When considering Boolean switching functions with n input variables, there are 2^(2^n) possible functions that can be realized by enumerating all possible combinations of input values and arrangements of output values. As is expected with double exponential growth, the number of functions becomes unmanageable very quickly as n increases. This thesis develops a new approach for computing the spectral classes where the spectral operations are performed by manipulating the truth tables rather than first moving to the spectral domain to manipulate the spectral coefficients. Additionally, a generic approach is developed for modeling these spectral operations within the functional domain. The results of this research match previous for n < or = to 4 but differ when n=5 is considered. This research indicates with a high level of confidence that there are in fact 15 previously unidentified classes, for a total of 206 spectral classes needed to represent all 2^(2^n) Boolean functions.

In this thesis, we work on the upper bound for nonlinearity of t-resilient Boolean functions given by Sarkar and Maitra, which is based on divisibility properties of spectral weights of resilient functions and study construction methods that achieve the upper bound. One of the construction methods, introduced by Maity and Johansson, starts with a bent function and complements some values of its truth table corresponding to a previously chosen set of inputs, S, which satisfies three criteria. In this thesis, we show that a fourth criterion is needed for t-resiliency of the resulting function, and prove that three criteria of Maity and Johansson do not guarantee resiliency. We also work on other constructions, one by Sarkar and Maitra, which uses a Maiorana-McFarland like technique to satisfy the upper bound and the other by Tarannikov, which satisfies the nonlinearity bound using a technique with low computational complexity. However, these methods have tendency to maximize the order of resiliency for a given number of variables, therefore one cannot construct functions for all possible resiliency values given the number of variables, using this method. We further go into details and compute the auto-correlation functions of the constructed Boolean functions to find the absolute indicator and sum-of-squared-errors for each of them. We also provide a comparison of Boolean functions constructed by other techniques given in the literature, together with the ones studied in this thesis.

The thesis consist of three independent parts. In the first part, we investigate the height sequence of an element of a partially ordered set. Let $x$ be an element of the partially ordered set $P$. Then $h_i(x)$ is the number of linear extensions of $P$ in which $x$ is in the $i$th lowest position. The sequence ${h_i(x)}$ is called the height sequence of $x$ in $P$. Stanley proved in 1981 that the height sequence is log-concave, but no combinatorial proof has been found, and Stanley's proof does not reveal anything about the deeper structure of the height sequence. In this part of the thesis, we provide a combinatorial proof of a special case of Stanley's theorem. The proof of the inequality uses the Ahlswede--Daykin Four Functions Theorem. In the second part, we study two classes of segment orders introduced by Shahrokhi. Both classes are natural generalizations of interval containment orders and interval orders. We prove several properties of the classes, and inspired by the observation, that the classes seem to be very similar, we attempt to find out if they actually contain the same partially ordered sets. We prove that the question is equivalent to a stretchability question involving certain sets of pseudoline arrangements. We also prove several facts about continuous universal functions that would transfer segment orders of the first kind into segments orders of the second kind. In the third part, we consider the lattice whose elements are the subsets of ${1,2,ldots,n}$. Trotter and Felsner asked whether this subset lattice always contains a monotone Hamiltonian path. We make progress toward answering this question by constructing a path for all $n$ that satisfies the monotone properties and covers every set of size at most $3$. This portion of thesis represents joint work with David M.~Howard.

Parameter variations, noise susceptibility, and increasing energy dissipation of CMOS devices have been recognized as major challenges in circuit and micro-architecture design in the nanometer regime. Among these, parameter variations and noise susceptibility are increasingly causing CMOS devices to behave in an "unreliable" or "probabilistic" manner. To address these challenges, a shift in design paradigm, from current day deterministic designs to "statistical" or "probabilistic" designs is deemed inevitable. Motivated by these considerations, I introduce and define probabilistic Boolean logic, whose logical operators are by definition "correct" with a probability 1/2 <= p <= 1. While most of the laws of conventional Boolean logic can be naturally extended to be valid in the probabilistic case, there are a few significant departures. We also show that computations realized using implicitly probabilistic Boolean operators are more energy efficient than their counterparts which use explicit sources of randomness, in the context of probabilistic Boolean circuits as well as probabilistic models with state, Rabin automata. To demonstrate the utility of implicitly probabilistic elements, we study a family of probabilistic architectures: the probabilistic system-on-a-chip PSOC, based on CMOS devices rendered probabilistic due to noise, referred to as probabilistic CMOS or PCMOS devices. These architectures yield significant improvements, both in the energy consumed as well as in the performance in the context of probabilistic or randomized applications with broad utility. Finally, we extend the consideration of probability of correctness to arithmetic operations, through probabilistic arithmetic. We show that in the probabilistic context, substantial savings in energy over correct arithmetic operations may be achieved. This is the theoretical basis of the energy savings reported in the video decoding and radar processing applications that has been demonstrated in prior work.

