P-Realizable Boolean Functions and Universal Binary Neurons

Introduction. The problem of mathematical safe arises in the theory of computer games and cryptographic applications. The article considers numerous variations of the mathematical safe problem and examples of its solution using systems of linear Diophantine equations in finite rings and fields. The purpose of the article. To present methods for solving the problem of a mathematical safe for its various variations, which are related both to the domain over which the problem is considered and to the structure of systems of linear equations over these domains. To consider the problem of a mathematical safe (in matrix and graph forms) in different variations over different finite domains and to demonstrate the work of methods for solving this problem and their efficiency (systems over finite simple fields, finite fields, ghost rings and finite associative-commutative rings). Results. Examples of solving the problem of a mathematical safe, the conditions for the existence of solutions in different areas, over which this problem is considered. The choice of the appropriate area over which the problem of the mathematical safe is considered, and the appropriate algorithm for solving it depends on the number of positions of the latches of the safe. All these algorithms are accompanied by estimates of their time complexity, which were considered in the first part of this paper. Conclusions. The considered methods and algorithms for solving linear equations and systems of linear equations in finite rings and fields allow to solve the problem of a mathematical safe in a large number of variations of its formulation (over finite prime field, finite field, primary associative-commutative ring and finite associative-commutative ring with unit). Keywords: mathematical safe, finite rings, finite fields, method, algorithm.

Multi-valued neurons (MVN) and universal binary neurons (UBN) are neural elements with complex-valued weights and high functionality. It is possible to implement the arbitrary mapping described by partially defined multiple-valued function on the single MVN and the arbitrary mapping described by Boolean function (which may not be threshold) on the single UBN. In this paper we consider some applications carried out using these wonderful features of MVN and UBN. Conception of cellular neural networks based on MVN and UBN becomes a base for nonlinear cellular neural filtering. Application of the corresponding filters for edge detection and solving of the super-resolution problem are considered in the paper.

We consider fast convergence learning algorithms for multi-valued and universal binary neurons. These neurons are suggested to be used for design of neural networks based on CNN paradigm. On the basis of such networks we offer to solve some problems of image processing. For instance, high efficient method for contours detection obtained by learning algorithm described in the paper is presented. Also solution of the XOR-problem on the single neuron is described. >

Algebraic immunity is an important cryptographic property for Boolean functions against algebraic attacks. Constructions of Boolean functions with the maximum algebraic immunity (MAI Boolean functions) by using univariate polynomial representation of Boolean functions over finite fields have received more and more attention. In this paper, how to obtain more MAI Boolean functions from a known MAI Boolean function under univariate polynomial representation is further investigated. The sufficient condition of Boolean functions having the maximum algebraic immunity obtained by changing a known MAI Boolean function under univariate polynomial representation is given. With this condition, more balanced MAI Boolean functions under univariate polynomial representation can be obtained. The algebraic degree and the nonlinearity of these Boolean functions are analyzed.

Letbe a field and[x,y] the ring of polynomials in two variables over. Letf∈[x,y] and consider the residue class ringR:=[x,y]/f[x,y]. Our first aim is to study digit representations inR, i.e., we ask for whichfeach element ofRadmits a digit representation of the formd0+d1x+ ⋅ ⋅ ⋅ +dℓxℓwith digitsdi∈[y] satisfying degy(di) < degy(f). These digit systems are motivated by the well-known notion of canonical number systems. Next we enlarge the ring in order to allow representations including negative powers of the “base”x. In particular, we define and characterize digit representations for the ringS:=((x−1,y−1))/f((x−1,y−1)) and give easy to handle criteria for finiteness and periodicity of such representations. Finally, we attach fundamental domains to our digit systems. The fundamental domain of a digit system is the set of all elements having only negative powers ofxin their “x-ary” representation. The translates of the fundamental domain induce a tiling ofS. Interestingly, the fundamental domains of our digit systems turn out to be unions of boxes. If we choose=qto be a finite field, these unions become finite.

