Code and ciphers: Julius Caesar, the Enigma and the internet

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x preface It is the author’s belief, based upon experience, that there is a middle way and that, without going into all the details, it is possible to convey to non-specialists the essentials of some of the mathematics involved even in the more modern cipher systems. My aim therefore has been to introduce the general reader to a number of codes and ciphers, starting with the ancient and elementary and progressing, via some of the wartime cipher machines, to systems currently in commercial use. Examples of the use, and methods of solution, of various cipher systems are given but in those cases where the solution of a realistically sized message would take many pages the method of solution is shown by scaled-down examples. In the main body of the text the mathematics, including mathematical notation and phraseology, is kept to a minimum. For those who would like to know more, however, further details and explanations are provided in the mathematical appendix where, in some cases, rather more information than is absolutely necessary is given in the hope of encouraging them to widen their acquaintance with some fascinating and useful areas of mathematics, which have applications in ‘codebreaking’ and elsewhere. I am grateful to Cardiff University for permission to reproduce Plates 9.1 to 9.4 inclusive, 10.1 and 10.2, and to my son John for permission to reproduce Plate 11.1. I am also grateful to Dr Chris Higley of Information Services, Cardiff University, for material relating to Chapter 13 and to the staff at CUP, particularly Roger Astley and Peter Jackson, for their helpfulness throughout the preparation of this book.
1 Introduction Some aspects of secure communication For at least two thousand years there have been people who wanted to send messages which could only be read by the people for whom they were intended. When a message is sent by hand, carried from the sender to the recipient, whether by a slave, as in ancient Greece or Rome, or by the Post Office today, there is a risk of it going astray. The slave might be captured or the postman might deliver to the wrong address. If the message is written in clear, that is, in a natural language without any attempt at concealment, anyone getting hold of it will be able to read it and, if they know the language, understand it. In more recent times messages might be sent by telegraph, radio, telephone, fax or e-mail but the possibility of them being intercepted is still present and, indeed, has increased enormously since, for example, a radio transmission can be heard by anyone who is within range and tuned to the right frequency whilst an e-mail message might go to a host of unintended recipients if a wrong key on a computer keyboard is pressed or if a ‘virus’ is lurking in the computer. It may seem unduly pessimistic but a good rule is to assume that any message which is intended to be confidential will fall into the hands of someone who is not supposed to see it and therefore it is prudent to take steps to ensure that they will, at least, have great difficulty in reading it and, preferably, will not be able to read it at all. The extent of the damage caused by unintentional disclosure may depend very much on the time that has elapsed between interception and reading of the message. There are occasions when a delay of a day or even a few hours in reading a message nullifies the damage; for example, a decision by a shareholder to [1]
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Page 5 and 6: R. F. Churchhouse Codes and ciphers
Page 7 and 8: Contents Preface ix 1 Introduction
Page 9 and 10: Cryptanalysisofalinearrecurrence 10
Page 11: Preface Virtually anyone who can re
Page 15 and 16: Not a very sophisticated method, pa
Page 17 and 18: change. As an example, anticipating
Page 19 and 20: Another form of encryption is the u
Page 21 and 22: and so is valid. On the other hand
Page 23 and 24: When the modulus is 10 only the num
Page 25 and 26: 2 From Julius Caesar to simple subs
Page 27 and 28: Table 2.3 Shift Message 12 OUI 12 Y
Page 29 and 30: worry about but, on the other hand,
Page 31 and 32: then it is probably THE and the unk
Page 33 and 34: From Julius Caesar to simple substi
Page 37 and 38: Table 2.6 From Julius Caesar to sim
Page 41 and 42: and the key which provides the enci
Page 43 and 44: Further examination reveals that th
Page 45 and 46: ALTHOUGH I AM AN OLD MAN NIGHT IS G
Page 47 and 48: Polyalphabetic systems 35 messages
Page 49 and 50: Before leaving Vigenère try the fo
Page 51 and 52: Polyalphabetic systems 39 blocks of
Page 53 and 54: eginning at the top, but it is then
Page 55 and 56: first row above, for example, we fi
Page 57 and 58: Table 4.4 Key 3 1 5 2 4 1 2 3 4 5 6
Page 59 and 60: giving B G L D I N A F K E J O C H
Page 61 and 62: then the cipher text is HORUX SXSEO
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The plaintext digraphs are now sepa
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ox increases. By enumerating the po
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Example 5.1 HAPPY BIRTHDAY encipher
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Encryption The ‘plaintext’ is A
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plaintext digraphs according to som
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Cryptanalytic aspects of Playfair P
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OURXSITUATI ONXISXDESPE RATEXSENDXS
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the alphabet are represented by up
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From a cryptographic point of view
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3 7 0 7 7 4 1 5 6 1 7 8 5 3 8 1 9 0
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If we generate the same sequence (m
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Stencil ciphers The example above i
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Book ciphers A spy must avoid arous
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Table 7.2 Encipher table for a book
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Letter frequencies in book ciphers
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If then we try subtracting THE from
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where row F (the row of the cipher
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Ciphers for spies 85 that were nece
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mistake can occur if the sender lea
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inks and ciphers were provided by h
