Recognizing Decomposition of a Partial Boolean Function

A hard combinatorial problem is investigated which has useful application in design of discrete devices: the two-block decomposition of a partial Boolean function. The key task is regarded: finding such a weak partition on the set of arguments, at which the considered function can be decomposed. Solving that task is essentially speeded up by the way of preliminary discovering traces of the sought-for partition. Efficient combinatorial operations are used by that, based on parallel execution of operations above adjacent units in the Boolean space.

Implementation of a Heuristic Method of Decomposition of Partial Boolean Functions

An original heuristic algorithm of sequential two-block decomposition of partial Boolean functions is researched. The key combinatorial task is considered: finding of suitable partition on the set of arguments, i. e. such one, on which the function is separable. The search for suitable partition is essentially accelerated by preliminary detection of its traces. Within the framework of the experimental system the efficiency of the algorithm is evaluated, the boundaries of its practical application are determined.

A linear programming code to determine whether a Boolean function is a threshold function,

Bibliography: p. 14.

Boolean logic device done with DFB laser diode

We present simulation results on how power output-input characteristic Instability in Distributed FeedBack -DFB semiconductor laser diode SLA can be employed to implemented Boolean logic device. Two configurations of DFB Laser diode under external optical injection, either in the transmission or in the reflective mode of operation, is used to implement different Optical Logic Cells (OLCs), called the Q- and the P-Device OLCs. The external optical injection correspond to two inputs data plus a cw control signal that allows to choose the Boolean logic function to be implement. DFB laser diode parameters are choosing to obtain an output-input characteristic with the values desired. The desired values are mainly the on-off contrast and switching power, conforming shape of hysteretic cycle. Two DFB lasers in cascade, one working in transmission operation and the other one in reflective operation, allows designing an inputoutput characteristic based on the same respond of a self-electrooptic effect device is obtained. Input power for a bit'T' is 35 uW(70uW) and a bit "0" is zero for all the Boolean function to be execute. Device control signal range to choose the logic function is 0-140 uW (280 uW). Q-device (P-device)

Vector Boolean Functions: Applications in Symmetric Cryptography

Esta tesis establece los fundamentos teóricos y diseña una colección abierta de clases C++ denominada VBF (Vector Boolean Functions) para analizar funciones booleanas vectoriales (funciones que asocian un vector booleano a otro vector booleano) desde una perspectiva criptográfica. Esta nueva implementación emplea la librería NTL de Victor Shoup, incorporando nuevos módulos que complementan a las funciones de NTL, adecuándolas para el análisis criptográfico. La clase fundamental que representa una función booleana vectorial se puede inicializar de manera muy flexible mediante diferentes estructuras de datas tales como la Tabla de verdad, la Representación de traza y la Forma algebraica normal entre otras. De esta manera VBF permite evaluar los criterios criptográficos más relevantes de los algoritmos de cifra en bloque y de stream, así como funciones hash: por ejemplo, proporciona la no-linealidad, la distancia lineal, el grado algebraico, las estructuras lineales, la distribución de frecuencias de los valores absolutos del espectro Walsh o del espectro de autocorrelación, entre otros criterios. Adicionalmente, VBF puede llevar a cabo operaciones entre funciones booleanas vectoriales tales como la comprobación de igualdad, la composición, la inversión, la suma, la suma directa, el bricklayering (aplicación paralela de funciones booleanas vectoriales como la empleada en el algoritmo de cifra Rijndael), y la adición de funciones coordenada. La tesis también muestra el empleo de la librería VBF en dos aplicaciones prácticas. Por un lado, se han analizado las características más relevantes de los sistemas de cifra en bloque. Por otro lado, combinando VBF con algoritmos de optimización, se han diseñado funciones booleanas cuyas propiedades criptográficas son las mejores conocidas hasta la fecha. ABSTRACT This thesis develops the theoretical foundations and designs an open collection of C++ classes, called VBF, designed for analyzing vector Boolean functions (functions that map a Boolean vector to another Boolean vector) from a cryptographic perspective. This new implementation uses the NTL library from Victor Shoup, adding new modules which complement the existing ones making VBF better suited for cryptography. The fundamental class representing a vector Boolean function can be initialized in a flexible way via several alternative types of data structures such as Truth Table, Trace Representation, Algebraic Normal Form (ANF) among others. This way, VBF allows the evaluation of the most relevant cryptographic criteria for block and stream ciphers as well as for hash functions: for instance, it provides the nonlinearity, the linearity distance, the algebraic degree, the linear structures, the frequency distribution of the absolute values of the Walsh Spectrum or the Autocorrelation Spectrum, among others. In addition, VBF can perform operations such as equality testing, composition, inversion, sum, direct sum, bricklayering (parallel application of vector Boolean functions as employed in Rijndael cipher), and adding coordinate functions of two vector Boolean functions. This thesis also illustrates the use of VBF in two practical applications. On the one hand, the most relevant properties of the existing block ciphers have been analysed. On the other hand, by combining VBF with optimization algorithms, new Boolean functions have been designed which have the best known cryptographic properties up-to-date.