Boolean functions are an important area of study for cryptography. These functions, consisting merely of one's and zero's, are the heart of numerous cryptographic systems and their ability to provide secure communication. Boolean functions have application in a variety of such systems, including block ciphers, stream ciphers and hash functions. The continued study of Boolean functions for cryptography is therefore fundamental to the provision of secure communication in the future. This thesis presents an investigation into the analysis of Boolean functions and in particular, analysis of affine transformations with respect to both the design and application of Boolean functions for cryptography. Past research has often been limited by the difficulties arising from the magnitude of the search space. The research presented in this thesis will be shown to provide an important step towards overcoming such restrictions and hence forms the basis for a new analysis methodology. The new perspective allows a reduced view of the Boolean space in which all Boolean functions are grouped into connected equivalence classes so that only one function from each class need be established. This approach is a significant development in Boolean function research with many applications, including class distinguishing, class structures, self mapping analysis and finite field based s-box analysis. The thesis will begin with a brief overview of Boolean function theory; including an introduction to the main theme of the research, namely the affine transformation. This will be followed by the presentation of a fundamental new theorem describing the connectivity that exists between equivalence classes. The theorem of connectivity will form the foundation for the remainder of the research presented in this thesis. A discussion of efficient algorithms for the manipulation of Boolean functions will then be presented. The ability of Boolean function research to achieve new levels of analysis and understanding is centered on the availability of computer based programs that can perform various manipulations. The development and optimisation of efficient algorithms specifically for execution on a computer will be shown to have a considerable advantage compared to those constructed using a more traditional approach to algorithm optimisation. The theorem of connectivety will be shown to be fundamental in the provision many avenues of new analysis and application. These applications include the first non-exhaustive test for determining equivalent Boolean functions, a visual representation of the connected equivalence class structure to aid in the understanding of the Boolean space and a self mapping constant that enables enumeration of the functions in each equivalence class. A detailed survey of the classes with six inputs is also presented, providing valuable insight into their range and structure. This theme is then continued in the application Boolean function construction. Two important new methodologies are presented; the first to yield bent functions and the second to yield the best currently known balanced functions of eight inputs with respect to nonlinearity. The implementation of these constructions is extremely efficient. The first construction yields bent functions of a variety of algebraic order and inputs sizes. The second construction provides better results than previously proposed heuristic techniques. Each construction is then analysed with respect to its ability to produce functions from a variety of equivalence classes. Finally, in a further application of affine equivalence analysis, the impact to both s-box design and construction will be considered. The effect of linear redundancy in finite field based s-boxes will be examined and in particular it will be shown that the AES s-box possesses complete linear redundancy. The effect of such analysis will be discussed and an alternative construction to s-box design that ensures removal of all linear redundancy will be presented in addition to the best known example of such an s-box.

