## An Introduction to Single-User Information Theory

provides the first comprehensive treatment of the theory of I-Measure, network coding theory, Shannon and non-Shannon type information inequalities, and a. A First Course in Information Theory is an up-to-date introduction to information Shannon's information measures refer to entropy, conditional entropy, mutual. Shannon's information theory deals with source coding [ ] Claude Shannon established the mathematical basis of information theory and published [ ].## Shannon Information Theory Support Science Journalism Video

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Eine übergreifende Frage für Nachrichtentechniker war, wie eine wirtschaftlich-effiziente und störungsfreie Montesino erreicht werden kann. Today we call that the bandwidth of the channel. Shannon demonstrated mathematically that even in a noisy channel with a low bandwidth, essentially perfect, error-free communication could be achieved by keeping the transmission rate within the channel's bandwidth and by using error-correcting schemes: the transmission of additional bits that would enable the data to be extracted from the noise-ridden signal.

Today everything from modems to music CDs rely on error-correction to function. A major accomplishment of quantum-information scientists has been the development of techniques to correct errors introduced in quantum information and to determine just how much can be done with a noisy quantum communications channel or with entangled quantum bits qubits whose entanglement has been partially degraded by noise.

The Unbreakable Code A year after he founded and launched information theory, Shannon published a paper that proved that unbreakable cryptography was possible.

He did this work in , but at that time it was classified. The scheme is called the one-time pad or the Vernam cypher, after Gilbert Vernam, who had invented it near the end of World War I.

The idea is to encode the message with a random series of digits--the key--so that the encoded message is itself completely random. The catch is that one needs a random key that is as long as the message to be encoded and one must never use any of the keys twice.

Shannon's contribution was to prove rigorously that this code was unbreakable. To this day, no other encryption scheme is known to be unbreakable.

The problem with the one-time pad so-called because an agent would carry around his copy of a key on a pad and destroy each page of digits after they were used is that the two parties to the communication must each have a copy of the key, and the key must be kept secret from spies or eavesdroppers.

Quantum cryptography solves that problem. More properly called quantum key distribution, the technique uses quantum mechanics and entanglement to generate a random key that is identical at each end of the quantum communications channel.

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.

Information theoretic concepts apply to cryptography and cryptanalysis. Turing's information unit, the ban , was used in the Ultra project, breaking the German Enigma machine code and hastening the end of World War II in Europe.

Shannon himself defined an important concept now called the unicity distance. Based on the redundancy of the plaintext , it attempts to give a minimum amount of ciphertext necessary to ensure unique decipherability.

Information theory leads us to believe it is much more difficult to keep secrets than it might first appear. A brute force attack can break systems based on asymmetric key algorithms or on most commonly used methods of symmetric key algorithms sometimes called secret key algorithms , such as block ciphers.

The security of all such methods currently comes from the assumption that no known attack can break them in a practical amount of time.

Information theoretic security refers to methods such as the one-time pad that are not vulnerable to such brute force attacks.

In such cases, the positive conditional mutual information between the plaintext and ciphertext conditioned on the key can ensure proper transmission, while the unconditional mutual information between the plaintext and ciphertext remains zero, resulting in absolutely secure communications.

In other words, an eavesdropper would not be able to improve his or her guess of the plaintext by gaining knowledge of the ciphertext but not of the key.

However, as in any other cryptographic system, care must be used to correctly apply even information-theoretically secure methods; the Venona project was able to crack the one-time pads of the Soviet Union due to their improper reuse of key material.

Pseudorandom number generators are widely available in computer language libraries and application programs. They are, almost universally, unsuited to cryptographic use as they do not evade the deterministic nature of modern computer equipment and software.

A class of improved random number generators is termed cryptographically secure pseudorandom number generators , but even they require random seeds external to the software to work as intended.

These can be obtained via extractors , if done carefully. Although related, the distinctions among these measures mean that a random variable with high Shannon entropy is not necessarily satisfactory for use in an extractor and so for cryptography uses.

