottwitz conjecture for unitary PEL-type Rapoport–Zink spaces

Abstract In this paper we study the cohomology of PEL-type Rapoport–Zink spaces associated to unramified unitary similitude groups over ℚ p {\operatorname{\mathbb{Q}}_{p}} in an odd number of variables. We extend the results of Kaletha–Minguez–Shin–White and Mok to construct a local Langlands correspondence for these groups and prove an averaging formula relating the cohomology of Rapoport–Zink spaces to this correspondence. We use this formula to prove the Kottwitz conjecture for the groups we consider.

Application of Election Functions to Estimate the Number of Monotone Self-Dual Boolean functions

One of the problems of modern discrete mathematics is R. Dedekind problem on the number of monotone boolean functions. For other precomplete classes, general formulas for the number of functions of the classes had been found, but it has not been found so far for the class of monotone boolean functions. Within the framework of this problem, there are problems of a lower level. One of them is the absence of a general formula for the number of boolean functions of intersection $MS$ of two classes --- the class of monotone functions and the class of self-dual functions. In the paper, new lower bounds are proposed for estimating the cardinality of the intersection for both an even and an odd number of variables. It is shown that the election function of an odd number of variables is monotone and self-dual. The election function of an even number of variables is determined. Free election functions, which are functions with fictitious variables similar in properties to election functions, are introduced. Then the union of a set of election functions and a set of free election functions is considered, and the cardinality of this union is calculated. The resulting value of the cardinality is proposed as a lower bound for $|MS|$. For the class $MS$ of monotone self-dual functions of an even number of variables, the lower bound is improved over the bounds proposed earlier, and for functions of an odd number of variables, the lower bound for $|MS|$ is presented for the first time.

On some invariants under the action of an extension of GA(n, 2) on the set of Boolean functions

Abstract Let G be the extension of a general affine group by the group of affine functions. We study the action of G on the set of Boolean functions. The action consists in nondegenerate affine transformations of variables and addition of affine Boolean functions. We introduce and examine some parameters of Boolean functions which are invariant with respect to the action of G. These are the amplitude (which is closely related to the nonlinearity), the dimension of a function, and some others. The invariants, together with some additionally proposed notions, could be used to obtain new bounds on cryptographic parameters of Boolean functions, including the maximum nonlinearity of functions in an odd number of variables.

Shifted plateaued functions and their differential properties

A bent4 function is a Boolean function with a flat spectrum with respect to a certain unitary transform $\mathcal {T}$ . It was shown previously that a Boolean function f in an even number of variables is bent4 if and only if f + σ is bent, where σ is a certain quadratic function depending on $\mathcal {T}$ . Hence bent4 functions are also called shifted bent functions. Similarly, a Boolean function f in an odd number of variables is bent4 if and only if f + σ is a semibent function satisfying some additional properties. In this article, for the first time, we analyse in detail the effect of the shifts on plateaued functions, on partially bent functions and on the linear structures of Boolean functions. We also discuss constructions of bent and bent4 functions from partially bent functions and study the differential properties of partially bent4 functions, unifying the previous work on partially bent functions.

High-Meets-Low: Construction of Strictly Almost Optimal Resilient Boolean Functions via Fragmentary Walsh Spectra

This paper considers the construction of resilient Boolean functions on an odd number of variables with strictly almost optimal (SAO) nonlinearity. Through introducing the fragmentary Walsh transform, a construction technique called “High-Meets-Low” is proposed. The detailed design procedures of a 39-variable 3-resilient Boolean function with SAO nonlinearity $2^{38}-2^{19}+2^{16}+2^{14}$ are given. It is shown that the nonlinearity of an $n$ -variable $t$ -resilient Boolean function can reach $2^{n-1}-2^{(n-1)/2}+5\cdot 2^{(n-11)/2}$ or $2^{n-1}-2^{(n-1)/2}+2^{(n-7)/2}$ , which are the largest known values for the corresponding $n$ and $t$ values. Finally, by constructing a 29-variable balanced Boolean function with SAO nonlinearity $2^{28}-2^{14}+2^{10}+2^{9}$ , we show an alternative method to realize the High-Meets-Low construction technique.

