The paper presents a discussion of Haar spectra for binary bent functions. The presentation is restricted to bent functions of n = 4 variables, since in this case the total of 896 bent functions allows an exhaustive search and discussion of their features. Possibilities for a straightforward extension to functions of a larger number of variables are illustrated by examples for n = 6. The definition and multiresolution feature of Haar coefficients and their organization into packets provides a deeper insight into bent functions, and in particular the relationships between values in their function vectors. In the spectral domain, these relationships are expressed as the appearance of Walsh spectra for smaller number of variables in the Haar spectra of bent functions for a given n. Reconstruction of function values from Haar coefficients shows that for bent functions, function values appear in pairs of values in certain orders. Destroying these pairs results in non-bent functions.

This paper studies the problem of computing the stochastic probability (shortest code length) of the encoded vectors containing cluster structure using Normalized Maximum Likelihood (NML) model. This is of great theoretical and practical importance in data clustering based on Minimum Description Length (MDL) principle, such as for estimating the best number of clusters and best cluster structure for the data. Straightforward computation of the shortest code length of the vector containing cluster structure based on the NML model requires polynomial time with respect to the size of the vector and number of clusters. We show that this is a tractable problem by introducing a recursion formula for the efficient computation of normalizing constant from the NML model. The time complexity of the new formula is linear opposed to previous polynomial time with respect to the size of the vector and number of clusters.

Ternary bent functions are defined in the spectral domain as ternary functions having flat Vilenkin-Chrestenson spectra. This is a global transform in the sense that all function values are involved in computing every spectral coefficient. The paper presents an analysis of characteristics of the spectra of ternary bent functions with respect to the ternary Haar transform, which is a multiresolution and local transform in the above sense. Due to this, it provides a further insight in the structure of ternary bent functions, their characteristics, and mutual relationships.

This paper presents a discussion of Haar spectra for Generalized Boolean bent functions. The definition and multiresolution feature of Haar coefficients and their organization into packets provides a deeper insight into bent functions and their properties. It is shown that manipulation of packets of Haar coefficients for binary bent functions leads to the Generalized Boolean bent functions. The presentation is restricted to functions in n=4 variables, since in this case a direct examination of examples is easily realisable. Possibilities for a straightforward extension to functions of a larger number of variables are illustrated by examples for n=6.

As already noticed in previous chapters, there are differences between binary bent functions and bent functions for $$p>2$$ . For example, the binary bent functions are defined as maximally non-linear binary functions. In the case of bent functions for other values of p, including the case $$p=4$$ this feature does not necessarily hold. As pointed out in [1] for quaternary functions, and discussed more generally for any p in [2], there are quaternary bent functions that are not maximally non-linear, and, vice versa, maximally non-linear quaternary functions that are not bent.

In this chapter, we present the notation and introduce basic concepts that will be used further in this book.

Binary bent functions are often used as a basis for deriving cryptographically interesting sequences. For instance, such sequences are used to secure safe communication among unsafe channels. Ternary bent functions are a mathematically interesting object defined as a direct extension of the concept by referring to definition of binary bent functions in terms of Walsh spectral coefficients. The degree of ternary bent functions is defined as the largest sum of powers of variables in a term in their Generalized Positive-polarity Reed-Muller expressions for ternary functions. In cryptography, the corresponding expression for the binary case is also called the Algebraic Normal Form (ANF). It is supposed that bent functions with larger degrees are more suitable for practical applications. We present a way to increase the degree of ternary bent functions by using a particular class of FFT-like permutation matrices implementing certain well suited substitutions of variables.

In theory and practice of bent functions, the term class is used in a twofold meaning. Most often, it is applied to sets of bent functions generated in a specified manner or by a particular algorithm. In this respect, several classes of bent functions are distinguished [1–3]. In another meaning, the term classes refers to sets of bent functions sharing certain specific properties or are mutually related by some transformations. For example, already Rothaus proved that for $$n=6$$ there are four classes of bent functions that are affine equivalent, i.e., functions from the same class can be converted to each other by affine transformations [4]. Table 6.1 shows representative functions for these classes [4].

Functions in a large number of variables necessarily require large function vectors to be represented. For a p-valued function, such a vector of length $$p^{n}$$ can be split into subvectors of equal size $$p^{n-k}$$ for various values of k which can be arranged as either rows or columns of a matrix. Sometimes, such matrices can be viewed as a more suitable data structure for manipulating and computing with given functions.

As in the binary case, ternary bent functions are a very small portion of the set of all ternary functions for a given number of variables. For example, for n = 2, there are 486 ternary bent functions out of 19683 ternary functions, which is 2, 47%, and this number reduces exponentially with the increase of n. However, finding, or alternatively, constructing them is a challenging task. A possible approach is based upon the manipulation of known ternary bent functions to construct other ternary bent functions. In this paper, we define Gibbs permutation matrices derived from the Gibbs derivative with respect to the Vilenkin-Chrestenson transform and propose their usage in constructing bent functions. The method can be extended to p-valued bent functions, where p is a prime larger than 3.

