 # Fundamentals Of Information Theory Coding And Cryptography Pdf

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Published: 27.04.2021  Quantum information science is a young field, its underpinnings still being laid by a large number of researchers [see "Rules for a Complex Quantum World," by Michael A.

## Fundamentals in Information Theory and Coding

Information theory is the scientific study of the quantification , storage , and communication of information. The field was fundamentally established by the works of Harry Nyquist and Ralph Hartley , in the s, and Claude Shannon in the s. The field is at the intersection of probability theory , statistics , computer science, statistical mechanics , information engineering , and electrical engineering. A key measure in information theory is entropy.

Entropy quantifies the amount of uncertainty involved in the value of a random variable or the outcome of a random process. For example, identifying the outcome of a fair coin flip with two equally likely outcomes provides less information lower entropy than specifying the outcome from a roll of a die with six equally likely outcomes. Some other important measures in information theory are mutual information , channel capacity, error exponents , and relative entropy.

Important sub-fields of information theory include source coding , algorithmic complexity theory , algorithmic information theory , and information-theoretic security. Applications of fundamental topics of information theory include lossless data compression e. ZIP files , lossy data compression e. Its impact has been crucial to the success of the Voyager missions to deep space, the invention of the compact disc , the feasibility of mobile phones and the development of the Internet.

The theory has also found applications in other areas, including statistical inference ,  cryptography , neurobiology ,  perception ,  linguistics, the evolution  and function  of molecular codes bioinformatics , thermal physics ,  quantum computing , black holes, information retrieval , intelligence gathering , plagiarism detection ,  pattern recognition , anomaly detection  and even art creation.

Information theory studies the transmission, processing, extraction, and utilization of information. Abstractly, information can be thought of as the resolution of uncertainty. In the case of communication of information over a noisy channel, this abstract concept was formalized in by Claude Shannon in a paper entitled A Mathematical Theory of Communication , in which information is thought of as a set of possible messages, and the goal is to send these messages over a noisy channel, and to have the receiver reconstruct the message with low probability of error, in spite of the channel noise.

Shannon's main result, the noisy-channel coding theorem showed that, in the limit of many channel uses, the rate of information that is asymptotically achievable is equal to the channel capacity, a quantity dependent merely on the statistics of the channel over which the messages are sent.

Information theory is closely associated with a collection of pure and applied disciplines that have been investigated and reduced to engineering practice under a variety of rubrics throughout the world over the past half-century or more: adaptive systems , anticipatory systems , artificial intelligence , complex systems , complexity science , cybernetics , informatics , machine learning , along with systems sciences of many descriptions.

Information theory is a broad and deep mathematical theory, with equally broad and deep applications, amongst which is the vital field of coding theory. Coding theory is concerned with finding explicit methods, called codes , for increasing the efficiency and reducing the error rate of data communication over noisy channels to near the channel capacity.

These codes can be roughly subdivided into data compression source coding and error-correction channel coding techniques. In the latter case, it took many years to find the methods Shannon's work proved were possible. A third class of information theory codes are cryptographic algorithms both codes and ciphers. Concepts, methods and results from coding theory and information theory are widely used in cryptography and cryptanalysis.

See the article ban unit for a historical application. The landmark event establishing the discipline of information theory and bringing it to immediate worldwide attention was the publication of Claude E. Prior to this paper, limited information-theoretic ideas had been developed at Bell Labs , all implicitly assuming events of equal probability.

The unit of information was therefore the decimal digit , which since has sometimes been called the hartley in his honor as a unit or scale or measure of information.

Alan Turing in used similar ideas as part of the statistical analysis of the breaking of the German second world war Enigma ciphers. Much of the mathematics behind information theory with events of different probabilities were developed for the field of thermodynamics by Ludwig Boltzmann and J.

Willard Gibbs. Connections between information-theoretic entropy and thermodynamic entropy, including the important contributions by Rolf Landauer in the s, are explored in Entropy in thermodynamics and information theory. In Shannon's revolutionary and groundbreaking paper, the work for which had been substantially completed at Bell Labs by the end of , Shannon for the first time introduced the qualitative and quantitative model of communication as a statistical process underlying information theory, opening with the assertion:.

