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1. Cryptography in the Context Of Communication Theory Another very important area where the Communication Theory introduced by C. E. Shannon has found its application is the science of cryptography and cryptanalysis. The theoretical foundations of how the theory can be used to analyse cryptographic systems in their most general and abstract form has been consolidated in the article called 'Communication Theory of Secrecy Systems' by Claude Shannon, which has been the main 'inspiration' of this chapter. To start with, we first introduce cryptography as a discipline in its own right, including historical remarks on its development, complemented by a brief description of some commonly used types of ciphers, which will be used for illustrative purposes later in the chapter. [To fill – the main body] The chapter will be summarized by the discussion of the state-of-the-art cryptographic methodologies available now. 1.1. Introduction to Cryptography The fundamental goal of cryptography is make communication over an insecure channel possible, in such a way that even if the message being passed between the participants is somehow obtained by the 'enemy', the information conveyed in this message could not be understood. In order to informally introduce the terminology subsequently required, let's consider a very simple cryptographic situation. The message generated by some information source, called plaintext, is encrypted using some algorithm and some encryption key into unintelligible ciphertext and then sent over the communication medium. When the ciphertext is received at the destination, it gets decrypted back into plaintext, using the reverse algorithm. The pair of algorithms, used by the message sender to create the encryption and then by the recipient for reversing it, is called a cipher. Hence, the transformation of plaintext into ciphertext is controlled by both the encryption algorithm and the key, the latter of which is of great importance. This is due to the fact that ciphers used in any particular cryptographic system stay constant, apart from this variable parameter, and consequently, even if initially unknown, can be assumed trivially broken over time. In fact, that is the treatment given to the situation by Claude Shannon in his theory, i.e. the system is assumed to be known to the enemy. Although this seems to be slightly on the pessimistic side, it can hardly be argued to be a totally unrealistic assumption. 1.1.1. Historical Overview Cryptography is thought to have appeared thousands of years ago in Ancient Egypt, possibly triggered by the spread of literacy among wider masses of people, so that the need to 'hide' written communication from unwanted eyes became more apparent.[2] The first forms were rather simple and looked at physically concealing the message. This ancient technique, the first recorded use of which is dates back to 440 BC, is called steganography ( derived from the Greek 'steganos' – covered and 'graphen' – to write) – the art of concealed writing. Quite a lot of descriptions of the actual ways to cover messages exist in literature, ranging from using invisible ink, which then could be developed by some means (e.g. heat or chemical reaction), to transcribing a message on the messenger's shaved head and allowing the hair to grow back on, in order to conceal the writing.[3] It must be noted that the weak spot of this form of cryptography is that it relies on some sort of secret or obscurity, known only to communication participants. If this secret either accidentally or purposefully gets discovered, the message is ultimately revealed. However, there does exist a digital form of steganography nowadays, as opposed to the classical pen and paper variety, which is mainly concerned with concealment of information in computer files. Very often digital images are used for this purpose due to their large size. E.g.: The fully recoverable lossless Tag Image File Format (TIF) uses 3 bytes to store every pixel (that is 8 bits per each of the primary colours – red, green and blue). Thus, even a relatively small 170x170 TIF image will contain 28900 pixels = 693600 bits = 86.7 KB. If we reserve just one bit in the representation of each of the primary colours – a change that would only be possible to detect programatically and not visible to a human eye – we have 3 bits per each pixel, or 86700 in total, spare for some other use. Thus, if we wanted to hide a message in the English language, we would need at most 5 bits (25 = 32 > 26 letters of the English alphabet) per character. This arrangement would allow us to hide a message consisting of up to 17340 English letters (that's the whole essay!) in such a seemingly small digital picture. Another historically notable method of cryptography is the so-called 'privacy systems'[1] – the systems requiring some special equipment in order to recover encrypted messages. Early examples can be found in some literary works, e.g. in the 'The Dark Castle Olshansky' by a Belarusian writer Uladzimir Karatkievich, where the author describes the following medieval encryption technique. A narrow paper tape was wrapped around an object, preferably of some rare and strange shape, and then the writing was done across different layers of the tape. The result of such a transcription was what appeared to be random symbols written under different angles and with some arbitrary in length gaps in between. However, for anyone in possession of the secret object, the reconstruction of the original message was just a matter of wrapping the tape around it. More modern examples of privacy systems include speech inversion – a voice encryption method based on reversing the signal around some fixed or variable frequencies.