Algebraic graph theoretic applications to cryptography.

Master of Science in Mathematics. University of KwaZulu-Natal, Durban, 2015.Abstract available in PDF file

The Potts model and the independence polynomial:Uniqueness of the Gibbs measure and distributions of complex zeros

Part 1 of this dissertation studies the antiferromagnetic Potts model, which originates in statistical physics. In particular the transition from multiple Gibbs measures to a unique Gibbs measure for the antiferromagnetic Potts model on the infinite regular tree is studied. This is called a uniqueness phase transition. A folklore conjecture about the parameter at which the uniqueness phase transition occurs is partly confirmed. The proof uses a geometric condition, which comes from analysing an associated dynamical system.Part 2 of this dissertation concerns zeros of the independence polynomial. The independence polynomial originates in statistical physics as the partition function of the hard-core model. The location of the complex zeros of the independence polynomial is related to phase transitions in terms of the analycity of the free energy and plays an important role in the design of efficient algorithms to approximately compute evaluations of the independence polynomial. Chapter 5 directly relates the location of the complex zeros of the independence polynomial to computational hardness of approximating evaluations of the independence polynomial. This is done by moreover relating the set of zeros of the independence polynomial to chaotic behaviour of a naturally associated family of rational functions; the occupation ratios. Chapter 6 studies boundedness of zeros of the independence polynomial of tori for sequences of tori converging to the integer lattice. It is shown that zeros are bounded for sequences of balanced tori, but unbounded for sequences of highly unbalanced tori

Halfway to Halfspace Testing

In this thesis I study the problem of testing halfspaces under arbitrary probability distributions, using only random samples. A halfspace, or linear threshold function, is a boolean function f : Rⁿ → {±1} defined as the sign of a linear function; that is, f(x) = sign(Σᵢ wᵢxᵢ - θ) where we refer to w ∈ Rⁿ as a weight vector and θ ∈ R as a threshold. These functions have been studied intensively since the middle of the 20th century; they appear in many places, including social choice theory (the theory of voting rules), circuit complexity theory, machine learning theory, hardness of approximation, and the analysis of boolean functions. The problem of testing halfspaces, in the sense of property testing, is to design an algorithm that, with high probability, decides whether an unknown function f is a halfspace function or far from a halfspace, using as few examples of labelled points (x, f (x)) as possible. In this work I focus on the problem of testing halfspaces using only random examples drawn from an arbitrary distribution, and the algorithm cannot choose the points it receives. This is in contrast with previous work on the problem, where the algorithm can query points of its choice, and the distribution was assumed to be uniform over the boolean hypercube. Towards a solution to this problem I present an algorithm that works for rotationally invariant probability distributions (under reasonable conditions), using roughly O(√n) random examples, which is close to the known lower bound of Ω(√n/ √log n) . I further develop the algorithm to work for mixtures of two such rotationally invariant distributions and provide a partial analysis. I also survey related machine learning results, and conclude with a survey of the theory of halfspaces over the boolean hypercube, which has recently received much attention

On applications of simulated annealing to cryptology

Boolean functions are critical building blocks of symmetric-key ciphers. In most cases, the security of a cipher against a particular kind of attacks can be explained by the existence of certain properties of its underpinning Boolean functions. Therefore, the design of appropriate functions has received significant attention from researchers for several decades. Heuristic methods have become very powerful tools for designing such functions. In this thesis, we apply simulated annealing methods to construct Boolean functions with particular properties. Our results meet or exceed the best results of available theoretical constructions and/or heuristic searches in the literature, including a 10-variable balanced Boolean function with resiliency degree 2, algebraic degree 7, and nonlinearity 488 for the first time. This construction affirmatively answers the open problem about the existence of such functions. This thesis also includes results of cryptanalysis for symmetric ciphers, such as Geffe cipher and TREYFER cipher

Structural Nonlinear Invariant Attacks on T-310: Attacking Arbitrary Boolean Functions

Recent papers show how to construct polynomial invariant attacks for block ciphers, however almost all such results are somewhat weak: invariants are simple and low degree and the Boolean functions tend by very simple if not degenerate. Is there a better more realistic attack, with invariants of higher degree and which is likely to work with stronger Boolean functions? In this paper we show that such attacks exist and can be constructed explicitly through on the one side, the study of Fundamental Equation of eprint/2018/807, and on the other side, a study of the space of Annihilators of any given Boolean function. The main contribution of this paper is that to show that the ``product attack\u27\u27 where the invariant polynomial is a product of simpler polynomials is interesting and quite powerful. Our approach is suitable for backdooring a block cipher in presence of an arbitrarily strong Boolean function not chosen by the attacker. The attack is constructed using excessively simple paper and pencil maths. We also outline a potential application to Data Encryption Standard (DES)

