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

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122 chapter 9 acquired a military-version Enigma the 10 17 possibilities would seem to make decryption of even a single message impossible. If a million decrypts a second could be tried it would still take several thousand years to try them all; and in the 1930s there were no computers. How then did it come about that, in 1932, a method was found for decrypting Enigma messages? The fundamental flaw which led to the decryption of Enigma messages was not due to the design of the machine itself, but to the method which was used by the Germans to send messages. As has been remarked before, if a cipher operator is sending a message on any cipher system with variable parameters he must somehow either let the recipient know the values of those parameters, or leave him to work them out. The latter situation rarely applies, although it may apply in the case of the Jefferson cylinder. In the case of the Enigma such an arrangement is quite impractical and the recipient must be provided with all the information needed for the decipherment of the message. What information does the recipient need? Assuming that both the Enigma sender and the recipient are ‘on the same net’, that is, they are using the same Enigma wheels, in the same order, and the same plugboard wirings, there is still the problem that the sender must somehow let the receiver know the positions of the three wheels at the start of the message, and this necessary information constitutes the ‘indicator’ in this case. The choice of these starting positions, or ‘settings’, should be ‘randomly’ chosen by the sender, for if a number of messages are enciphered with the same initial setting the cryptanalyst would be presented with a ‘depth’ and the messages might then be readable, even though the identity of the wheels and/or their initial settings might not be discovered. Since there are 17 576 possible starting positions for the three wheels the probability that any pair of messages will be given the same starting settings is less than 0.000 06 if the settings are chosen randomly but, in practice, cipher operators are liable to develop habits, such as choosing the three letters for the wheel settings from the same row of the keyboard, which make the probability of a depth much greater. Even with randomly chosen settings if a large number of messages are sent with the same machine set-up (same wheel order and plugboard) a depth may well occur by chance: 200 messages, for example, would make it more likely than not that at least one pair of messages will be in depth (for an explanation see M16). So, in order to reduce the possibility of depths occurring, the set-up needs to be changed sufficiently often that large numbers of messages on the same set-up are very unlikely to occur.
The Enigma cipher machine 123 The method initially adopted by the Germans for letting the recipient know the starting positions of the three wheels was based upon the use of a common ground setting and the procedure for the sender was as follows. (1) Choose a ‘random’ set of three letters for the actual wheel settings to be used for the encipherment of the message; let us suppose that this is FMZ (say). (2) Look up the list provided to all users of the ‘net’ and note the ground setting for the day (or period, if the ground setting is changed more than once a day); let us suppose that this is BLE (say). (3) Turn the wheels to the ground setting, BLE. (4) Encipher the three letters of your chosen random setting twice; that is, encipher the six letters FMZFMZ and note the six resulting cipher letters; suppose that these are LOCWHQ (say). (5) Turn the wheels to the chosen random setting, FMZ, and encipher the message. (6) Precede the cipher text of the message with the six cipher letters of the enciphered double indicator, LOCWHQ. (7) Transmit the cipher message with the enciphered double indicator at the front. On receiving the cipher message the recipient would proceed as follows. (1) Set the three wheels to the ground setting for the day or period, BLE. (2) Type in the six letters, LOCWHQ, of the enciphered double indicator. (Remember that it is a characteristic of the Enigma that encipherment and decipherment are equivalent. Thus, since F in the first encipherment became L so L would have become F, which is what happens when we decipher L.) (3) This should yield the chosen random indicator twice, viz: FMZFMZ. If this does not happen there has been a mistake and the message may have to be re-sent; a dangerous situation which may cause a breach of security. (4) If the six letters do decipher to the same repeated trigraph, FMZ in this case, turn the wheels to these positions and decipher the text of the message. Note that the word ‘twice’ is in italics. As we shall see, it was this operational procedure, introduced to ensure correct receipt of the three-letter settings, that proved to be the Achilles heel of the Enigma. We begin by noting that