One early commercial application of information theory was in the field of seismic oil exploration. Work in this field made it possible to strip off and separate the unwanted noise from the desired seismic signal.

Information theory and digital signal processing offer a major improvement of resolution and image clarity over previous analog methods.

Semioticians Doede Nauta and Winfried Nöth both considered Charles Sanders Peirce as having created a theory of information in his works on semiotics.

Concepts from information theory such as redundancy and code control have been used by semioticians such as Umberto Eco and Ferruccio Rossi-Landi to explain ideology as a form of message transmission whereby a dominant social class emits its message by using signs that exhibit a high degree of redundancy such that only one message is decoded among a selection of competing ones.

Information theory also has applications in Gambling and information theory , black holes , and bioinformatics. From Wikipedia, the free encyclopedia.

Theory dealing with information. Not to be confused with Information science. This article may contain indiscriminate , excessive , or irrelevant examples.

Please improve the article by adding more descriptive text and removing less pertinent examples. See Wikipedia's guide to writing better articles for further suggestions.

May Main article: History of information theory. Main article: Quantities of information. Main article: Coding theory. Main article: Channel capacity.

Mathematics portal. Active networking Cryptanalysis Cryptography Cybernetics Entropy in thermodynamics and information theory Gambling Intelligence information gathering Seismic exploration.

Hartley, R. History of information theory Shannon, C. Timeline of information theory Yockey, H. Coding theory Detection theory Estimation theory Fisher information Information algebra Information asymmetry Information field theory Information geometry Information theory and measure theory Kolmogorov complexity List of unsolved problems in information theory Logic of information Network coding Philosophy of information Quantum information science Source coding.

Rieke; D. Spikes: Exploring the Neural Code. The MIT press. Scientific Reports. Bibcode : NatSR Bibcode : Sci Bibcode : PhRv..

And all over the world, as displayed below:. Now, in the first page of his article, Shannon clearly says that the idea of bits is J.

And, surely enough, the definition given by Shannon seems to come out of nowhere. But it works fantastically.

Meanwhile, in Vietnam, people rather use my full first name. A context corresponds to what messages you expect. More precisely, the context is defined by the probability of the messages.

Thus, the context of messages in Vietnam strongly differs from the context of western countries. But this is not how Shannon quantified it, as this quantification would not have nice properties.

Because of its nice properties. But mainly, if you consider a half of a text, it is common to say that it has half the information of the text in its whole.

This is due to the property of logarithm to transform multiplication which appears in probabilistic reasonings into addition which we actually use.

This is an awesome remark! Indeed, if the fraction of the text you read is its abstract, then you already kind of know what the information the whole text has.

It does! And the reason it does is because the first fraction of the message modifies the context of the rest of the message.

In other words, the conditional probability of the rest of the message is sensitive to the first fraction of the message.

This updating process leads to counter-intuitive results, but it is an extremely powerful one. Find out more with my article on conditional probabilities.

The whole industry of new technologies and telecommunications! But let me first present you a more surprising application to the understanding of time perception explain in this TedED video by Matt Danzico.

As Shannon put it in his seminal paper, telecommunication cannot be thought in terms of information of a particular message.

Indeed, a communication device has to be able to work with any information of the context. This has led Shannon to re -define the fundamental concept of entropy , which talks about information of a context.

You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name.

In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage.

In , Ludwig Boltzmann shook the world of physics by defining the entropy of gases, which greatly confirmed the atomic theory.

He defined the entropy more or less as the logarithm of the number of microstates which correspond to a macrostate. For instance, a macrostate would say that a set of particles has a certain volume, pressure, mass and temperature.

Meanwhile, a microstate defines the position and velocity of every particle. This is explained in the following figure, where each color stands for a possible message of the context:.

The average amount of information is therefore the logarithm of the number of microstates. This is another important interpretation of entropy.

For the average information to be high, the context must allow for a large number of unlikely events.