Fundamental to the electronic security of information and communication systems, is the correct use and application of appropriate ciphers. The strength of these ciphers, particularly in their ability to resist cryptanalytic attacks, directly in uences the overall strength of the entire system. The strength of the underlying cipher is reliant upon a robust structure and the carefully designed interaction between components in its architecture. Most importantly, however, cipher strength is critically dependent on the strength of the individual components of which it is comprised. Boolean functions and substitution boxes (s-boxes) are among the most common and essential components of ciphers. This is because they are able to provide a cipher with strengthening properties to resist known and potential cryptanalytic attacks. Thus, it is not surprising that significant research effort has been made in trying to develop ways of obtaining boolean functions and substitution boxes with optimal achievable measures of desirable cryptographic properties. Three of the main cryptographic properties required by strong boolean functions and s-boxes are nonlinearity, correlation immunity and propagation criteria, with different cryptographic applications requiring different acceptable measures of these and other properties. As combinations of cryptographic properties exhibited by functions can be conicting, finding cryptographically strong functions often means that a trade-off needs to be made when optimizing property values. Throughout this thesis, the term "optimization" specifically refers to seeking to obtain the best achievable combination of target property values which may be exhibited by boolean functions and s-boxes, regardless of whether the relevant properties are conflicting or complementary. This thesis focusses on a particular class of techniques for obtaining strong functions for cryptographic applications, referred to as heuristic methods or, simply, heuristics. Three new heuristic methods, each aimed at generating boolean functions optimizing one or more of the main cryptographic properties mentioned above, in addition to other desirable properties, are presented. The first of the new heuristic methods developed for this thesis focusses on generating boolean functions which are balanced and exhibit very high nonlinearities. Highly nonlinear balanced functions are critical to many cryptographic applications, as they provide good resistance to linear cryptanalytic attacks. This first method is based on the recursive modification of a starting bent function and is shown to be highly successful and efficient at generating numerous such functions, which also exhibit low autocorrelation values, in a very short computational time. The generation of balanced, correlation immune boolean functions that also exhibit the confl icting property of high nonlinearity is the focus of the second new heuristic method developed for this thesis. By concatenating selected pairs of lower-dimensional boolean functions together in the Walsh Hadamard transform domain, direct optimization for both resilience and nonlinearity was able to take place at each level towards and for the final function. This second method was able to generate examples of boolean functions with almost all of the best known optimal combinations of target property values. Experiments have shown the success of this method in consistently generating highly nonlinear resilient boolean functions, for a range of orders of resilience, with such functions possessing optimal algebraic degree. A third new heuristic method, which searches for balanced boolean functions which satisfy a non-zero degree of propagation criteria and exhibit high nonlinearity, is presented. Intelligent bit manipulations in the truth table of starting functions, based on fundamental relationships between boolean function transforms and measures, provide the design rationale for this method. Two new function generation schemes have been proposed for this method, to efficiently satisfy the requirements placed on the starting functions utilized in the computational process. An optional process attempts to increase the algebraic degree of the resulting functions, without sacrificing the optimalities that are achievable. The validity of this method is demonstrated through the success of various experimental trials. Switching the focus from single output boolean functions to multiple output boolean functions (s-boxes), the effectiveness of existing heuristic techniques (namely Genetic Algorithm, Hill Climbing Method and combined Genetic Algorithm/Hill Climbing) in primarily being applied to improve the nonlinearity of s-boxes of various dimensions, is investigated. The prior success of these heuristic techniques for improving the nonlinearity of boolean functions has been previously demonstrated, as has the success of hill climbing in isolation when applied to bijective s-boxes. An extension to the bijective s-box optimization work is presented in this thesis. In this new research, a Genetic Algorithm, Hill Climbing Method and the two in combination are applied to the nonlinearity and autocorrelation optimization of regular NxM s-boxes (N > M) to investigate the effectiveness and efficiency of each of these heuristics. A new breeding scheme, utilized in the Genetic Algorithm and combined Genetic Algorithm/Hill Climbing trials, is also presented. The success of experimental results compared to random regular s-box generation is demonstrated. New research in applying the Hill Climbing Method to construct NxM sboxes (N > M) required to meet specific property criteria is presented. The consideration of the characteristics desired by the constructed s-boxes largely dictated the generation process. A discussion on the generation process of the component functions is included. Part of the results produced by experimental trials were incorporated into a commonly used family of stream ciphers, thus further supporting the use of heuristic techniques as a useful means of obtaining strong functions suitable for incorporation into practical ciphers. An analysis of the cryptographic properties of the s-box used in the MARS block cipher, the method of generation and the computational time taken to obtain this s-box, led to the new research reported in this thesis on the generation of MARS-like s-boxes. It is shown that the application of the Hill Climbing Method, with suitable requirements placed on the component boolean functions, was able to generate multiple MARS-like s-boxes which satisfied the MARS sbox requirements and provided additional properties. This new work represented an alternative approach to the generation of s-boxes satisfying the MARS sbox property requirements but which are cryptographically superior and can be obtained in a fraction of the time than that which was taken to produce the MARS s-box. An example MARS-like s-box is presented in this thesis. The overall value of heuristic methods in generating strong boolean functions and substitution boxes is clearly demonstrated in this thesis. This thesis has made several significant contributions to the field, both in the development of new, specialized heuristic methods capable of generating strong boolean functions, and in the analysis and optimization of substitution boxes, the latter achieved through applying existing heuristic techniques.