Permutation polynomials on matrices

Diestelkamp, Hartke and Kenney recently studied local permutation polynomials over finite field $F_s$ and proved that $2s-4$ is the sharp bound of their degrees when $s \\gt 3$. In this paper, we focus on local permutation polynomials over residue class rings. We prove two specific formulas computing the number and the sharp bound of degrees of local permutation polynomials modulo $p^n$ respectively.

Separations among the first order logic Ring(0,+, ∗) of finite residue class rings, its extensions with generalized quantifiers, and in the presence of a built-in order are shown, using algebraic methods from class field theory. These methods include classification of spectra of sentences over finite residue classes as systems of congruences, and the study of their h-densities over the set of all prime numbers, for various functions h on the natural numbers. Over ordered structures the logic of finite residue class rings and extensions are known to capture DLOGTIMEuniform circuit complexity classes ranging from AC to TC. Separating these circuit complexity classes is directly related to classifying the h-density of spectra of sentences in the corresponding logics of finite residue classes. We further give general conditions under which a logic over the finite residue class rings has a sentence whose spectrum has no h-density. One application of this result is that Supported by MICINN project TIN2011-27479-C04-03 (BASMATI), MINECO project TIN201457226-P (APCOM) and Gen. Cat. project SGR2014-890 (MACDA).

A variety of Lie rings is called just-non-Cross if it itself is not Cross, but each of its proper subvarieties is Cross, i.e. is generated by a finite ring. In this paper, we completely describe the solvable just-non-Cross varieties both of Lie rings and of Lie R-algebras where R is a finite commutative ring with identity and, in particular, where R is a finite field. We find algorithms which allow us to determine whether a given identity defines a Cross variety of Lie algebras; also, using the multiplication and addition tables of a finite Lie algebra, we find algorithms for extracting its identities.Bibliography: 10 titles.

Multi-valued and universal binary neurons (MVN and UBN) are the neural processing elements with complex-valued weights and high functionality. It is possible to implement an arbitrary mapping described by partial-defined multiple-valued function on the single MVN and an arbitrary mapping described by partial-defined or fully-defined Boolean function (which can be not threshold) on the single UBN. The fast-converged learning algorithms are existing for both types of neurons. Such features of the MVN and UBN may be used for solution of the different kinds of problems. One of the most successful applications of the MVN and UBN is their usage as basic neurons in the Cellular Neural Networks (CNN) for solution of the image processing and image analysis problems. Another effective application of the MVN is their use as the basic neurons in the neural networks oriented to the image recognition.

The object of research is the method of figurative transformations for Boolean functions minimization. One of the most problematic places to minimize Boolean functions is the complexity of the minimization algorithm and the guarantee of obtaining a minimal function. During the study, the method of equivalent figurative transformations was used, which is based on the laws and axioms of the algebra of logic; minimization protocols for Boolean functions that are used when the truth table of a given function has a complete binary combinatorial system with repetition or an incomplete binary combinatorial system with repetition. A reduction in the complexity of the minimization process for Boolean functions is obtained, new criteria for finding minimal functions are established. This is due to the fact that the proposed method of Boolean functions minimization has a number of peculiarities of solving the problem of finding minimal logical functions, in particular: – mathematical apparatus of the block diagram with repetition makes it possible to obtain more information about the orthogonality, adjacency, uniqueness of the truth table blocks; – equivalent figurative transformations due to the greater information capacity are capable of replacing verbal procedures of algebraic transformations; – result of minimization is estimated based on the sign of the minimum function; – minimum DNF or CNF functions are obtained regardless of the given normal form of the logical function, which means that it is necessary to minimize the given function for two normal forms — DNF and CNF using the full truth table; This ensures that it is possible to obtain an optimal reduction in the number of variables of a given function without losing its functionality. The effectiveness of the use of equivalent figurative transformations for Boolean functions minimization is demonstrated by examples of minimization of functions borrowed from other methods for the purpose of comparison. Compared with similar well-known methods of Boolean functions minimization, this provides: – less complexity of the minimization procedure for Boolean functions; – guaranteed Boolean functions minimization; – self-sufficiency of the specified method of Boolean functions minimization due to the introduction of features of the minimal function and minimization of two normal forms – DNF and CNF on the complete truth table of a given Boolean function