Another way of phrasing this is to say that there is a lot of uncertainties in the context. In other words, entropy is a measure of the spreading of a probability.

In some sense, the second law of thermodynamics which states that entropy cannot decrease can be reinterpreted as the increasing impossibility of defining precise contexts on a macroscopic level.

It is essential! The most important application probably regards data compression. Indeed, the entropy provides the theoretical limit to the average number of bits to code a message of a context.

It also gives an insight into how to do so. Data compression has been applied to image, audio or file compressing, and is now essential on the Web.

Youtube videos can now be compressed enough to surf all over the Internet! For any given introduction, the message can be described with a conditional probability.

This defines a entropy conditional to the given introduction. When you add in a space, which is required for communication in words, the English alphabet creates 27 total characters.

This results in 4. Thanks to the mathematics of the information theory, we can know with certainty that any transmission or storage of information in digital code requires a multiplication of 4.

Probabilities help us to further reduce the uncertainty that exists when evaluating the equations of information that we receive every day.

It also means we can transmit less data, further reducing our uncertainty we face in solving the equation.

Once all of these variables are taken into account, we can reduce the uncertainty which exists when attempting to solve informational equations.

With enough of these probabilities in place, it becomes possible to reduce the 4. That means less time is needed to transmit the information, less storage space is required to keep it, and this speeds up the process of communicating data to one another.

Claude Shannon created the information theory in order to find a more practical way to create better and more efficient codes for communication.

This has allowed us to find the limits of how fast data can be processed. Through digital signals, we have discovered that not only can this information be processed extremely quickly, but it can be routed globally with great consistency.

It can even be translated, allowing one form of information to turn into another form of information digitally.

Think of it like using Google Translate to figure out how to say something in Spanish, but you only know the English language. The information you receive occurs because bits of information were used to reduce the uncertainty of your request so that you could receive a desired outcome.

It is why computers are now portable instead of confined to one very large room.

Consider the communications process over a discrete channel. Mindy Block says:. Other bases are also possible, but less commonly used. In other words, 20000 Meilen Unter Dem Meer Film 2021 eavesdropper would not be able to Spielautomaten Spielen his Social Trading Vergleich her guess of the plaintext by gaining knowledge of the ciphertext but not of the key. Don't worry, we are not going to deal with the actual details of the equation here, but instead I will give you a flavour of what it means.### MГchte, kannst du entweder deinen ganzen oder **Quoten Samstag.** - Navigationsmenü

Kategorien : Informationstheorie Kybernetik Kognitionswissenschaft. Die Informationstheorie ist eine mathematische Theorie aus dem Bereich der Claude E. Shannon: A mathematical theory of communication. Bell System Tech. In diesem Jahr veröffentlichte Shannon seine fundamentale Arbeit A Mathematical Theory of Communication und prägte damit die moderne Informationstheorie. Der Claude E. Shannon Award, benannt nach dem Begründer der Informationstheorie Claude E. Shannon, ist eine seit von der IEEE Information Theory. provides the first comprehensive treatment of the theory of I-Measure, network coding theory, Shannon and non-Shannon type information inequalities, and a. **Quoten Samstag**ciphertext remains zero, resulting in absolutely secure communications. Indeed, Lotto Lottery 24 your professors may have forgotten to tell you is that this law connects today's world to its first instant, the Big Bang! Illinois: University of Illinois Press. He raised the right questions, which no one else even thought of asking. Encryption based on the Vernam cypher and quantum key distribution is perfectly secure: quantum physics guarantees security of the key and Shannon's theorem proves that the encryption method is unbreakable. Pseudorandom number generators are widely available in computer language libraries and application programs. Inwith a Ph. The structure of information also lies in the concatenation into longer texts. Interpreter Middleware Virtual machine Operating system Software quality. May Artificial intelligence Biological cybernetics Biomedical cybernetics Biorobotics Biosemiotics Pinnaclesport Catastrophe theory

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