Information theory is based on probability theory and statistics. Information theory often concerns itself with measures of information of the distributions associated with random variables. Important quantities of information are entropy, a measure of information in a single random variable, and mutual information, a measure of information in common between two random variables. The former quantity is a property of the probability distribution of a random variable and gives a limit on the rate at which data generated by independent samples with the given distribution can be reliably compressed.

The latter is a property of the joint distribution of two random variables, and is the maximum rate of reliable communication across a noisy channel in the limit of long block lengths, when the channel statistics are determined by the joint distribution. The choice of logarithmic base in the following formulae determines the unit of information entropy that is used. A common unit of information is the bit, based on the binary logarithm. Other units include the nat , which is based on the natural logarithm , and the decimal digit , which is based on the common logarithm.

Based on the probability mass function of each source symbol to be communicated, the Shannon entropy H , in units of bits per symbol , is given by. This equation gives the entropy in the units of "bits" per symbol because it uses a logarithm of base 2, and this base-2 measure of entropy has sometimes been called the shannon in his honor.

Entropy is also commonly computed using the natural logarithm base e , where e is Euler's number , which produces a measurement of entropy in nats per symbol and sometimes simplifies the analysis by avoiding the need to include extra constants in the formulas.

Other bases are also possible, but less commonly used. Intuitively, the entropy H X of a discrete random variable X is a measure of the amount of uncertainty associated with the value of X when only its distribution is known.

If one transmits bits 0s and 1s , and the value of each of these bits is known to the receiver has a specific value with certainty ahead of transmission, it is clear that no information is transmitted. If, however, each bit is independently equally likely to be 0 or 1, shannons of information more often called bits have been transmitted. Between these two extremes, information can be quantified as follows.

The special case of information entropy for a random variable with two outcomes is the binary entropy function, usually taken to the logarithmic base 2, thus having the shannon Sh as unit:. The joint entropy of two discrete random variables X and Y is merely the entropy of their pairing: X , Y. This implies that if X and Y are independent , then their joint entropy is the sum of their individual entropies.

For example, if X , Y represents the position of a chess piece— X the row and Y the column, then the joint entropy of the row of the piece and the column of the piece will be the entropy of the position of the piece. Despite similar notation, joint entropy should not be confused with cross entropy. The conditional entropy or conditional uncertainty of X given random variable Y also called the equivocation of X about Y is the average conditional entropy over Y : .

Because entropy can be conditioned on a random variable or on that random variable being a certain value, care should be taken not to confuse these two definitions of conditional entropy, the former of which is in more common use. A basic property of this form of conditional entropy is that:. Mutual information measures the amount of information that can be obtained about one random variable by observing another.

It is important in communication where it can be used to maximize the amount of information shared between sent and received signals. The mutual information of X relative to Y is given by:. Mutual information is symmetric :. Mutual information can be expressed as the average Kullback—Leibler divergence information gain between the posterior probability distribution of X given the value of Y and the prior distribution on X :. In other words, this is a measure of how much, on the average, the probability distribution on X will change if we are given the value of Y.

This is often recalculated as the divergence from the product of the marginal distributions to the actual joint distribution:. It is thus defined. Although it is sometimes used as a 'distance metric', KL divergence is not a true metric since it is not symmetric and does not satisfy the triangle inequality making it a semi-quasimetric. The KL divergence is the objective expected value of Bob's subjective surprisal minus Alice's surprisal, measured in bits if the log is in base 2.

In this way, the extent to which Bob's prior is "wrong" can be quantified in terms of how "unnecessarily surprised" it is expected to make him.

Coding theory is one of the most important and direct applications of information theory. It can be subdivided into source coding theory and channel coding theory. Using a statistical description for data, information theory quantifies the number of bits needed to describe the data, which is the information entropy of the source.

This division of coding theory into compression and transmission is justified by the information transmission theorems, or source—channel separation theorems that justify the use of bits as the universal currency for information in many contexts.