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Example 7.6 Encipher the message AG
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Ciphers for spies 93 simply a ‘ra
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going into the mathematical criteri
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numbers, 0 to 31 inclusive, into bi
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Linear recurrences The sequences lo
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Having converted the characters of
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What about sequences of higher orde
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combining the keys of two or more l
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Example 8.3 (1) Use the mid-square
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Problem 8.3 Starting with U 0 �1
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The Enigma cipher machine 111 syste
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The Enigma cipher machine 113 Plate
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The Enigma cipher machine 115 Plate
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Figure 9.2. wheel. For example, if
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wheel and then through the three wh
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first day of usage and so the asses
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The Enigma cipher machine 123 The m
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show how, in a typical encipherment
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interested reader can find it in [9
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great deal of data and many pages o
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It should be realised, of course, t
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10 The Hagelin cipher machine Histo
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ut there was no cryptographic advan
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of each cipher period, to which sid
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value can be expected to occur two
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Since some cages are obviously very
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2, there are 26! possible simple su
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The Hagelin cipher machine 145 Exam
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values are repeating at the appropr
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Overlapping will obviously affect t
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Table 10.6 The Hagelin cipher machi
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11 Beyond the Enigma The SZ42: a pr
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Description of the SZ42 machine The
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P1 41 43 Z1 (4) the five bits from
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which is approximately 1.6�10 19
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12 Public key cryptography Historic
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part of the story of what has been
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Public key cryptography 165 graphic
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(4) Now (k x ) y �(k y ) x �m x
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Public key cryptography 169 Despite
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If the cryptographer were prepared
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number of tests increased by a fact
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In this case the remainder is 4, no
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key, d, we use the Euclidean Algori
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the decipher key, d, is used instea
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and and, since Table 13.1 n N�2 n
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The Data Encryption Standard (DES)
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the message were known to relate to
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whereas in block cipher systems, su
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(1) X precedes M with information w
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ways, since choosing to pair, say,
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For example; if someone claims that
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depth, mentioned in Chapter 3. If w
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M9 Combining two biased streams of
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and so 2�4�6�12 �(4095)�4
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Verification Let the recurrence be
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the lettersatsetting2ofthewheel,and
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M16 Probability of a ‘depth’ in
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(2) In how many ways can N be repre
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Chapter 13 M21 (Rate of increase of
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Multiply each of these by M: M, 2M,
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If x 3 �0 divide x 2 by x 3 to gi
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then k�[ log 2 n], where [z] deno
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Mathematical aspects 217 problem un
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RHAPSODY and SYMPHONY agree in posi
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4.2 (Number of possible transpositi
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Table S.5 R H A P S O D Y B C E F G
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Chapter 8 8.1 (Recurrences of order
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columns in each case, are full of c
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Chapter 11 11.1 (Pin-setting errors
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[2.4] Moroney, M.J.: Facts from Fig
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[10.3] Almost any elementary book o
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Name index Adelman, L. 171, 234 And
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Subject index Abwehr Enigma 124, 13
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active pin 136 cage:‘good’ 141;
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Code and ciphers: Julius Caesar, the Enigma and the internet

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