ON THE NONEXISTENCE of BENT FUNCTIONS

The class of affine Boolean functions is rich enough to express constant bits and dependencies between different bits of different words. For example, the function (x0) ∧ (¬y1) ∧ (x4 ⇔ y7) ∧ (x5 ⇔ ¬y9) is affine and expresses the invariant that the low bit (bit 0) of the variable x is true, that bit 1 of y is false, that the bits 4 and 7 of x and y coincide whereas bits 5 and 9 of x and y differ. This class of Boolean function is amenable to bit-precise reasoning since it satisfies strong chain properties which bound the number of times a system of semantic fixpoint equations need to be reapplied when reasoning about loops. This paper address the key problem of abstracting an arbitrary Boolean function to either a general affine function or a so-called affine function of width 2, when the function is represented as an ROBDD. Novel algorithms are presented for this task: one that manipulates Boolean vectors and another which is inspired by anti-unification. The speed and precision of both algorithms are compared on benchmark circuits, to draw conclusions on the tractability of affine abstraction.

This paper investigates the optimum OBDD representation problem based on two classes of Boolean functions. The first class is defined by OBDDs, in which the number of non-terminal nodes is equal to the number of input variables. We refer to such OBDDs and their corresponding Boolean functions as thin OBDDs and thin Boolean functions. The second class is the thin factored Boolean functions, which is defined by factored forms with each variable appearing exactly once. Many interesting properties of these two classes of functions are presented. Based on which, a revised dynamic shortest cube first OBDD variable ordering algorithm is developed. This algorithm is shown to be optimum for thin factored Boolean functions.

Chapter 2 - Fourier Analysis of Boolean Functions

Two Boolean functions are affine equivalent if one can be obtained from the other by applying an affine transformation to the input variables. For a long time, there have been efforts to investigate the affine equivalence of Boolean functions. Due to the complexity of the general problem, only affine equivalence under certain groups of permutations is usually considered. Boolean functions which are invariant under the action of cyclic rotation of the input variables are known as rotation symmetric (RS) Boolean functions. Due to their speed of computation and the prospect of being good cryptographic Boolean functions, this class of Boolean functions has received a lot of attention from cryptographic researchers. In this paper, we study affine equivalence for the simplest rotation symmetric Boolean functions, called MRS functions, which are generated by the cyclic permutations of a single monomial. Using Pólya’s enumeration theorem, we compute the number of equivalence classes, under certain large groups of permutations, for these MRS functions in any number

Several underlying structural and functional factors that determine the fault behavior of a network are not yet well understood. In this paper, we show that there exists a large class of Boolean functions, called root functions, which can never appear as faulty response in an irredundant two-level circuit even when any arbitrary multiple stuck-at faults are injected. Conversely, we show that any other Boolean function can appear as a faulty response in an irredundant realization of some root function under certain stuck-at faults. We characterize this new class of functions and show that for n variables, their number is exactly equal to the number of independent dominating sets (Harary and Livingston, Appl. Math. Lett., 1993) in a Boolean n-cube. Similar properties are observed for multiple-valued logic functions as well. Finally, we discuss its application to logic design and point out some open problems.

Information processing in dendrites: II. Information theoretic complexity

Rotation Symmetric Boolean Functions –; Count and Cryptographic Properties

This problem has been around for a while but is one of my favorites. I will state it here in three forms, discuss a number of known results (some easy and some more intricate), and finally end with small financial incentives for various kinds of partial progress. This problem appears in various guises in [2, 3, 10]. To begin we need the following standard definition: a boolean function f over {0,1} n has (at most) r relevant variables if there exist r indices i 1, ..., i r such that $f(x) = g(x_{i_1}, \ldots, x_{i_r})$ for some boolean function g over {0,1} r . In other words, the value of f is determined by only a subset of r of its n input variables. For instance, the function $f(x) = x_1\bar{x}_2 \vee x_2\bar{x}_5 \vee x_5\bar{x}_1$ has three relevant variables. The “class of boolean functions with r relevant variables” is the set of all such functions, over all possible g and sets {i 1, ..., i r }.

The properties of the 2m-variable symmetric Boolean functions with maximum algebraic immunity are studied in this paper. Their value vectors, algebraic normal forms, and algebraic degrees and weights are all obtained. At last, some necessary conditions for a symmetric Boolean function on even number variables to have maximum algebraic immunity are introduced.

The algebraic degree is an important parameter of Boolean functions used in cryptography. When a function in a large number of variables is not given explicitly in algebraic normal form, it is usually not feasible to compute its degree, so we need to estimate it. We propose a probabilistic test for deciding whether the algebraic degree of a Boolean function f is below a certain value k. If the degree is indeed below k, then f will always pass the test, otherwise f will fail each instance of the test with a probability dtk(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{dt}_k(f)$$\\end{document}, which is closely related to the average number of monomials of degree k of the polynomials which are affine equivalent to f. The test has a good accuracy only if this probability dtk(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{dt}_k(f)$$\\end{document} of failing the test is not too small. We initiate the study of dtk(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{dt}_k(f)$$\\end{document} by showing that in the particular case when the degree of f is actually equal to k, the probability will be in the interval (0.288788, 0.5], and therefore a small number of runs of the test will be sufficient to give, with very high probability, the correct answer. Exact values of dtk(f)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ extrm{dt}_k(f)$$\\end{document} for all the polynomials in 8 variables were computed using the representatives listed by Hou and by Langevin and Leander.