Phase transitions and memory effects in the dynamics of Boolean networks

The generating functional method is employed to investigate the synchronous dynamics of Boolean networks, providing an exact result for the system dynamics via a set of macroscopic order parameters. The topology of the networks studied and its constituent Boolean functions represent the system's quenched disorder and are sampled from a given distribution. The framework accommodates a variety of topologies and Boolean function distributions and can be used to study both the noisy and noiseless regimes; it enables one to calculate correlation functions at different times that are inaccessible via commonly used approximations. It is also used to determine conditions for the annealed approximation to be valid, explore phases of the system under different levels of noise and obtain results for models with strong memory effects, where existing approximations break down. Links between Boolean networks and general Boolean formulas are identified and results common to both system types are highlighted. © 2012 Copyright Taylor and Francis Group, LLC.

On reliable computation by noisy random Boolean formulas

We study noisy computation in randomly generated k-ary Boolean formulas. We establish bounds on the noise level above which the results of computation by random formulas are not reliable. This bound is saturated by formulas constructed from a single majority-like gate. We show that these gates can be used to compute any Boolean function reliably below the noise bound.

Decomposition of Boolean Functions – Recognizing a Good Solution by Traces

The problem of sequent two-block decomposition of a Boolean function is regarded in case when a good solution does exist. The problem consists mainly in finding an appropriate weak partition on the set of arguments of the considered Boolean function, which should be decomposable at that partition. A new fast heuristic combinatorial algorithm is offered for solving this task. At first the randomized search for traces of such a partition is fulfilled. The recognized traces are represented by some "triads" - the simplest weak partitions corresponding to non-trivial decompositions. After that the whole sought-for partition is restored from the discovered trace by building a track initialized by the trace and leading to the solution. The results of computer experiments testify the high practical efficiency of the algorithm.

equivalence of almost bent and almost perfect nonlinear functions and their generalizations

Vectorial Boolean function, almost bent, almost perfect nonlinear, affine equivalence, CCZ-equivalence

