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Public key cryptography is a major interdisciplinary subject with many real-world applications, such as digital signatures. A strong background in the mathematics underlying public key cryptography is essential for a deep understanding of the subject, and this book provides exactly that for students and researchers in mathematics, computer science and electrical engineering. Carefully written to communicate the major ideas and techniques of public key cryptography to a wide readership, this text is enlivened throughout with historical remarks and insightful perspectives on the development of the subject. Numerous examples, proofs and exercises make it suitable as a textbook for an advanced course, as well as for self-study. For more experienced researchers it serves as a convenient reference for many important topics: the Pollard algorithms, Maurer reduction, isogenies, algebraic tori, hyperelliptic curves and many more.

In this paper we study three domination-like problems, namely identifying codes, locating-dominating codes, and locating-total-dominating codes. We are interested in finding the minimum cardinality of such codes in circular and infinite grid graphs of given height. We provide an alternate proof for already known results, as well as new results. These were obtained by a computer search based on a generic framework, that we developed earlier, for the search of a minimum labeling satisfying a pseudo-d-local property in rotagraphs.

This book presents hypergraph theory and covers traditional elements of the theory as well as original concepts such as entropy of hypergraph, similarities and kernels. It details applications in telecommunications and parallel data structure modeling.

This paper is devoted to the study of particular geometrically defined intersection classes of graphs. Those were previously studied by Magnant and Martin, who proved that these graphs have arbitrary large chromatic number, while being triangle-free. We give several structural properties of these graphs, and we raise several questions.

New symmetric primitives are being designed to address a novel set of design criteria. Instead of being executed on regular processors or smartcards, they are instead intended to be run in abstract settings such as multi-party computations or zero-knowledge proof systems. This implies in particular that these new primitives are described using operations over large finite fields. As the number of such primitives grows, it is important to better understand the properties of their underlying operations. In this paper, we investigate the algebraic degree of one of the first such block ciphers, namely MiMC. It is composed of many iterations of a simple round function, which consists of an addition and of a low-degree power permutation applied to the full state, usually x ↦ x 3 . We show in particular that, while the univariate degree increases predictably with the number of rounds, the algebraic degree (a.k.a multivariate degree) has a much more complex behaviour, and simply stays constant during some rounds. Such plateaus slightly slow down the growth of the algebraic degree. We present a full investigation of this behaviour. First, we prove some lower and upper bounds for the algebraic degree of an arbitrary number of iterations of MiMC and of its inverse. Then, we combine theoretical arguments with simulations to prove that the upper bound is tight for up to 16,265 rounds. Using these results, we slightly improve the higher-order differential attack presented at Asiacrypt 2020 to cover one or two more rounds. More importantly, our results provide some precise guarantees on the algebraic degree of this cipher, and then on the minimal complexity for a higher-order differential attack.

Professor Stephen A. Cook is a pioneer of the theory of computational complexity. His work on NP-completeness and the P vs. NP problem remains a central focus of this field. Cook won the 1982 Turing Award for his advancement of our understanding of the complexity of computation in a significant and profound way. This volume includes a selection of seminal papers embodying the work that led to this award, exemplifying Cook's synthesis of ideas and techniques from logic and the theory of computation including NP-completeness, proof complexity, bounded arithmetic, and parallel and space-bounded computation. These papers are accompanied by contributed articles by leading researchers in these areas, which convey to a general reader the importance of Cook's ideas and their enduring impact on the research community. The book also contains biographical material, Cook's Turing Award lecture, and an interview. Together these provide a portrait of Cook as a recognized leader and innovator in mathematics and computer science, as well as a gentle mentor and colleague.