Le sepsis est un syndrome observé chez des patients associant une infection documentée (microbiologiquement ou cliniquement suspectée) à une réponse inflammatoire systémique (Systemic Inflammatory Response Syndrome : SIRS).Celui-ci peut évoluer vers un sepsis sévère s'il est associé à la défaillance d'un ou de plusieurs organes. Le choc septique est l'association d'un état septique grave et d'une défaillance hémodynamique caractérisée par une chute aiguë de la pression artérielle ne pouvant pas être corrigée par une procédure standard de remplissage vasculaire. Les syndromes septiques sont aujourd'hui la première cause de mortalité en unités de réanimation. Cette mortalité élevée, en particulier pour les cas les plus graves tels que les chocs septiques, témoigne d'une absence de traitements curatifs pour cette pathologie. Partant de l'hypothèse que les syndromes septiques graves sont la conséquence d'une perte de contrôle précoce de la régulation de la réponse inflammatoire, nous avons étudié, via deux démarches complémentaires, l'initiation de la voie de signalisation TLR4 et les mécanismes intracellulaires contribuant à sa régulation. Dans un premier temps, l'utilisation d'approches transcriptomiques nous a permis d'identifier la voie de signalisation mTOR comme discriminante entre des patients sains (SIRS induit par l'injection d'endotoxines) et des patients souffrant de syndromes septiques graves. Nous avons ensuite développé et utilisé des techniques de modélisation logique pour simuler in silico le rôle joué par la voie mTOR dans la résolution d'une réponse inflammatoire. Ces résultats encourageants ouvrent des perspectives pour de nouvelles applications thérapeutiques dans le domaine du sepsis / Sepsis is a syndrome observed in patients combining a documented infection (microbiologically or clinically suspected) with a systemic inflammatory response (Systemic Inflammatory Response Syndrome : SIRS). It may progress to severe sepsis if it is associated with failures of one or more organs. Septic shock is the combination of a severe sepsis and a hemodynamic dysfunction characterized by an acute fall in blood pressure that cannot be corrected by a procedure of vascular filling.Sepsis syndromes represent today the first cause of mortality in intensive care units around the world. This poor survival rate, in particular for the most severe cases, such as septic shock, testifies a real curative therapeutic demand.Based on the assumption that severe sepsis syndromes are the consequence of a loss of control in early mechanisms of inflammatory response regulation, we studied via two complementary approaches the initiation of TLR4 signaling pathway and the intracellular mechanisms contributing to its regulation.First, the use of transcriptomic approaches allowed us to identify the mTOR signaling pathway as discriminating between healthy patients (SIRS induced by the infusion of endotoxins) and patients with severe septic syndromes. We then developed and used logic modeling approaches to in silico simulate the role played by the mTOR signaling pathway in the resolution of an inflammatory response. These encouraging results open perspectives for new therapeutic applications in the field of sepsis

Algoritmos exatos para fatoração estão limitados a funções Booleanas read-once, onde cada variável aparece uma vez na equação final. No entanto, estes algoritmos apresentam duas restrições principais: (1) eles não consideram funções Booleanas incompletamente especificadas, e (2) eles não são adequados para as funções binate. Para superar o primeiro inconveniente, é proposto um algoritmo que encontra equações read-once para funções Booleanas incompletamente especificadas, sempre que possível, é proposto. Com respeito à segunda limitação, é apresentada uma transformação de domínio que divide variáveis binate existentes em duas variáveis unate independentes. Tal transformação de domínio conduz a funções Booleanas incompletamente especificadas, que podem ser eficientemente fatoradas mediante a aplicação do algoritmo proposto. A combinação das duas contribuições dá resultados ótimos para uma nova classe de funções Booleanas chamada read-polarity-once, onde cada polaridade (positiva ou negativa) de uma variável aparece no máximo uma vez na forma fatorada da expressão Booleana. Resultados experimentais sobre circuitos ISCAS'85 mostrou que funções read-polarity-once são significativamente mais frequentes em circuitos reais quando comparado com a classe de funções read-once, a qual muitos trabalhos já foram dedicados na literatura. / Efficient exact factoring algorithms are limited to read-once functions, in which each variable appears once in the final Boolean equation. However, those algorithms present two main constraints: (1) they do not consider incompletely specified Boolean functions; and (2) they are not suitable for binate functions. To overcome the first drawback, it is proposed an algorithm that finds read-once formulas for incompletely specified Boolean functions, whenever possible. With respect to the second limitation, a domain transformation that splits existing binate variables into two independent unate variables is presented. Such domain transformation leads to incompletely specified Boolean functions, which can be efficiently factored by applying the proposed algorithm. The combination of both contributions gives optimal results for a novel broader class of Boolean functions named as read-polarity-once functions, where each polarity (positive or negative) of a variable appears at most once in the factored form. Experimental results over ISCAS'85 benchmark circuits have shown that read-polarityonce functions are significantly more frequent than read-once functions, for which many works have already been devoted in the literature.

