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Unformatted text preview: Coding and Cryptography T. W. K¨orner February 14, 2011 Transmitting messages is an important practical problem. Coding theory includes the study of compression codes which enable us to send messages cheaply and error correcting codes which ensure that messages remain legible even in the presence of errors. Cryptography on the other hand, makes sure that messages remain unreadable — except to the intended recipient. These techniques turn out to have much in common. Many Part II courses go deeply into one topic so that you need to un derstand the whole course before you understand any part of it. They often require a firm grasp of some preceding course. Although this course has an underlying theme, it splits into parts which can be understood separately and, although it does require knowledge from various earlier courses, it does not require mastery of that knowledge. All that is needed is a little probability, a little algebra and a fair amount of common sense. On the other hand, the variety of techniques and ideas probably makes it harder to understand everything in the course than in a more monolithic course. Small print The syllabus for the course is defined by the Faculty Board Schedules (which are minimal for lecturing and maximal for examining). I should very much appreciate being told of any corrections or possible improvements however minor . This document is written in L A T E X2e and should be available from my home page http://www.dpmms.cam.ac.uk/˜twk in latex, dvi, ps and pdf formats. Supervisors can obtain comments on the exercises at the end of these notes from the secretaries in DPMMS or by email from me. My email address is [email protected] . These notes are based on notes taken in the course of a previous lecturer Dr Pinch, on the excellent set of notes available from Dr Carne’s home page and on Dr Fisher’s collection of examples. Dr Parker and Dr Lawther produced two very useful list of corrections. Any credit for these notes belongs to them, any discredit to me. This is a course outline. A few proofs are included or sketched in these notes but most are omitted. Please note that vectors are row vectors unless otherwise stated. 1 Contents 1 Codes and alphabets 3 2 Huffman’s algorithm 5 3 More on prefixfree codes 9 4 Shannon’s noiseless coding theorem 11 5 Nonindependence 16 6 What is an error correcting code? 18 7 Hamming’s breakthrough 20 8 General considerations 24 9 Some elementary probability 27 10 Shannon’s noisy coding theorem 29 11 A holiday at the race track 31 12 Linear codes 33 13 Some general constructions 38 14 Polynomials and fields 43 15 Cyclic codes 47 16 Shift registers 53 17 A short homily on cryptography 59 18 Stream ciphers 61 19 Asymmetric systems 68 20 Commutative public key systems 71 21 Trapdoors and signatures 76 22 Quantum cryptography 78 2 23 Further reading 82 24 Exercise Sheet 1 85 25 Exercise Sheet 2 90 26 Exercise Sheet 3 95 27 Exercise Sheet 4 99 1 Codes and alphabets Originally, a code was a device for making messages hard to read. The studyOriginally, a code was a device for making messages hard to read....
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This note was uploaded on 02/18/2012 for the course MATH 533 taught by Professor Drewarmstrong during the Fall '11 term at University of Miami.
 Fall '11
 DrewArmstrong
 Cryptography, Ulysses

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