[4] And finally, the class of cryptosystems that is of primary interest in relation to the Communication Theory, due to being essentially a mathematical problem, are the ones where the message is encrypted using codes and ciphers (called 'true secrecy systems' by Claude Shannon[1]). The honour of inventing the first known ciphers belongs to the Greeks of Classical times, who are believed to have used, for example, the scytale transposition cipher for their military campaigns or the Polybius cipher as an aid to telegraphy[2]. These are the secrecy systems that will be the main focus of this chapter. It's also worth mentioning that alongside the developments in cryptography, there appeared another science trying to constantly keep up – cryptanalysis, or the study of how to 'break' codes and ciphers. An example of this would be a discovery of frequency analysis in the Medieval ages that could be employed to break some simple ciphers, like substitution cipher[2], as will be demonstrated shortly below. Keeping this 'counter-cryptography' science in mind, the natural question one might ask is whether it is possible to create an ideal secrecy system, which cannot be broken, even if infinite resources are assumed, e.g. computation power, time, etc. This is the motivational question that we will leave until the end to answer. 1.1.2. Standard Ciphers In this sub-section, we present some widely known ciphers that will later serve for the purpose of illustrating the theoretical concepts behind secrecy systems as well some practical considerations. Simple Substitution Ciphers In a simple substitution cipher, as the name suggests, each symbol in the message is substituted by another symbol in such a manner that the original message can be uniquely and unambiguously recovered, once the key is known. I.e. speaking in mathematical terms, this cipher is defined as a finite set of transformations Ti, each of which corresponds to some encryption key i and is applied to a finite set of messages M: E= TiM Hence, the message M = m1m2m3m4… becomes the encryption E = e1e2e3e4… with e1 = f(m1), e2 = f(m2), e3 = f(m3), e4 = f(m4)… For this kind of coding, the key can be thought of as a permutation of the substitution alphabet being used. So, if we are encrypting a message in English by substituting for each letter another letter of the English alphabet, we have 26! possible encryption keys and transformations corresponding to the number of ways we can substitute for 26 different symbols. One instance of the encryption key could be: Q U C Z R K F A S Y E V M O N J W P X D G I B H T L where letter Q is used as a substitute for A, U – a substitute for B and so on. Using the above key, the text extract becomes Although the encryption looks like a complete gibberish, in fact, the ciphertext is not that hard to reverse. All it takes, in this case, is the knowledge of the letter frequency analysis of the English language: a typical distribution of letters in the English text is not uniform, with certain letters or combinations of letters more likely to appear than others (this is extremely likely to be true for all natural languages???). For instance, it is common knowledge that in English the most frequently occurring letter is 'E', while the least frequent is 'Z'. Or, there are very few words in the English language, where the letter 'Q' is not followed immediately by 'U' (e.g. qadi - a Muslim judge, qanat - an irrigation channel), of which practically all are of foreign origin. Hence, if we manage to decode 'Q' in some piece of ciphertext, the probability that the next letter is 'U' is nearly unity, with the probabilities of all other solutions effectively approaching zero. In the general case, of course, the success of reversing a passage of simple substitution ciphertext depends on the number of intercepted letters: the greater this number is, the more likely it is that there will only be one possible message that could have produced the cryptogram. For the demonstration of the weaknesses of simple substitution ciphers discussed in this section, please see a separate program (to be developed). (Need to decide on which of the well-known ciphers would be most interesting to pick) Transposition Ciphers Vigenere Ciphers The Playfair Cipher Others 1.2. Mathematical Foundations of Cryptographic Systems From Shannon's paper 1.3. State-of-the-art developments, or where the cryptography is heading now Quantum Cryptography [1] – Claude Shannon, “The Theory of Secrecy Systems” [2] - http://en.wikipedia.org/wiki/History_of_cryptography [3] - http://en.wikipedia.org/wiki/Steganography [4] - http://seussbeta.tripod.com/crypt.html [5] – Charles Dickens, “Great Expectations” http://mathcircle.berkeley.edu/BMC3/crypto/node2.html