Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
33,249
result(s) for
"Discrete mathematics"
Sort by:
Mathematics of Public Key Cryptography
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.
eBook
Frameworks, tensegrities, and symmetry
\"This introduction to the theory of rigid structures explains how to analyze the performance of built and natural structures under loads, paying special attention to the role of geometry. The book unifies the engineering and mathematical literatures by exploring different notions of rigidity -- local, global, and universal -- and how they are interrelated. Important results are stated formally, but also clarified with a wide range of revealing examples. An important generalization is to tensegrities, where fixed distances are replaced with \"cables\" not allowed to increase in length and \"struts\" not allowed to decrease in length. A special feature is the analysis of symmetric tensegrities, where the symmetry of the structure is used to simplify matters and allows the theory of group representations to be applied. Written for researchers and graduate students in structural engineering and mathematics, this work is also of interest to computer scientists and physicists\"-- Provided by publisher.
Book
Exact values for three domination-like problems in circular and infinite grid graphs of small height
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.
Journal Article
Contact graphs of boxes with unidirectional contacts
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.
Journal Article
Hypergraph Theory
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.
eBook
On Monotonicity Testing and the 2-to-2 Games Conjecture
This book discusses two questions in Complexity Theory: the Monotonicity Testing problem and the 2-to-2 Games Conjecture. Monotonicity testing is a problem from the field of property testing, first considered by Goldreich et al. in 2000. The input of the algorithm is a function, and the goal is to design a tester that makes as few queries to the function as possible, accepts monotone functions and rejects far-from monotone functions with a probability close to 1. The first result of this book is an essentially optimal algorithm for this problem. The analysis of the algorithm heavily relies on a novel, directed, and robust analogue of a Boolean isoperimetric inequality of Talagrand from 1993. The probabilistically checkable proofs (PCP) theorem is one of the cornerstones of modern theoretical computer science. One area in which PCPs are essential is the area of hardness of approximation. Therein, the goal is to prove that some optimization problems are hard to solve, even approximately. Many hardness of approximation results were proved using the PCP theorem; however, for some problems optimal results were not obtained. This book touches on some of these problems, and in particular the 2-to-2 games problem and the vertex cover problem. The second result of this book is a proof of the 2-to-2 games conjecture (with imperfect completeness), which implies new hardness of approximation results for problems such as vertex cover and independent set. It also serves as strong evidence towards the unique games conjecture, a notorious related open problem in theoretical computer science. At the core of the proof is a characterization of small sets of vertices in Grassmann graphs whose edge expansion is bounded away from 1.
eBook
Mutually orthogonal latin squares based on cellular automata
We investigate sets of mutually orthogonal latin squares (MOLS) generated by cellular automata (CA) over finite fields. After introducing how a CA defined by a bipermutive local rule of diameter d over an alphabet of q elements generates a Latin square of order qd-1 , we study the conditions under which two CA generate a pair of orthogonal Latin squares. In particular, we prove that the Latin squares induced by two Linear Bipermutive CA (LBCA) over the finite field Fq are orthogonal if and only if the polynomials associated to their local rules are relatively prime. Next, we enumerate all such pairs of orthogonal Latin squares by counting the pairs of coprime monic polynomials with nonzero constant term and degree n over Fq . Finally, we present a construction for families of MOLS based on LBCA, and prove that their cardinality corresponds to the maximum number of pairwise coprime polynomials with nonzero constant term. Although our construction does not yield all such families of MOLS, we show that the resulting lower bound is asymptotically close to their actual number.
Journal Article
On the algebraic degree of iterated power functions
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.
Journal Article