Context. Such perfect algebraic constructions of many-valued logic as ternary bent-functions and their truth tables which are called as 3-bent-sequences, are used very often in modern cryptographic algorithms, in particular, in pseudorandom sequence generators. However, today there are no methods for synthesizing the ternary bent-functions class for a number of variables greater than two, which significantly limits the ability to scale the number of protection levels of the of pseudorandom sequence generators based on the ternary bent-functions. This circumstance generates the task of developing methods for the synthesis of ternary bent-functions, which is solved in this paper for the case of ternary bent-functions of three variables. The object of this research is the process of efficiency increasing of the cryptographic algorithms based on the functions of many-valued logic.Objective. The purpose of the paper is to construct a method for the synthesis of the set of ternary bent-functions of three variables.Method. The mathematical apparatus of the Reed-Muller transform (algebraic normal form) was used as the basis of the proposed constructive method for the synthesis of ternary bent-functions of three variables. So, on the basis of the established properties of the algebraic normal form of ternary bent-functions and limited enumeration, the search for ternary bent-functions up to affine terms is performed, after which we apply the procedure of reproduction.Results. As a result of using of the proposed method for the synthesis of ternary bent-functions of three variables, 155844 3-bentsequences were found up to an affine term, while the cardinality of the full set of found 3-bent-sequences is 12623364. The research performed made it possible to determine that in this set there are 3-bent-sequences of six different weight structures, on the basis of which 12 different triple sets can be compiled for use in pseudorandom sequence generators. A scheme for a cryptographically stable pseudorandom sequence generator based on the found set of 3-bent-sequences of length N = 27 is proposed. It is shown that the protection levels number of such a generator of pseudorandom sequences is 41 7.041 10    which is comparable with the protection levels number of modern block symmetric cryptographic algorithms, for example, AES-128.Conclusions. The further development of modern cryptographic algorithms, in particular, cryptographically stable pseudorandom sequence generators, is largely based on the use of perfect algebraic constructions of many-valued logic. For the first time, a constructive method for the synthesis of ternary bent-functions of three variables is proposed. For the found set of ternary bent-functions, the distribution of weight structures is found, and the possible triple sets are established. Based on the constructed set of ternary bentfunctions, a pseudorandom sequence generator scheme is proposed that has a protection levels number that is comparable with modern block symmetric cryptographic algorithms. We note that the constructed class of ternary bent-functions can also be used for the synthesis of cryptographically strong S-boxes, codes of constant amplitude, as well as error correction codes. As an actual area of further research, we can note the development of methods for the synthesis of ternary bent-functions of a larger number of variables.

In this work, our aim is to achieve a high throughput compact AES S-box with minimal power consumption. In most VLSI implementations, there exist a definite trade off between hardware performance and its operating requirements. In this work, we propose a novel pipelining arrangement over the compact composite field AES S-box such that both high throughput and low power are optimized. Our S-box outperformed the conventional pipelined AES S-box from three perspectives, (i) the most optimum (compact and short critical path) composite field AES S-box is used, which has different arithmetic properties compared to previous works; (ii) Algebraic Normal Form (ANF) representation is utilized to induce consistent and optimal pipelining arrangement; and (iii) Fine-grain pipelining is applied in the GF (24) multiplier. As such, a higher throughput rate is attained and at the same time the dynamic hazards is mitigated. A high throughput of 3.3Gbps with a low power consumption of 34.98mW and total of 95 LE (Logic Element) composite field AES S-box is reported in this work.

Random Boolean networks (RBNs) have been widely explored as models for understanding complex systems, in particular gene regulatory networks (GRNs). Stability (order) and instability (chaos), critical characteristics of any dynamic system, have naturally been a central focus in RBN research. Several measures have previously been introduced to assess stability, e.g., via damage spreading or via statistical properties of attractors such as their number, lengths or basin sizes. Undoubtedly, network topology plays an important role in shaping the dynamics of RBN. Another key factor influencing the dynamics is the Boolean functions (BFs) assigned to each node. In this work, we conduct a systematic investigation to examine the influence of five different classes of BFs on the dynamics of RBNs. By employing various dynamical stability measures, we show that biologically meaningful BFs consistently drive the dynamics toward the stable regime compared to random BFs. This observation holds across networks with varying size, connectivity, and degree distributions. Additionally, we show that the change in the values of our stability measures with increasing connectivity or size of RBNs is different across the BF classes. For most observables, random BFs change the values significantly, pushing the dynamics toward a more chaotic regime with increasing connectivity or network size. In contrast, biologically meaningful BFs show very minimal fluctuations for most stability measures. These findings emphasize the advantage of restricting to biologically meaningful classes in the reconstruction and modeling of biological systems within Boolean framework.

We consider an approach to the classification of n-variable Boolean bent functions of the nonlinearity degree 3. We utilize the apparatus of bent rectangles introduced by S. V. Agievich. This apparatus was used for the classification of 8-variable Boolean cubic bent functions. The results of our research allow to construct cubic bent functions that depend on an arbitrary even number of variables; the construction is based on well studied quadratic bent functions.

We prove a lower bound of Ω(n1/2-c), for all c> 0, on the query complexity of (two-sided error) non-adaptive algorithms for testing whether an n-variable Boolean function is monotone versus constant-far from monotone. This improves a ~Ω(n1/5) lower bound for the same problem that was obtained in [6], and is very close to the recent upper bound of ~O(n1/2/e2) by Khot et al. [13].