Modeling and Analysis of Some Heavy Tailed Time Series

The thesis has covered various aspects of modeling and analysis of finite mean time series with symmetric stable distributed innovations. Time series analysis based on Box and Jenkins methods are the most popular approaches where the models are linear and errors are Gaussian. We highlighted the limitations of classical time series analysis tools and explored some generalized tools and organized the approach parallel to the classical set up. In the present thesis we mainly studied the estimation and prediction of signal plus noise model. Here we assumed the signal and noise follow some models with symmetric stable innovations.We start the thesis with some motivating examples and application areas of alpha stable time series models. Classical time series analysis and corresponding theories based on finite variance models are extensively discussed in second chapter. We also surveyed the existing theories and methods correspond to infinite variance models in the same chapter. We present a linear filtering method for computing the filter weights assigned to the observation for estimating unobserved signal under general noisy environment in third chapter. Here we consider both the signal and the noise as stationary processes with infinite variance innovations. We derived semi infinite, double infinite and asymmetric signal extraction filters based on minimum dispersion criteria. Finite length filters based on Kalman-Levy filters are developed and identified the pattern of the filter weights. Simulation studies show that the proposed methods are competent enough in signal extraction for processes with infinite variance.Parameter estimation of autoregressive signals observed in a symmetric stable noise environment is discussed in fourth chapter. Here we used higher order Yule-Walker type estimation using auto-covariation function and exemplify the methods by simulation and application to Sea surface temperature data. We increased the number of Yule-Walker equations and proposed a ordinary least square estimate to the autoregressive parameters. Singularity problem of the auto-covariation matrix is addressed and derived a modified version of the Generalized Yule-Walker method using singular value decomposition.In fifth chapter of the thesis we introduced partial covariation function as a tool for stable time series analysis where covariance or partial covariance is ill defined. Asymptotic results of the partial auto-covariation is studied and its application in model identification of stable auto-regressive models are discussed. We generalize the Durbin-Levinson algorithm to include infinite variance models in terms of partial auto-covariation function and introduce a new information criteria for consistent order estimation of stable autoregressive model.In chapter six we explore the application of the techniques discussed in the previous chapter in signal processing. Frequency estimation of sinusoidal signal observed in symmetric stable noisy environment is discussed in this context. Here we introduced a parametric spectrum analysis and frequency estimate using power transfer function. Estimate of the power transfer function is obtained using the modified generalized Yule-Walker approach. Another important problem in statistical signal processing is to identify the number of sinusoidal components in an observed signal. We used a modified version of the proposed information criteria for this purpose.

Multilevel Training of Binary Morphological Operators

The design of binary morphological operators that are translation-invariant and locally defined by a finite neighborhood window corresponds to the problem of designing Boolean functions. As in any supervised classification problem, morphological operators designed from a training sample also suffer from overfitting. Large neighborhood tends to lead to performance degradation of the designed operator. This work proposes a multilevel design approach to deal with the issue of designing large neighborhood-based operators. The main idea is inspired by stacked generalization (a multilevel classifier design approach) and consists of, at each training level, combining the outcomes of the previous level operators. The final operator is a multilevel operator that ultimately depends on a larger neighborhood than of the individual operators that have been combined. Experimental results show that two-level operators obtained by combining operators designed on subwindows of a large window consistently outperform the single-level operators designed on the full window. They also show that iterating two-level operators is an effective multilevel approach to obtain better results.

Likelihood approximations and discrete models for tied survival data

Ties among event times are often recorded in survival studies. For example, in a two week laboratory study where event times are measured in days, ties are very likely to occur. The proportional hazards model might be used in this setting using an approximated partial likelihood function. This approximation works well when the number of ties is small. on the other hand, discrete regression models are suggested when the data are heavily tied. However, in many situations it is not clear which approach should be used in practice. In this work, empirical guidelines based on Monte Carlo simulations are provided. These recommendations are based on a measure of the amount of tied data present and the mean square error. An example illustrates the proposed criterion.

A new approach to a model of the mammalian retina with optically programmable logic cells

This paper reports a model of the mammalian retina as well as an interpretation of some functions of the visual cortex. Its main objective is to simulate some of the behaviors observed at the different retina cells depending on the characteristics of the light impinging onto the photoreceptors. This simulation is carried out with a simple structure employed previously as basic building block of some optical computer architectures. Its possibility to perform any type of Boolean function allows a wide range of behaviors.

