A novel algorithm enumerating bent functions

Based on the relationship between the Walsh spectra of a Boolean function at partial points and the Walsh spectra of its subfunctions, and on the binary Möbius transform, a novel algorithm is developed, which can theoretically construct all bent functions. Practically we enumerate all bent functions in 6 variables. With the restriction on the algebraic normal form, the algorithm is also efficient in more variables case. For example, enumeration of all homogeneous bent functions of degree 3 in 8 variables can be done in one minute with a P4 1.7 GHz computer; the nonexistence of homogeneous bent functions in 10 variables of degree 4 is computationally proved.     

On the arithmetic Walsh coefficients of Boolean functions

We generalize to the arithmetic Walsh transform (AWT) some results which were previously known for the Walsh–Hadamard transform of Boolean functions. We first generalize the classical Poisson summation formula to the AWT. We then define a generalized notion of resilience with respect to an arbitrary statistical measure of Boolean functions. We apply the Poisson summation formula to obtain a condition equivalent to resilience for one such statistical measure. Last, we show that the AWT of a large class of Boolean functions can be expressed in terms of the AWT of a Boolean function of algebraic degree at most three in a larger number of variables.     

New results on non-existence of generalized bent functions (II)

Several new results on non-existence of generalized bent functions are presented by using the class group of related imaginary abelian number fields.     

New results on non-existence of generalized bent functions (Ⅱ)

Several new results on non-existence of generalized bent functions are presented by using the class group of related imaginary abelian number fields.     

Towards the classification of self-dual bent functions in eight variables

In this paper, we classify quadratic and cubic self-dual bent functions in eight variables with the help of computers. There are exactly four and 45 non-equivalent self-dual bent functions of degree two and three, respectively. This result is achieved by enumerating all eigenvectors with ± 1 entries of the Sylvester Hadamard matrix with an integer programming algorithm based on lattice basis reduction. The search space has been reduced by breaking the symmetry of the problem with the help of additional constraints. The final number of non-isomorphic self-dual bent functions has been determined by exploiting that EA-equivalence of Boolean functions is related to the equivalence of linear codes.     

A construction of bent functions from plateaued functions

In this presentation, a technique for constructing bent functions from plateaued functions is introduced and analyzed. This generalizes earlier techniques for constructing bent from near-bent functions. Using this construction, we obtain a big variety of inequivalent bent functions, some weakly regular and some non-weakly regular. Classes of bent functions having some additional properties that enable the construction of strongly regular graphs are formed, and explicit expressions for bent functions with maximal degree are presented.     

Walsh-function analysis of a certain class of time series

This paper is concerned with investigating the use of the orthonormal system of Walsh functions in the analysis of a dyadic-stationary series. The main emphasis is on the finite Walsh transform of a sequence of values coming from such a series. Under a certain mixing condition, given in terms of the dyadic auto-covariance function of the series, we derive a central limit theorem for the finite Walsh transform. This, in turn, allows us to consider estimates of the Walsh spectrum, and we discuss briefly the Walsh periodogram.     

Results on Bent Functions

In this paper, we present three results on bent functions: a construction, a restriction, and a characterization. Starting with a single bent function, in a simple but very effective way, the construction produces a large number of new bent functions in the same number of variables. The restriction imposes new conditions on the directional derivatives of bent functions. Certain non-existence results that were previously obtained through computer search follow easily from these conditions. The characterization describes bent functions as certain solutions of a system of quadratic equations. Interesting new properties of bent functions are obtained using the characterization.     

On some boundary value problems for a system of two second-order equations

For a two-point homogeneous boundary value problem for a system of two nonlinear second-order differential equations, we suggest sufficient solvability conditions (in particular, stated, like Bernstein conditions, in terms of the growth of the absolute values of the right-hand sides of the system with respect to the derivatives of the unknown functions). We obtain a priori estimates for solutions.     