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R. F. Churchhouse Codes and ciphers
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Contents Preface ix 1 Introduction
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Cryptanalysisofalinearrecurrence 10
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Preface Virtually anyone who can re
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1 Introduction Some aspects of secu
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Not a very sophisticated method, pa
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change. As an example, anticipating
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Another form of encryption is the u
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and so is valid. On the other hand
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When the modulus is 10 only the num
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2 From Julius Caesar to simple subs
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Table 2.3 Shift Message 12 OUI 12 Y
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worry about but, on the other hand,
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then it is probably THE and the unk
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From Julius Caesar to simple substi
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Table 2.6 From Julius Caesar to sim
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and the key which provides the enci
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Further examination reveals that th
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ALTHOUGH I AM AN OLD MAN NIGHT IS G
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Polyalphabetic systems 35 messages
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Before leaving Vigenère try the fo
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Polyalphabetic systems 39 blocks of
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eginning at the top, but it is then
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first row above, for example, we fi
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Table 4.4 Key 3 1 5 2 4 1 2 3 4 5 6
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giving B G L D I N A F K E J O C H
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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
Page 83 and 84: If we generate the same sequence (m
Page 85 and 86: Stencil ciphers The example above i
Page 87 and 88: Book ciphers A spy must avoid arous
Page 89 and 90: Table 7.2 Encipher table for a book
Page 91 and 92: Letter frequencies in book ciphers
Page 93 and 94: If then we try subtracting THE from
Page 95 and 96: where row F (the row of the cipher
Page 97 and 98: Ciphers for spies 85 that were nece
Page 99 and 100: mistake can occur if the sender lea
Page 101 and 102: inks and ciphers were provided by h
Page 103 and 104: Example 7.6 Encipher the message AG
Page 105 and 106: Ciphers for spies 93 simply a ‘ra
Page 107 and 108: going into the mathematical criteri
Page 109 and 110: numbers, 0 to 31 inclusive, into bi
Page 111 and 112: Linear recurrences The sequences lo
Page 113 and 114: Having converted the characters of
Page 115 and 116: What about sequences of higher orde
Page 117 and 118: combining the keys of two or more l
Page 119 and 120: Example 8.3 (1) Use the mid-square
Page 121 and 122: Problem 8.3 Starting with U 0 �1
Page 123 and 124: The Enigma cipher machine 111 syste
Page 125 and 126: The Enigma cipher machine 113 Plate
Page 127 and 128: The Enigma cipher machine 115 Plate
Page 129 and 130: Figure 9.2. wheel. For example, if
Page 131 and 132: wheel and then through the three wh
Page 133: first day of usage and so the asses
Page 137 and 138: show how, in a typical encipherment
Page 139 and 140: interested reader can find it in [9
Page 141 and 142: great deal of data and many pages o
Page 143 and 144: It should be realised, of course, t
Page 145 and 146: 10 The Hagelin cipher machine Histo
Page 147 and 148: ut there was no cryptographic advan
Page 149 and 150: of each cipher period, to which sid
Page 151 and 152: value can be expected to occur two
Page 153 and 154: Since some cages are obviously very
Page 155 and 156: 2, there are 26! possible simple su
Page 157 and 158: The Hagelin cipher machine 145 Exam
Page 159 and 160: values are repeating at the appropr
Page 161 and 162: Overlapping will obviously affect t
Page 163 and 164: Table 10.6 The Hagelin cipher machi
Page 165 and 166: 11 Beyond the Enigma The SZ42: a pr
Page 167 and 168: Description of the SZ42 machine The
Page 169 and 170: P1 41 43 Z1 (4) the five bits from
Page 171 and 172: which is approximately 1.6�10 19
Page 173 and 174: 12 Public key cryptography Historic
Page 175 and 176: part of the story of what has been
Page 177 and 178: Public key cryptography 165 graphic
Page 179 and 180: (4) Now (k x ) y �(k y ) x �m x
Page 181 and 182: Public key cryptography 169 Despite
Page 183 and 184: 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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