This accessible book provides an introduction to the analysis and design of dynamic multiagent networks. Such networks are of great interest in a wide range of areas in science and engineering, including: mobile sensor networks, distributed robotics such as formation flying and swarming, quantum networks, networked economics, biological synchronization, and social networks. Focusing on graph theoretic methods for the analysis and synthesis of dynamic multiagent networks, the book presents a powerful new formalism and set of tools for networked systems. The book's three sections look at foundations, multiagent networks, and networks as systems. The authors give an overview of important ideas from graph theory, followed by a detailed account of the agreement protocol and its various extensions, including the behavior of the protocol over undirected, directed, switching, and random networks. They cover topics such as formation control, coverage, distributed estimation, social networks, and games over networks. And they explore intriguing aspects of viewing networks as systems, by making these networks amenable to control-theoretic analysis and automatic synthesis, by monitoring their dynamic evolution, and by examining higher-order interaction models in terms of simplicial complexes and their applications. The book will interest graduate students working in systems and control, as well as in computer science and robotics. It will be a standard reference for researchers seeking a self-contained account of system-theoretic aspects of multiagent networks and their wide-ranging applications. This book has been adopted as a textbook at the following universities: University of Stuttgart, GermanyRoyal Institute of Technology, SwedenJohannes Kepler University, AustriaGeorgia Tech, USAUniversity of Washington, USAOhio University, USA

The (bitwise) complement$\\overline{x}$of a binary word$x$is obtained by changing each$0$in$x$to$1$and vice versa. An$\\textit{antisquare}$is a nonempty word of the form$x\\, \\overline{x}$ . In this paper, we study infinite binary words that do not contain arbitrarily large antisquares. For example, we show that the repetition threshold for the language of infinite binary words containing exactly two distinct antisquares is$(5+\\sqrt{5})/2$ . We also study repetition thresholds for related classes, where \"two\" in the previous sentence is replaced by a larger number. We say a binary word is$\\textit{good}$if the only antisquares it contains are$01$and$10$ . We characterize the minimal antisquares, that is, those words that are antisquares but all proper factors are good. We determine the growth rate of the number of good words of length$n$and determine the repetition threshold between polynomial and exponential growth for the number of good words.

Introduction to Recognition and Deciphering of Patterns aims to get STEM and non-STEM students acquainted with different patterns, as well as where and when specific patterns arise. In addition, the book seeks to get students to learn how to recognize patterns and distinguish the similarities and differences between them. Patterns emerge on an everyday basis, such as weather patterns, traffic patterns, behavioral patterns, geometric patterns, linguistic patterns, structural patterns, digital patterns, etc. Recognizing patterns and studying their unique traits is essential for the development and enhancement of our intuitive skills and in strengthening our analytical skills. Mathematicians often apply patterns to get acquainted with new concepts, but this is a technique that can be applied across many disciplines. Throughout this book we will encounter assorted patterns that emerge from various geometrical configurations of squares, circles, right triangles and equilateral triangles that either repeat at the same scale or at different scales. The book will also focus on describing linear patterns, geometric patterns, alternating patterns, piecewise patterns, summation-type patterns and factorial-type patterns analytically. Deciphering the details of these distinct patterns will lead to the proof by induction method. Furthermore, the book will render properties of Pascal’s triangle and provide supplemental practice in deciphering specific patterns and verifying them. The book will adjourn with first-order recursive relations: describing sequences as recursive relations, obtaining the general solution by solving an initial value problem and determining the periodic traits.

Efficient treatment strategies for stress urinary incontinence (SUI) in elite female athletes (EFAs) are crucial for their timely return to sports. This study evaluates the effectiveness and potential drawbacks of non-ablative Er: YAG laser therapy combined with pelvic floor muscle training (PFMT) in treating SUI among EFAs. We employ a discrete mathematics analytical approach using network graphs to identify key factors influencing treatment outcomes and to address the challenges of small sample sizes and unknown variables in this population. Our results demonstrate significant improvements in urinary incontinence symptoms and increased return rates to elite sports activities in the laser treatment group compared to the PFMT-only group. The discrete mathematics approach effectively visualizes the complex relationships between variables and supports the development of personalized treatment plans. This study highlights the potential of laser therapy as an effective treatment option for SUI in EFAs while emphasizing the importance of tailored treatment strategies.