Constructing Odd‐Variable Rotation Symmetric Boolean Functions with Optimal Algebraic Immunity and High Nonlinearity

Rotation symmetric Boolean functions (RSBFs) have attracted widespread attention due to their good cryptographic properties. We present a new construction of RSBFs with optimal algebraic immunity on odd number of variables. The nonlinearity of the new function is much higher than other best known RSBFs with optimal algebraic immunity. The algebraic degree of the constructed n-variable RSBF can achieve the upper bound n-1 when n/2 is odd or when n/2 is a power of 2 for n-11. In addition, the constructed function can possess almost perfect immunity to fast algebraic attacks for n=11, 13, 15

Further Results on Generalized Bent Functions and Their Complete Characterization

This paper contributes to increase our knowledge on generalized bent functions (including generalized bent Boolean functions and generalized $p$ -ary bent functions with odd prime $p$ ) by bringing new results on their characterization and construction in arbitrary characteristic. More specifically, we first investigate relations between generalized bent functions and bent functions by the decomposition of generalized bent functions. This enables us to completely characterize generalized bent functions and $\mathbb Z_{p^{k}}$ -bent functions by some affine space associated with the generalized bent functions. We also present the relationship between generalized bent Boolean functions with an odd number of variables and generalized bent Boolean functions with an even number of variables. Based on the well-known Maiorana-McFarland class of Boolean functions, we present some infinite classes of generalized bent Boolean functions. In addition, we introduce a class of generalized hyperbent functions that can be seen as generalized Dillon’s $PS$ functions. Finally, we solve an open problem related to the description of the dual function of a weakly regular generalized bent Boolean function with an odd number of variables via the Walsh–Hadamard transform of their component functions, and we generalize these results to the case of odd prime.

Constructions of balanced odd-variable rotation symmetric Boolean functions with optimal algebraic immunity and high nonlinearity

Rotation symmetric Boolean functions have been used as components of different cryptosystems. In this paper, two classes of balanced rotation symmetric Boolean functions having optimal algebraic immunity on odd number of variables are constructed. We give a lower bound on the algebraic degree of the first class of functions, and prove that the n-variable functions in the second class has optimal algebraic degree if n≠2m+1 for m>2. Moreover, it is shown that both classes of functions have much better nonlinearity than all the previously obtained rotation symmetric Boolean functions with optimal algebraic immunity, and have good behavior against fast algebraic attacks at least for small numbers of input variables.

A Super-Set of Patterson--Wiedemann Functions: Upper Bounds and Possible Nonlinearities

Constructing Boolean functions on odd number of variables with nonlinearity exceeding the bent concatenation bound is one of the most difficult combinatorial problems in the domain of Boolean functions and it has deep implications to coding theory and cryptology. After demonstration of such functions by Patterson and Wiedemann in 1983, for more than three decades the efforts have been channelized in obtaining the instances only. For the first time, in this paper, we try to explore non-trivial upper bounds on nonlinearity of such functions which are invariant under several group actions. In fact, we consider much larger sets of functions than what have been considered so far and obtain tight upper bounds on the nonlinearity in several cases. To support our claims, we present computational results for functions on n variables where n is an odd composite integer, \(9\le n\le 39\). In particular, our results for \(n = 15\) and 21 are of immediate interest given recent research results in this domain. Not only the upper bounds, we also identify what are the nonlinearities that can actually be achieved above the bent concatenation bound for such class of functions.

Classification and Construction of quaternary self-dual bent functions

Quaternary self-dual bent functions are studied from the viewpoints of existence, construction, and symmetry. A search algorithm is described to classify their orbits under the orthogonal group in low dimensions. A connection with self-dual bent Boolean functions shows that they do not exist in odd number of variables.

Shimura lifting of modular forms of weight 3/2

We show that there is the Shimura lifting map of \(\mathbf {M}_{3/2}(N,\chi _{0})\) to \(\mathbf {M}_{2}(N/2,\chi _{0}^{2})\) for any natural number N divisible by 4 and for any odd Dirichlet character \(\chi _{0}\), which was already proved in the case of higher weight (Tsuyumine, Tsukuba J Math 23:465–483, 1999). As the application, we show that the lift of the weighted average of theta series of quadratic forms of an odd number of variables in a class, is an Eisenstein series of integral weight.

Constructions of Generalized Bent Boolean Functions on Odd Number of Variables

In this paper, we investigate the constructions of generalized bent Boolean functions defined on with values in Z4. We first present a construction of generalized bent Boolean functions defined on with values in Z4. The main technique is to utilize bent functions to derive generalized bent functions on odd number of variables. In addition, by using Boolean permutations, we provide a specific method to construct generalized bent functions on odd number of variables.