Submitted by JULIANA DE FÁTIMA FRANCISCANI null (ifsp.juliana@gmail.com) on 2016-10-28T18:39:46Z No. of bitstreams: 1 Juliana de Fátima Franciscani.pdf: 3657600 bytes, checksum: dfdbe82d43ba74271de101385cdbbf6e (MD5) / Approved for entry into archive by Felipe Augusto Arakaki (arakaki@reitoria.unesp.br) on 2016-11-03T19:07:45Z (GMT) No. of bitstreams: 1 franciscani_jf_me_ilha.pdf: 3638504 bytes, checksum: 6ca7f15a8be8ef019afd3f8e0ecc1e52 (MD5) / Made available in DSpace on 2016-11-03T19:07:45Z (GMT). No. of bitstreams: 1 franciscani_jf_me_ilha.pdf: 3638504 bytes, checksum: 6ca7f15a8be8ef019afd3f8e0ecc1e52 (MD5) Previous issue date: 2016-08-31 / Com a evolução e difusão do desenvolvimento de equipamentos utilizando microtecnologia e nanotecnologia, circuitos cada vez menores, mais eficientes e que consomem menos energia, são necessários. Os métodos de minimização de funções booleanas tornam-se relevantes por possibilitarem a otimização de circuitos lógicos, através da geração de circuitos que possuam a mesma funcionalidade, porém, minimizados. Estudos na área de minimização de funções booleanas são realizados há muito tempo, e estão sendo adaptados às novas tecnologias. A geração de implicantes primos de uma função booleana é um dos passos para a cobertura dos mintermos da função e, consequentemente, para a obtenção da função de custo mínimo. Neste trabalho, a Primeira Fase do Método de Quine-McCluskey para Funções Booleanas com Múltiplas Saídas (QMM) foi implementada para posterior comparação com os Métodos Propostos GPMultiplo e MultiGeraPlex (baseados na filosofia do algoritmo GeraPlex). Os métodos propostos geram os implicantes primos de uma função booleana com múltiplas saídas e utilizam a operação de consenso iterativo para comparar dois termos. Os resultados obtidos, através da comparação do GPMultiplo, MultiGeraPlex e da Primeira Fase do Método de QMM, puderam comprovar que a aplicação dos métodos propostos torna-se mais viável e vantajosa por permitir menor tempo de execução e uso de memória, menor quantidade de implicantes gerados e de comparações entre os termos. / With the evolution and spread of the development of equipment using microtechnology and nanotechnology, circuits in need are smaller, more efficient and consume less power. Methods of Minimizing Boolean Functions become important as they allow optimization of logic circuits by generating circuits having the same functionality, but minimized. Studies in Minimizing Boolean Functions area are carried out long ago, and are being adapted to new technologies. The generation of prime implicants of a Boolean function is one of the steps for covering the function of the minterms, and consequently to obtain the minimum cost function. In this work, the first phase of the Quine-McCluskey Method for Booleans Functions with Multiple Output (QMM) was implemented for comparison with Proposed Methods GPMultiplo and MultiGeraPlex (based on the philosophy of GeraPlex algorithm). The proposed methods generates the prime implicants of a Boolean Function with Multiple Output and using the iterative consensus operation to compare two terms. The results obtained by comparing the GPMultiplo, MultiGeraPlex and the first phase of the QMM Method, were able to prove that the application of the proposed methods becomes more feasible and advantageous, by allowing smaller execution time, number of implicants and number of comparisons.

