Codes: An Introduction to Information Communication and CryptographyMany people do not realise that mathematics provides the foundation for the devices we use to handle information in the modern world. Most of those who do know probably think that the parts of mathematics involvedare quite ‘cl- sical’, such as Fourier analysis and di?erential equations. In fact, a great deal of the mathematical background is part of what used to be called ‘pure’ ma- ematics, indicating that it was created in order to deal with problems that originated within mathematics itself. It has taken many years for mathema- cians to come to terms with this situation, and some of them are still not entirely happy about it. Thisbookisanintegratedintroductionto Coding.Bythis Imeanreplacing symbolic information, such as a sequence of bits or a message written in a naturallanguage,byanother messageusing (possibly) di?erentsymbols.There are three main reasons for doing this: Economy (data compression), Reliability (correction of errors), and Security (cryptography). I have tried to cover each of these three areas in su?cient depth so that the reader can grasp the basic problems and go on to more advanced study. The mathematical theory is introduced in a way that enables the basic problems to bestatedcarefully,butwithoutunnecessaryabstraction.Theprerequisites(sets andfunctions,matrices,?niteprobability)shouldbefamiliartoanyonewhohas taken a standard course in mathematical methods or discrete mathematics. A course in elementary abstract algebra and/or number theory would be helpful, but the book contains the essential facts, and readers without this background should be able to understand what is going on. vi Thereareafewplaceswherereferenceismadetocomputeralgebrasystems. |
Contents
| 1 | |
Prefixfree codes 13 | 12 |
Economical coding | 27 |
Data compression | 47 |
Noisy channels | 73 |
The problem of reliable communication | 89 |
The noisy coding theorems | 107 |
Linear codes | 123 |
Algebraic coding theory | 141 |
Coding natural languages | 163 |
The development of cryptography | 179 |
Cryptography in theory and practice | 191 |
The RSA cryptosystem | 207 |
Cryptography and calculation | 221 |
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Codes: An Introduction to Information Communication and Cryptography Norman L. Biggs No preview available - 2008 |
Common terms and phrases
algebraic algorithm Alice and Bob alphabet arithmetic coding average word-length BCH code binary code binary symmetric channel bit-error probability bits blocks calculate channel matrix Chapter check matrix choose ciphertext codeword construct corresponding coset cryptosystem cyclic code cyclic group decision rule decryption function defined Definition denote digrams elements ElGamal elliptic curve encoded stream encryption entropy equation error error-correction example extended BSC follows frequency given Hamming code Hence information rate input integer inverse Lemma letters linear code MD rule memoryless memoryless source method minimum distance multiplication natural language optimal output parameters permutation plaintext polynomial possible prefix-free previous exercise primitive root private key probability distribution Proof public key received represented result Sender sequence Show Solution source emits stationary source Suppose symbols syndrome Theorem transmitted uncertainty words of length