“Correct” versus “incorrect” doesn’t fully capture information content, but can provide bounds, and using this experiment Shannon estimated the “true entropy of English” to be between 0.6 and 1.3 bits per letter. In particular, he provided the subjects with a fragment of English text, and asked them to predict the next letter, and statistics were gathered on how often the subject guessed correctly. What is the best model for a source generating English? Shannon had a great answer for this: The adult human mind has been trained for years to work with English, so he used an experiment with human subjects to estimate the entropy of English. For example, if you see the letters “algori” then the next letter is almost certainly a “t”. We can go even farther by considering longer contexts when selecting the “next letter” in English. In the same experiment described in the previous paragraph, English was modeled as a source that produces pairs of letters, and the entropy of this source model was only 7.18 bits per pair of characters, or 3.59 bits per character. Therefore, the probability of the pair “qu” is much higher than would be expected when looking at letters independently, and hence the entropy is lower for modeling pairs of letters rather than individual letters. For example, the letter “q” is almost always followed by the letter “u” (the only exceptions being words derived from a foreign language, like “Iraqi,” “Iqbal,” and “Chongqing”). We can make our model of English more accurate by considering pairs of letters rather than individual letters (what cryptographers call “digrams”). From these statistics, the entropy of English (from a source generating single letters) was calculated to be 4.47 bits. In one such study, the probability of a space is 0.1741, the probability of an ‘e’ is 0.0976, the probability of a ‘t’ is 0.0701, and so on. Many experiments by cryptographers and data compression researchers have shown that the frequency of individual letters is remarkably constant among different pieces of English writing. Obviously this isn’t a great model of English, since letters in English writing aren’t independent, but this is a good starting point to think about the information content of English. One way we can view English language writing is as a sequence of individual letters, selected independently. This can have serious consequences for security, which we’ll see below.
DEFINE TROPY GENERATOR
In other words, using a biased random number generator reduces the entropy of the source from 128 bits to around 60 bits.
We’re not saying that data is really just randomly chosen, but this is a good model that has proved its usefulness over time.ĭefinition: A source is an ordered pair \(\right) How are messages selected? We take the source to be probabilistic. For now we focus just on how messages are selected, not how they are encoded. The question of encoding is very involved, and we can encode for the most compact representation (data compression), for the most reliable transmission (error detection/correction coding), to make the information unintelligible to an eavesdropper (encryption), or perhaps with other goals in mind. We consider an information source as having access to a set of possible messages, from which it selects a message, encodes it somehow, and transmits it across a communication channel. To talk precisely about the information content in messages, we first need a mathematical model that describes how information is transmitted. In honor of his work, this use of “entropy” is sometimes called “Shannon entropy.” Entropy - Basic Model and Some Intuition The fundamental question of how information is represented is a common and deep thread connecting issues of communication, data compression, and cryptography, and information theory is a key component of all three of these areas. Information theory grew out of the initial work of Claude Shannon at Bell Labs in the 1940’s, where he also did some significant work on the science of cryptography. In settings that deal with computer and communication systems, entropy refers to its meaning in the field of information theory, where entropy has a very precise mathematical definition measuring randomness and uncertainty. In general English usage, entropy is used as a less precise term but still refers to disorder or randomness.
For example, in physics the term is a measure of a system’s thermal energy, which is related to the level of random motion of molecules and hence is a measure of disorder. The word entropy is used in several different ways in English, but it always refers to some notion of randomness, disorder, or uncertainty.
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