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Initially, the encryption of message was based on symmetric key cryptography where sender and receiver of message use the same key for encryption and decryption.But, to use the same key, sender and receiver must share the key in advance. And if their locations are diﬀerent than there is risk in transmission of the key. Later in 1976,a cryprosystem,which is known as Diﬃe-hellman key-exchange, was published by Whiteﬁeld Diﬃe and Martin Hellman and concept behind the cryptosystem is known as public key encryption. In public key cryptosystem, each one gets a pair of keys, public key and private key. The pubic key is freely available to everyone while the private key remains secret. The sender, who wants to send a message securely to someone, use public key of receiver to encrypt the message and receiver use his private key to decrypt the message.This system doesn’t require secure key transmission.So, it resolves the one of the problem faced by symmetric key cryptosystem. If someone is able to compute respective private key from a given public key, then this system is no more secure. So, Public key cryptosystem requires that calculation of respective private key is computationally impossible from given public key. In most of the Public key cryptosystem, private key is related to public key via Discrete Logarithm. Examples are Diﬃe-Hellman Key Exchange, Digital Signature Algorithm (DSA), Elgamal which are based on DLP in ﬁnite multiplicative group.

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2. Discrete logarithm problem

The Discrete Logarithm Problem (DLP)is the problem of ﬁnding an exponent x such that g x ≡ h (mod p) where, g is a primitive root for Fp and h is a non-zero element of Fp . Let, n be the order of g. Then solution x is unique up to multiples of n and x is called discrete logarithm of h to the base g (i.e.) x = logg h. In cryptosystem based on Discrete Logarithm , x is used…...

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...Course Design Guide MTH/221 Version 1 1 Course Design Guide College of Information Systems & Technology MTH/221 Version 1 Discrete Math for Information Technology Copyright © 2010 by University of Phoenix. All rights reserved. Course Description Discrete (as opposed to continuous) mathematics is of direct importance to the fields of Computer Science and Information Technology. This branch of mathematics includes studying areas such as set theory, logic, relations, graph theory, and analysis of algorithms. This course is intended to provide students with an understanding of these areas and their use in the field of Information Technology. Policies Faculty and students/learners will be held responsible for understanding and adhering to all policies contained within the following two documents: University policies: You must be logged into the student website to view this document. Instructor policies: This document is posted in the Course Materials forum. University policies are subject to change. Be sure to read the policies at the beginning of each class. Policies may be slightly different depending on the modality in which you attend class. If you have recently changed modalities, read the policies governing your current class modality. Course Materials Grimaldi, R. P. (2004). Discrete and combinatorial mathematics: An applied introduction. (5th ed.). Boston, MA: Pearson Addison Wesley. Article References Albert, I. Thakar, J., Li, S., Zhang, R., & Albert, R...

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...Log-Pearson Type III Distribution What is it? The Log-Pearson Type III distribution is a statistical technique for fitting frequency distribution data to predict the design flood for a river at some site. Once the statistical information is calculated for the river site, a frequency distribution can be constructed. The probabilities of floods of various sizes can be extracted from the curve. The advantage of this particular technique is that extrapolation can be made of the values for events with return periods well beyond the observed flood events. This technique is the standard technique used by Federal Agencies in the United States. How is it calculated? The Log-Pearson Type III distribution is calculated using the general equation: where x is the flood discharge value of some specified probability, is the average of the log x discharge values, K is a frequency factor, and is the standard deviation of the log x values. The frequency factor K is a function of the skewness coefficient and return period and can be found using the frequency factor table. The Log-Pearson Type III distribution tells you the likely values of discharges to expect in the river at various recurrence intervals based on the available historical record. This is helpful when designing structures in or near the river that may be affected by floods. It is also helpful when designing structures to protect against the largest expected event. For this reason, it is customary to perform the......