C编号Walsh函数及Manz算法

沃尔什编号Walsh函数是Walsh在1923年给出的;1931年,Paley定义了佩利编号Walsh函数;哈德玛编号Walsh函数是根据Hadamard 1893年的工作,由专用     

Smooth PONS

PONS^tm is a basis which satisfies all of the fundamental properties of the Walsh functions (each element is piecewise constant, takes on only the values ±1, and can be efficiently computed via a fast transform) plus three additional properties that are false for the Walsh functions: PONS is optimal with respect to a global uncertainty principle; all PONS elements have uniformly bounded crest factors; and all PONS elements are QMF’s. In 1991, Ingrid Daubechies asked whether there exists a smooth basis satisfying the global uncertainty principle property. In this article we show how to transform any basis into another basis by applying the PONS construction, thereby providing an affirmative answer to this question.     

迭代方程与离散叙率Walsh函数

在J.L.Shanks的基础上,给出了产生离散叙率Walsh函数的迭代方程,由迭代 方程推出了离散Walsh 函数的表达式和Walsh变换的速算 法(FWWT).     

Plateaued functions,partial geometric difference sets,and partial geometric designs

Plateaued functions on finite fields have been studied in many papers in recent years. As a generalization of plateaued functions on finite fields, we introduce the notion of a plateaued function on a finite abelian group. We will give a characterization of a plateaued function in terms of an equation of the matrix associated to the function. Then we establish a one‐to‐one correspondence between the ${\mathbb{Z}}_2$‐valued plateaued functions and partial geometric difference sets (with specific parameters) in finite abelian groups. We will also discuss two general methods (extension and lifting) for the construction of new partial geometric difference sets from old ones in (abelian or nonabelian) finite groups, and construct many partial geometric difference sets and plateaued functions. A one‐to‐one correspondence between partial geometric difference sets (in arbitrary finite groups) and partial geometric designs will be proved.     

Second-order invex functions in nonlinear programming

We introduce a notion of a second-order invex function. A Fréchet differentiable invex function without any further assumptions is second-order invex. It is shown that the inverse claim does not hold. A Fréchet differentiable function is second-order invex if and only if each second-order stationary point is a global minimizer. Two complete characterizations of these functions are derived. It is proved that a quasiconvex function is second-order invex if and only if it is second-order pseudoconvex. Further, we study the nonlinear programming problem with inequality constraints whose objective function is second-order invex. We introduce a notion of second-order type I objective and constraint functions. This class of problems strictly includes the type I invex ones. Then we extend a lot of sufficient optimality conditions with generalized convex functions to problems with second-order type I invex objective function and constraints. Additional optimality results, which concern type I and second-order type I invex data are obtained. An answer to the question when a kernel, which is not identically equal to zero, exists is given.     

一些特殊类型广义Bent函数的不存在性

本文利用分圆域中的素理想分解特性和一些特殊的不定方程的解性质,得到一些特殊类型的广义Bent函数的不存在性结果。     

广义Walsh变式与一极值问题

<正> 设p为大于1的整数,t为非负实数,t的p进表示为     

Walsh函数的一个迭代方程体系

Shanks曾用迭代方程产生离散佩利编号Walsh函数。作者在[4]中,给出了产生离散沃尔什编号Wa1sh函数的迭代方程. 本文在上述基础上,提出了一个产生离散哈德玛编号Walsh函数的迭代方程,推出了离散哈德玛编号Walsh函数的表示式及其变换(FWHT)的快速计算公式. 上述三个极为类似的迭代方程,已构成了离散Walsh函数的迭代方程体系.连续的Walsh函数,也能用迭代方程这种形式来描述.     

Non-Existence of Some Generalized Bent Functions

Several new results on the non-existence of some generalized bent functions are proved by using properties of the decomposition law of primes in cyclotomic fields and properties of the solutions of some special Diophantine equations.     

The eight variable homogeneous degree three bent functions

We determine the affine equivalence classes of the eight variable degree three homogeneous bent functions using a new algorithm. Our algorithm applies to general bent functions and can systematically determine the automorphism groups. We provide a partial verification of the enumeration of eight variable degree three homogeneous bent functions obtained by Meng et al. We determine the affine equivalence classes of these functions.