Circulant matrices form an important class of matrices that have been continuously studied due to their nice algebraic structures and wide applications. In this study, we focus specifically on negacirculant matrices, which are known as extensions of the classical circulant matrices. The algebraic structures of the rings of n × n n\times n negacirculant matrices over finite fields and over finite commutative chain rings are presented. Subsequently, the algebraic structures and enumeration of the unit groups of such matrix rings are established. Additionally, the number of non-singular n × n n\times n negacirculant matrices with prescribed determinant is given in some cases. Conjectures and open problems are proposed as well as a brief discussion in the case where the underlying ring is a finite commutative ring with identity is also presented.

Coded Distributed Matrix Multiplication (CDMM) is a distributed matrix multiplication (DMM) for large-scale matrices through a coding scheme such that any <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> worker nodes among all <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> worker nodes can recover the final product, where <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> corresponds to the length of the code and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">R</i> ≤ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i> is called the recovery threshold. The state-of-art CDMM schemes, such as EP codes for Single DMM and GCSA codes for batch DMM, are defined over a Galois field GF(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i>) of size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ≥ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>. These are inefficient for small Galois fields such as GF(2) and the integer residue ring Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i></sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> due to the lack of invertible elements for interpolation. DMM over Z<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">_p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> (such as Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">64</sup> ) is well-motivated in practice due to their direct compatibility with hardware. In this work, we construct efficient CDMM over the Galois ring GR(p<sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup>, <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i>) which is an extension ring over Z<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i></sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup> of degree <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</i>, particularly, GR(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p, d</i>) = GF(<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</sup>) is the Galois field and GR(<sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><i>p</i></sub><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup>, 1) = Z<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">_p</i><sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ℓ</sup>. We first give a general CDMM framework for the batch of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> matrix multiplications via the famous RMFE (Cascudo et al. Crypto’18). Compared with GCSA, our construction has a smaller recovery threshold by a factor of 1/<italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i>. Next, we optimize EP codes via batch preprocessing of the input matrices. We give two types of Single CDMM, which can achieve almost the same performance as EP codes over a Galois field with size <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q</i> ≥ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">N</i>. Finally, we present the experimental analysis of our CDMM on Galois rings.

Every second year, the GEOCRYPT conference brings together researchers from arithmetic geometry and cryptography. After Guadeloupe in 2009 and Corsica in 2011, French Polynesia was the host of its 2013 edition which took place in Punaauia, on the island of Tahiti, October 7-11. The main topic of GEOCRYPT has always been the application of pure mathematical techniques to the safety and efficiency of modern communication systems, with particular interest in the fields of arithmetic and algebraic geometry over finite fields, algorithms for finite fields, error correcting codes, cryptology, boolean functions, discrete dynamical systems, and their interactions. The 2013 edition had the honour to feature six invited and eleven contributed talks from renowned international experts presenting strong results recently obtained on topics ranging from pure mathematics to cryptographic algorithms. The GEOCRYPT 2013 Program Committee carefully evaluated the submitted abstracts and selected the best contributions for presentation at the conference.

A recent work of Chattopadhyay et al. (CCC 2018) introduced a new framework for the design of pseudorandom generators for Boolean functions. It works under the assumption that the Fourier tails of the Boolean functions are uniformly bounded for all levels by an exponential function. In this work, we design an alternative pseudorandom generator that only requires bounds on the second level of the Fourier tails. It is based on a derandomization of the work of Raz and Tal (ECCC 2018) who used the above framework to obtain an oracle separation between BQP and PH. As an application, we give a concrete conjecture for bounds on the second level of the Fourier tails for low degree polynomials over the finite field F_2. If true, it would imply an efficient pseudorandom generator for AC^0[oplus], a well-known open problem in complexity theory. As a stepping stone towards resolving this conjecture, we prove such bounds for the first level of the Fourier tails.