Bent functions, either binary or multiple-valued, are mathematical objects attractive for studying since besides highest non-linearity as their primary characteristic, express some other interesting properties, some of which can be used to devise construction algorithms for bent functions. In particular, binary bent functions have a strictly specified number of non-zero values. In the same way, ternary bent functions satisfy certain requirements on the distribution of values of elements of their value vectors. These requirements can be used to specify six classes of ternary bent functions. Classes are mutually related by encoding of function values. Functions within a class are mutually related by permutation of elements in their function vectors. Given a basic ternary bent function, other functions in the same class can be constructed by permutation matrices having a block structure similar to that of the factor matrices appearing in the Good-Thomas decomposition of the Cooley-Tukey Fast Fourier transform and related algorithms. Conversion of function vectors into matrices or equivalent matrix-valued vectors of smaller length leads to the redistribution of space complexity of related manipulation algorithms. In this matrix representations, construction of other bent functions from a given known bent function is performed by using either properties of bent functions or by manipulation of such representations in terms of FFT-like permutation matrices of dimensions smaller than the length of the function vector of the initial bent functions.

This chapter deals with lossy compression of images that have been corrupted by additive noise. The chapter's key contribution is that the analysis is done from the perspective of compressed image visual quality. Several coders are explored for which the compression ratio is regulated in various ways. Lossless coding usually does not produce sufficient compression ratios for many practical applications. Visual quality metrics that are the most adequate for the considered application (WSNR, MS-SSIM, PSNR-HVS-M and PSNR-HVS) are used. The objective is to analyze is optimal operation point (OOP) possible according to visual quality metrics. It is demonstrated that, under certain conditions, visual quality of compressed images can be slightly better than quality of original noisy images due to image filtering through lossy compression, i.e., OOP might exist. The “optimal” parameters of coders for which this positive effect can be observed depend upon standard deviation of the noise. We propose an algorithm to determine coder parameters in OOP. This enables the development of automated techniques for compressing noisy images at the vicinity of the optimal operation point, i.e. when visual quality improves or declines insufficiently. Another advantage is that compression ratio for this case is quite high. The results of a series of grayscale test images with various noise variations are shown.

It is entirely justified to use the attribute digital in describing contemporary era due to omnipresence of digital technologies and devices based on them and their strong influence to almost all aspects of human activities. The present paper is a tribute to a man whose work in logic and mathematics, leading to the mathematical logic, set theoretical foundations for the development and establishing of digital era.

As in the binary case, ternary bent functions are defined as most non-linear ternary functions meaning that they are at the largest possible Hamming distance from affine ternary functions. It is therefore interesting to observe that some ternary bent functions can be constructed as various combinations of linear ternary functions. By starting from the combination of linear functions corresponding to the basic ternary bent functions the construction of different other ternary bent functions can be performed by the application of different combinations of FFT-like permutation matrices for ternary functions.

Recently, two-dimensional polynomial predictors have been introduced. Some of their properties have been presented when simultaneous predictions in both directions are involved. In this paper we discuss some properties of twodimensional polynomial predictors when only one-way prediction is taken into account. We show that its behavior is similar to one-dimensional polynomial predictors. We also present how oneway prediction can be used to obtain prediction in almost any direction and that the results are not always a two-dimensional polynomial predictor of minimum area.

Binary bent functions have a strictly specified number of non-zero values. In the same way, ternary bent functions satisfy certain requirements on the elements of their value vectors. These requirements can be used to specify six classes of ternary bent functions. Classes are mutually related by encoding of function values. Functions within a class are mutually related by permutation of elements in their function vectors. Given a basic ternary bent function, other functions in the same class can be constructed by permutation matrices having a block structure similar to that of the factor matrices appearing in the Good-Thomas decomposition of Cooley-Tukey Fast Fourier transform and related algorithms. Conversion of function vectors into matrices or equivalent matrix-valued vectors of smaller length leads to the redistribution of space complexity of related manipulation algorithms. In this matrix representations, construction of other bent functions from a given known bent function is performed by using either properties of bent functions or by manipulation of such representations in terms of FFT-like permutation matrices of dimensions smaller than the length of the function vector of the initial bent functions.

Binary bent functions have a strictly specified number of non-zero values. In the same way, ternary bent functions satisfy certain requirements on the elements of their value vectors. These requirements can be used to specify six classes of ternary bent functions. Classes are mutually related by encoding of function values. Given a basic ternary bent function, other functions in the same class can be constructed by permutation matrices having a block structure similar to that of the factor matrices appearing in the Good-Thomas decomposition of Cooley-Tukey Fast Fourier transform and related algorithms.