Algoritmos para mineria de datos con redes de neuronas

Este trabajo propone una serie de algoritmos con el objetivo de extraer información de conjuntos de datos con redes de neuronas. Se estudian dichos algoritmos con redes de neuronas Enhenced Neural Networks (ENN), debido a que esta arquitectura tiene algunas ventajas cuando se aproximan funciones mediante redes neuronales. En la red ENN los pesos de la matriz principal varián con cada patrón, por lo que se comete un error menor en la aproximación. Las redes de neuronas ENN reúnen la información en los pesos de su red auxiliar, se propone un método para obtener información de la red a través de dichos pesos en formas de reglas y asignando un factor de certeza de dichas reglas. La red ENN obtiene un error cuadrático medio menor que el error teórico de una aproximación matemática por ejemplo mediante polinomios de Taylor. Se muestra como una red ENN, entrenada a partir un conjunto de patrones obtenido de una función de variables reales, sus pesos asociados tienen unas relaciones similares a las que se veri_can con las variables independientes con dicha función de variables reales. Las redes de neuronas ENN aproximan polinomios, se extrae conocimiento de un conjunto de datos de forma similar a la regresión estadística, resolviendo de forma más adecuada el problema de multicolionalidad en caso de existir. Las relaciones a partir de los pesos asociados de la matriz de la red auxiliar se obtienen similares a los coeficientes de una regresión para el mismo conjunto numérico. Una red ENN entrenada a partir de un conjunto de datos de una función boolena extrae el conocimiento a partir de los pesos asociados, y la influencia de las variables de la regla lógica de la función booleana, queda reejada en esos pesos asociados a la red auxiliar de la red ENN. Se plantea una red de base radial (RBF) para la clasificación y predicción en problemas forestales y agrícolas, obteniendo mejores resultados que con el modelo de regresión y otros métodos. Los resultados con una red RBF mejoran al método de regresión si existe colinealidad entre los datos que se dispone y no son muy numerosos. También se detecta que variables tienen más importancia en virtud de la variable pronóstico. Obteniendo el error cuadrático medio con redes RBF menor que con otros métodos, en particular que con el modelo de regresión. Abstract A series of algorithms is proposed in this study aiming at the goal of producing information about data groups with a neural network. These algorithms are studied with Enheced Neural Networks (ENN), owing to the fact that this structure shows sever advantages when the functions are approximated by neural networks. Main matrix weights in th ENN vary on each pattern; so, a smaller error is produced when approximating. The neural network ENN joins the weight information contained in their auxiliary network. Thus, a method to obtain information on the network through those weights is proposed by means of rules adding a certainty factor. The net ENN obtains a mean squared error smaller than the theorical one emerging from a mathematical aproximation such as, for example, by means of Taylor's polynomials. This study also shows how in a neural network ENN trained from a set of patterns obtained through a function of real variables, its associated weights have relationships similar to those ones tested by means of the independent variables connected with such functions of real variables. The neural network ENN approximates polynomials through it information about a set of data may be obtained in a similar way than through statistical regression, solving in this way possible problems of multicollinearity in a more suitable way. Relationships emerging from the associated weights in the auxiliary network matrix obtained are similar to the coeficients corresponding to a regression for the same numerical set. A net ENN trained from a boolean function data set obtains its information from its associated weights. The inuence of the variables of the boolean function logical rule are reected on those weights associated to the net auxiliar of the ENN. A radial basis neural networks (RBF) for the classification and prediction of forest and agricultural problems is proposed. This scheme obtains better results than the ones obtained by means of regression and other methods. The outputs with a net RBF better the regression method if the collineality with the available data and their amount is not very large. Detection of which variables are more important basing on the forecast variable can also be achieved, obtaining a mean squared error smaller that the ones obtained through other methods, in special the one produced by the regression pattern.

Bit noise from an optical logic gate with laser diodes

Output bits from an optical logic cell present noise due to the type of technique used to obtain the Boolean functions of two input data bits. We have simulated the behavior of an optically programmable logic cell working with Fabry Perot-laser diodes of the same type employed in optical communications (1550nm) but working here as amplifiers. We will report in this paper a study of the bit noise generated from the optical non-linearity process allowing the Boolean function operation of two optical input data signals. Two types of optical logic cells will be analyzed. Firstly, a classical "on-off" behavior, with transmission operation of LD amplifier and, secondly, a more complicated configuration with two LD amplifiers, one working on transmission and the other one in reflection mode. This last configuration has nonlinear behavior emulating SEED-like properties. In both cases, depending on the value of a "1" input data signals to be processed, a different logic function can be obtained. Also a CW signal, known as control signal, may be apply to fix the type of logic function. The signal to noise ratio will be analyzed for different parameters, as wavelength signals and the hysteresis cycles regions associated to the device, in relation with the signals power level applied. With this study we will try to obtain a better understanding of the possible effects present on an optical logic gate with Laser Diodes.