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...Introduction to Discrete Structures --- Whats and Whys What is Discrete Mathematics ? Discrete mathematics is mathematics that deals with discrete objects. Discrete objects are those which are separated from (not connected to/distinct from) each other. Integers (aka whole numbers), rational numbers (ones that can be expressed as the quotient of two integers), automobiles, houses, people etc. are all discrete objects. On the other hand real numbers which include irrational as well as rational numbers are not discrete. As you know between any two different real numbers there is another real number different from either of them. So they are packed without any gaps and can not be separated from their immediate neighbors. In that sense they are not discrete. In this course we will be concerned with objects such as integers, propositions, sets, relations and functions, which are all discrete. We are going to learn concepts associated with them, their properties, and relationships among them among others. Why Discrete Mathematics ? Let us first see why we want to be interested in the formal/theoretical approaches in computer science. Some of the major reasons that we adopt formal approaches are 1) we can handle infinity or large quantity and indefiniteness with them, and 2) results from formal approaches are reusable. As an example, let us consider a simple problem of investment. Suppose that we invest $1,000 every year with expected return of 10% a year. How much...

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...PART ONE Introduction to Discrete-Event System Simulation 1 Introduction to Simulation A simulation is the imitation of the operation of a real-world process or system over time. Whether done by hand or on a computer, simulation involves the generation of an artiﬁcial history of a system, and the observation of that artiﬁcial history to draw inferences concerning the operating characteristics of the real system. The behavior of a system as it evolves over time is studied by developing a simulation model. This model usually takes the form of a set of assumptions concerning the operation of the system. These assumptions are expressed in mathematical, logical, and symbolic relationships between the entities, or objects of interest, of the system. Once developed and validated, a model can be used to investigate a wide variety of “what-if” questions about the realworld system. Potential changes to the system can ﬁrst be simulated in order to predict their impact on system performance. Simulation can also be used to study systems in the design stage, before such systems are built. Thus, simulation modeling can be used both as an analysis tool for predicting the effect of changes to existing systems, and as a design tool to predict the performance of new systems under varying sets of circumstances. In some instances, a model can be developed which is simple enough to be “solved” by mathematical methods. Such solutions may be found by the use of differential calculus,......

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...Continued on ﬁrst page of back endpapers. Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. DISCRETE MATHEMATICS Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. DISCRETE MATHEMATICS WITH APPLICATIONS FOURTH EDITION SUSANNA S. EPP DePaul University Australia · Brazil · Japan · Korea · Mexico · Singapore · Spain · United Kingdom · United States Copyright 2010 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect......

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...Discrete Mathematics Lecture Notes, Yale University, Spring 1999 L. Lov´sz and K. Vesztergombi a Parts of these lecture notes are based on ´ ´ L. Lovasz – J. Pelikan – K. Vesztergombi: Kombinatorika (Tank¨nyvkiad´, Budapest, 1972); o o Chapter 14 is based on a section in ´ L. Lovasz – M.D. Plummer: Matching theory (Elsevier, Amsterdam, 1979) 1 2 Contents 1 Introduction 2 Let 2.1 2.2 2.3 2.4 2.5 us count! A party . . . . . . . . Sets and the like . . . The number of subsets Sequences . . . . . . . Permutations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 7 7 9 12 16 17 21 21 23 24 27 27 28 29 30 32 33 35 35 38 45 45 46 47 51 51 52 53 55 55 56 58 59 63 64 69 3 Induction 3.1 The sum of odd numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Subset counting revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Counting regions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Counting subsets 4.1 The number of ordered subsets . . . . 4.2 The number of subsets of a given size 4.3 The Binomial Theorem . . . . . . . . 4.4 Distributing presents . . . . . . . . . . 4.5 Anagrams . . . . . . . . . . . . . . . . 4.6 Distributing money . . . . . . . . . . ...

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...Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. In contrast to real numbers that have the property of varying "smoothly", the objects studied in discrete mathematics – such as integers, graphs, and statements in logic[1] – do not vary smoothly in this way, but have distinct, separated values.[2] Discrete mathematics therefore excludes topics in "continuous mathematics" such as calculus and analysis. Discrete objects can often be enumerated by integers. More formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets[3] (sets that have the same cardinality as subsets of the natural numbers, including rational numbers but not real numbers). However, there is no exact definition of the term "discrete mathematics."[4] Indeed, discrete mathematics is described less by what is included than by what is excluded: continuously varying quantities and related notions. The set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of discrete mathematics that deals with finite sets, particularly those areas relevant to business. Research in discrete mathematics increased in the latter half of the twentieth century partly due to the development of digital computers which operate in discrete steps and store data in discrete bits. Concepts and notations from discrete mathematics are...

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