A POWER INDEX BASED FRAMEWORKFOR FEATURE SELECTION PROBLEMS

One of the most challenging tasks in the Machine Learning context is the feature selection. It consists in selecting the best set of features to use in the training and prediction processes. There are several benefits from pruning the set of actually operational features: the consequent reduction of the computation time, often a better quality of the prediction, the possibility to use less data to create a good predictor. In its most common form, the problem is called single-view feature selection problem, to distinguish it from the feature selection task in Multi-view learning. In the latter, each view corresponds to a set of features and one would like to enact feature selection on each view, subject to some global constraints. A related problem in the context of Multi-View Learning, is Feature Partitioning: it consists in splitting the set of features of a single large view into two or more views so that it becomes possible to create a good predictor based on each view. In this case, the best features must be distributed between the views, each view should contain synergistic features, while features that interfere disruptively must be placed in different views. In the semi-supervised multi-view task known as Co-training, one requires also that each predictor trained on an individual view is able to teach something to the other views: in classification tasks for instance, one view should learn to classify unlabelled examples based on the guess provided by the other views. There are several ways to address these problems. A set of techniques is inspired by Coalitional Game Theory. Such theory defines several useful concepts, among which two are of high practical importance: the concept of power index and the concept of interaction index. When used in the context of feature selection, they take the following meaning: the power index is a (context-dependent) synthesis measure of the prediction\u2019s capability of a feature, the interaction index is a (context-dependent) synthesis measure of the interaction (constructive/disruptive interference) between two features: it can be used to quantify how the collaboration between two features enhances their prediction capabilities. An important point is that the powerindex of a feature is different from the predicting power of the feature in isolation: it takes into account, by a suitable averaging, the context, i.e. the fact that the feature is acting, together with other features, to train a model. Similarly, the interaction index between two features takes into account the context, by suitably averaging the interaction with all the other features. In this work we address both the single-view and the multi-view problems as follows. The single-view feature selection problem, is formalized as the problem of maximization of a pseudo-boolean function, i.e. a real valued set function (that maps sets of features into a performance metric). Since one has to enact a search over (a considerable portion of) the Boolean lattice (without any special guarantees, except, perhaps, positivity) the problem is in general NP-hard. We address the problem producing candidate maximum coalitions through the selection of the subset of features characterized by the highest power indices and using the coalition to approximate the actual maximum. Although the exact computation of the power indices is an exponential task, the estimates of the power indices for the purposes of the present problem can be achieved in polynomial time. The multi-view feature selection problem is formalized as the generalization of the above set-up to the case of multi-variable pseudo-boolean functions. The multi-view splitting problem is formalized instead as the problem of maximization of a real function defined over the partition lattice. Also this problem is typically NP-hard. However, candidate solutions can be found by suitably partitioning the top power-index features and keeping in different views the pairs of features that are less interactive or negatively interactive. The sum of the power indices of the participating features can be used to approximate the prediction capability of the view (i.e. they can be used as a proxy for the predicting power). The sum of the feature pair interactivity across views can be used as proxy for the orthogonality of the views. Also the capability of a view to pass information (to teach) to other views, within a co-training procedure can benefit from the use of power indices based on a suitable definition of information transfer (a set of features { a coalition { classifies examples that are subsequently used in the training of a second set of features). As to the feature selection task, not only we demonstrate the use of state of the art power index concepts (e.g. Shapley Value and Banzhaf along the 2lines described above Value), but we define new power indices, within the more general class of probabilistic power indices, that contains the Shapley and the Banzhaf Values as special cases. Since the number of features to select is often a predefined parameter of the problem, we also introduce some novel power indices, namely k-Power Index (and its specializations k-Shapley Value, k-Banzhaf Value): they help selecting the features in a more efficient way. For the feature partitioning, we use the more general class of probabilistic interaction indices that contains the Shapley and Banzhaf Interaction Indices as members. We also address the problem of evaluating the teaching ability of a view, introducing a suitable teaching capability index. The last contribution of the present work consists in comparing the Game Theory approach to the classical Greedy Forward Selection approach for feature selection. In the latter the candidate is obtained by aggregating one feature at time to the current maximal coalition, by choosing always the feature with the maximal marginal contribution. In this case we show that in typical cases the two methods are complementary, and that when used in conjunction they reduce one another error in the estimate of the maximum value. Moreover, the approach based on game theory has two advantages: it samples the space of all possible features\u2019 subsets, while the greedy algorithm scans a selected subspace excluding totally the rest of it, and it is able, for each feature, to assign a score that describes a context-aware measure of importance in the prediction process