Characterization of Negabent Functions and Construction of Bent-Negabent Functions With Maximum Algebraic Degree

We present necessary and sufficient conditions for a Boolean function to be a negabent function for both an even and an odd number of variables, which demonstrates the relationship between negabent functions and bent functions. By using these necessary and sufficient conditions for Boolean functions to be negabent, we obtain that the nega spectrum of a negabent function has at most four values. We determine the nega spectrum distribution of negabent functions. Further, we provide a method to construct bent-negabent functions in <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> variables ( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> even) of algebraic degree ranging from 2 to [( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> )/2], which implies that the maximum algebraic degree of an <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -variable bent-negabent function is equal to [( <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> )/2]. Thus, we answer two open problems proposed by Parker and Pott and by Stănică et al.

Secondary constructions of Boolean functions with maximum algebraic immunity

The algebraic immunity of cryptographic Boolean functions with odd number of variables is studied in this paper. Proper modifications of functions with maximum algebraic immunity are proved that yield new functions whose algebraic immunity is also maximum. Several results are provided for both the multivariate and univariate representation, and their applicability is shown on known classes of Boolean functions. Moreover, new efficient algorithms to produce functions of guaranteed maximum algebraic immunity are developed, which further extend and generalize well-known constructions in this area. It is shown that high nonlinearity as well as good behavior against fast algebraic attacks are also achievable in several cases.

A Family of Near-Exact Distributions Based on Truncations of the Exact Distribution for the Generalized Wilks Lambda Statistic

For the case where at least two sets have an odd number of variables we do not have the exact distribution of the generalized Wilks Lambda statistic in a manageable form, adequate for manipulation. In this article, we develop a family of very accurate near-exact distributions for this statistic for the case where two or three sets have an odd number of variables. We first express the exact characteristic function of the logarithm of the statistic in the form of the characteristic function of an infinite mixture of Generalized Integer Gamma distributions. Then, based on truncations of this exact characteristic function, we obtain a family of near-exact distributions, which, by construction, match the first two exact moments. These near-exact distributions display an asymptotic behaviour for increasing number of variables involved. The corresponding cumulative distribution functions are obtained in a concise and manageable form, relatively easy to implement computationally, allowing for the computation of virtually exact quantiles. We undertake a comparative study for small sample sizes, using two proximity measures based on the Berry-Esseen bounds, to assess the performance of the near-exact distributions for different numbers of sets of variables and different numbers of variables in each set.

An Index Theorem for Modules on a Hypersurface Singularity

A topological interpretation of Hochster’s Theta pairing of two modules on a hypersurface ring is given in terms of linking numbers. This generalizes results of M. Hochster and proves a conjecture of J. Steenbrink. As a corollary we get that the Theta pairing vanishes for isolated hypersurface singularities in an odd number of variables, as was conjectured by H. Dao.

Constructing Rotation Symmetric Boolean Functions with Maximum Algebraic Immunity on an Odd Number of Variables

Constructing Rotation Symmetric Boolean Functions with Maximum Algebraic Immunity on an Odd Number of Variables

Constructions of 1‐resilient Boolean functions on odd number of variables with a high nonlinearity

ABSTRACTIn this paper, we concentrate on the design of 1‐resilient Boolean functions with desirable cryptographic properties. Firstly, we put forward a novel secondary construction to obtain 1‐resilient functions. Next, we present the relationships between the properties of these constructed 1‐resilient functions and that of the initial functions. Based on the construction and a class of bent functions on n variables, we can obtain a class of (n + 3)‐variable 1‐resilient non‐separable cryptographic functions with a high algebraic immunity, whose nonlinearity is equal to the bent concatenation bound 2n + 2 − 2(n + 2)/2. Furthermore, we propose a set of 1‐resilient non‐separable functions on odd number of variables with an optimal algebraic degree, a high algebraic immunity, and a high nonlinearity. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Near-exact distributions for the generalized wilks lambda statistic

Two near-exact distributions for the generalized Wilks Lambda statistic, used to test the independence of several sets of variables with a multivariate normal distribution, are developed for the case where two or more of these sets have an odd number of variables. Using the concept of near-exact distribution and based on a factorization of the exact characteristic function we obtain two approximations, which are very close to the exact distribution but far more manageable. These near-exact distributions equate, by construction, some of the first exact moments and correspond to cumulative distribution functions which are practical to use, allowing for an easy computation of quantiles. We also develop three asymptotic distributions which also equate some of the first exact moments. We assess the proximity of the asymptotic and near-exact distributions obtained to the exact distribution using two measures based on the Berry-Esseen bounds. In our comparative numerical study we consider different numbers of sets of variables, different numbers of variables per set and different sample sizes.