However, these theorems only hold in the situation where one transmitting user wishes to communicate to one receiving user. In scenarios with more than one transmitter the multiple-access channel , more than one receiver the broadcast channel or intermediary "helpers" the relay channel , or more general networks , compression followed by transmission may no longer be optimal.

Network information theory refers to these multi-agent communication models. Any process that generates successive messages can be considered a source of information.

A memoryless source is one in which each message is an independent identically distributed random variable , whereas the properties of ergodicity and stationarity impose less restrictive constraints. All such sources are stochastic. These terms are well studied in their own right outside information theory. Information rate is the average entropy per symbol. For memoryless sources, this is merely the entropy of each symbol, while, in the case of a stationary stochastic process, it is.

For the more general case of a process that is not necessarily stationary, the average rate is. For stationary sources, these two expressions give the same result. It is common in information theory to speak of the "rate" or "entropy" of a language. This is appropriate, for example, when the source of information is English prose.

The rate of a source of information is related to its redundancy and how well it can be compressed, the subject of source coding. Communications over a channel—such as an ethernet cable—is the primary motivation of information theory. However, such channels often fail to produce exact reconstruction of a signal; noise, periods of silence, and other forms of signal corruption often degrade quality.

Consider the communications process over a discrete channel. A simple model of the process is shown below:. Here X represents the space of messages transmitted, and Y the space of messages received during a unit time over our channel. Let p y x be the conditional probability distribution function of Y given X.

We will consider p y x to be an inherent fixed property of our communications channel representing the nature of the noise of our channel. Then the joint distribution of X and Y is completely determined by our channel and by our choice of f x , the marginal distribution of messages we choose to send over the channel.

Under these constraints, we would like to maximize the rate of information, or the signal , we can communicate over the channel. The appropriate measure for this is the mutual information, and this maximum mutual information is called the channel capacity and is given by:.

This capacity has the following property related to communicating at information rate R where R is usually bits per symbol. Channel coding is concerned with finding such nearly optimal codes that can be used to transmit data over a noisy channel with a small coding error at a rate near the channel capacity. In practice many channels have memory. In such a case the capacity is given by the mutual information rate when there is no feedback available and the Directed information rate in the case that either there is feedback or not   if there is no feedback the dircted informationj equals the mutual information. ## information theory, coding and cryptography

This textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like algebraic-geometry codes and elliptic-curve cryptosystems as well as material not treated by other books, including function-field codes, digital nets, code-based public-key cryptosystems, and frameproof codes. Combining a systematic development of theory with a broad selection of real-world applications, this is the most comprehensive yet accessible introduction to the field available. ## Introduction to coding and information theory by steven roman pdf

Explore a preview version of Information Theory, Coding and Cryptography right now. Kythe, Prem K. Using a simple yet rigorous approach, Algebraic and Stochastic Coding Theory makes the subject of coding …. Learn algorithms for solving classic computer science problems with this concise guide covering everything from fundamental ….

The understanding of the theoretical matter is supported by many examples. One particular emphasis is put on the explanation of Genomic Coding.

Whenever we come across the term cryptography, the first thing and probably the only thing that comes to our mind is private communication through encryption. The lecture notes contain 10 chapters, each with a pages of an introductory summary of some concepts and results of the corresponding lecture and about exercises and their solutions as well as an appendix of. Until recently most abstract algebra texts included few if any. History of cryptography, encryption conventional, public key , digital signatures, hash functions, message authentication codes, identification, authentication, applications. Credit Hours: 3.

Introduction to coding and information theory by steven roman pdf Introduction to coding and information theory by steven roman pdf. Coding and Information Theory by. Introduction to Coding and Information Theory by. Free shipping and pickup in store on eligible orders. This book is intended to introduce coding theory and information theory to undergraduate students of mathematics and computer science.

Information theory is the scientific study of the quantification , storage , and communication of information. The field was fundamentally established by the works of Harry Nyquist and Ralph Hartley , in the s, and Claude Shannon in the s. The field is at the intersection of probability theory , statistics , computer science, statistical mechanics , information engineering , and electrical engineering.

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