Statistical analysis of grouped text documents

L'argomento di questa tesi sono i modelli statistici per l'analisi dei dati testuali, con particolare attenzione ai contesti in cui i campioni di testo sono raggruppati. Quando si ha a che fare con dati testuali, il primo problema è quello di elaborarli, per renderli compatibili dal punto di vista computazionale e metodologico con i metodi matematici e statistici prodotti e continuamente sviluppati dalla comunità scientifica. Per questo motivo, la tesi passa in rassegna i metodi esistenti per la rappresentazione analitica e l'elaborazione di campioni di dati testuali, compresi i "Vector Space Models", le "rappresentazioni distribuite" di parole e documenti e i "contextualized embeddings". Questa rassegna comporta la standardizzazione di una notazione che, anche all'interno dello stesso approccio di rappresentazione, appare molto eterogenea in letteratura. Vengono poi esplorati due domini di applicazione: i social media e il turismo culturale. Per quanto riguarda il primo, viene proposto uno studio sull'autodescrizione di gruppi diversi di individui sulla piattaforma StockTwits, dove i mercati finanziari sono gli argomenti dominanti. La metodologia proposta ha integrato diversi tipi di dati, sia testuali che variabili categoriche. Questo studio ha agevolato la comprensione sul modo in cui le persone si presentano online e ha trovato stutture di comportamento ricorrenti all'interno di gruppi di utenti. Per quanto riguarda il turismo culturale, la tesi approfondisce uno studio condotto nell'ambito del progetto "Data Science for Brescia - Arts and Cultural Places", in cui è stato addestrato un modello linguistico per classificare le recensioni online scritte in italiano in quattro aree semantiche distinte relative alle attrazioni culturali della città di Brescia. Il modello proposto permette di identificare le attrazioni nei documenti di testo, anche quando non sono esplicitamente menzionate nei metadati del documento, aprendo così la possibilità di espandere il database relativo a queste attrazioni culturali con nuove fonti, come piattaforme di social media, forum e altri spazi online. Infine, la tesi presenta uno studio metodologico che esamina la specificità di gruppo delle parole, analizzando diversi stimatori di specificità di gruppo proposti in letteratura. Lo studio ha preso in considerazione documenti testuali raggruppati con variabile di "outcome" e variabile di gruppo. Il suo contributo consiste nella proposta di modellare il corpus di documenti come una distribuzione multivariata, consentendo la simulazione di corpora di documenti di testo con caratteristiche predefinite. La simulazione ha fornito preziose indicazioni sulla relazione tra gruppi di documenti e parole. Inoltre, tutti i risultati possono essere liberamente esplorati attraverso un'applicazione web, i cui componenti sono altresì descritti in questo manoscritto. In conclusione, questa tesi è stata concepita come una raccolta di studi, ognuno dei quali suggerisce percorsi di ricerca futuri per affrontare le sfide dell'analisi dei dati testuali raggruppati.The topic of this thesis is statistical models for the analysis of textual data, emphasizing contexts in which text samples are grouped. When dealing with text data, the first issue is to process it, making it computationally and methodologically compatible with the existing mathematical and statistical methods produced and continually developed by the scientific community. Therefore, the thesis firstly reviews existing methods for analytically representing and processing textual datasets, including Vector Space Models, distributed representations of words and documents, and contextualized embeddings. It realizes this review by standardizing a notation that, even within the same representation approach, appears highly heterogeneous in the literature. Then, two domains of application are explored: social media and cultural tourism. About the former, a study is proposed about self-presentation among diverse groups of individuals on the StockTwits platform, where finance and stock markets are the dominant topics. The methodology proposed integrated various types of data, including textual and categorical data. This study revealed insights into how people present themselves online and found recurring patterns within groups of users. About the latter, the thesis delves into a study conducted as part of the "Data Science for Brescia - Arts and Cultural Places" Project, where a language model was trained to classify Italian-written online reviews into four distinct semantic areas related to cultural attractions in the Italian city of Brescia. The model proposed allows for the identification of attractions in text documents, even when not explicitly mentioned in document metadata, thus opening possibilities for expanding the database related to these cultural attractions with new sources, such as social media platforms, forums, and other online spaces. Lastly, the thesis presents a methodological study examining the group-specificity of words, analyzing various group-specificity estimators proposed in the literature. The study considered grouped text documents with both outcome and group variables. Its contribution consists of the proposal of modeling the corpus of documents as a multivariate distribution, enabling the simulation of corpora of text documents with predefined characteristics. The simulation provided valuable insights into the relationship between groups of documents and words. Furthermore, all its results can be freely explored through a web application, whose components are also described in this manuscript. In conclusion, this thesis has been conceived as a collection of papers. It aimed to contribute to the field with both applications and methodological proposals, and each study presented here suggests paths for future research to address the challenges in the analysis of grouped textual data