Made available in DSpace on 2015-05-14T12:11:59Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 891353 bytes, checksum: 2d056c7f53fdfb7c20586b64874e848d (MD5) Previous issue date: 2010-12-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The basic concepts of what later became called category theory were introduced in 1945 by Samuel Eilenberg and Saunders Mac Lane. In 1940s, the main applications were originally in the fields of algebraic topology and algebraic abstract. During the 1950s and 1960s, this theory became an important conceptual framework in other many areas of mathematical research, especially in algrebraic homology and algebraic geometry, as shows the works of Daniel M. Kan (1958) and Alexander Grothendieck (1957). Late, questions mathematiclogics about the category theory appears, in particularly, with the publication of the Functorial Semantics of Algebraic Theories (1963) of Francis Willian Lawvere. After, other works are done in the category logic, such as the the current Makkai (1977), Borceux (1994), Goldblatt (2006), and others. As introduction of application of the category theory in logic, this work presents a study on the logic category propositional. The first section of this work, shows to the reader the important concepts to a better understanding of subject: (a) basic components of category theory: categorical constructions, definitions, axiomatic, applications, authors, etc.; (b) certain structures of abstract algebra: monoids, groups, Boolean algebras, etc.; (c) some concepts of mathematical logic: pre-order, partial orderind, equivalence relation, Lindenbaum algebra, etc. The second section, it talk about the properties, structures and relations of category propositional logic. In that section, we interpret the logical connectives of the negation, conjunction, disjunction and implication, as well the Boolean connectives of complement, intersection and union, in the categorical language. Finally, we define a categorical boolean propositional semantics through a Boolean category algebra. / Os conceitos básicos do que mais tarde seria chamado de teoria das categorias são introduzidos no artigo General Theory of Natural Equivalences (1945) de Samuel Eilenberg e Saunders Mac Lane. Já em meados da década de 1940, esta teoria é aplicada com sucesso ao campo da topologia. Ao longo das décadas de 1950 e 1960, a teoria das categorias ostenta importantes mudanças ao enfoque tradicional de diversas áreas da matemática, entre as quais, em especial, a álgebra geométrica e a álgebra homológica, como atestam os pioneiros trabalhos de Daniel M. Kan (1958) e Alexander Grothendieck (1957). Mais tarde, questões lógico-matemáticas emergem em meio a essa teoria, em particular, com a publica ção da Functorial Semantics of Algebraic Theories (1963) de Francis Willian Lawvere. Desde então, diversos outros trabalhos vêm sendo realizados em lógica categórica, como os mais recentes Makkai (1977), Borceux (1994), Goldblatt (2006), entre outros. Como inicialização à aplicação da teoria das categorias à lógica, a presente dissertação aduz um estudo introdutório à lógica proposicional categórica. Em linhas gerais, a primeira parte deste trabalho procura familiarizar o leitor com os conceitos básicos à pesquisa do tema: (a) elementos constitutivos da teoria das categorias : axiomática, construções, aplicações, autores, etc.; (b) algumas estruturas da álgebra abstrata: monóides, grupos, álgebra de Boole, etc.; (c) determinados conceitos da lógica matemática: pré-ordem; ordem parcial; equivalência, álgebra de Lindenbaum, etc. A segunda parte, trata da aproximação da teoria das categorias à lógica proposicional, isto é, investiga as propriedades, estruturas e relações próprias à lógica proposicional categórica. Nesta passagem, há uma reinterpreta ção dos conectivos lógicos da negação, conjunção, disjunção e implicação, bem como dos conectivos booleanos de complemento, interseção e união, em termos categóricos. Na seqüência, estas novas concepções permitem enunciar uma álgebra booleana categórica, por meio da qual, ao final, é construída uma semântica proposicional booleana categórica.