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

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96 chapter 8 the spin, the sequence may be biased. It is, in addition, a very slow way of producing a random sequence that would presumably only be used if no other method were available. It is said that a prisoner of war carried out such a procedure for many thousand spins, to keep himself occupied, and analysed the resulting sequence with a variety of tests. Throwing dice A less laborious procedure can be based on throwing two dice. The dice must be distinguishable one from the other; let us assume that one is coloured red and the other is coloured blue. Throw both dice and compute the number then 6�(number on the red die)�(number on the blue die)�7 (i) reject the number if it exceeds 29, (ii) write down the remainder when the number is divided by 10. The resultant sequence of decimal digits should be random. The rather odd-looking rules are necessary because the faces of the dice are numbered 1 to 6 and not 0 to 5 and because there are 36 combinations which can be produced. Consequently values from 0 to 35 inclusive for the number can be produced and so we must reject any number above 29, in order to ensure that all the digits from 0 to 9 have an equal chance of appearing. More than two dice may be used and then more than one random digit can be generated at each throw. With four dice, for example, there are 1296 possible outcomes and if we colour the dice red, blue, green and white and compute the number 216�red�36�blue�6�green�white�259 and reject any number above 999 we can take the three-digit number so obtained as the next three digits of the random decimal sequence. There are many possible variations on this type of approach; for example, the two dice could be replaced by a roulette wheel, which has 37 sectors numbered 0 to 36. Sectors 30 to 36 would be ignored and the second digit of the ‘winning’ sector would provide the next random decimal digit. This is obviously rather wasteful and a more efficient use in this case would be to ignore sectors 32 to 36 and convert the other
numbers, 0 to 31 inclusive, into binary, thus providing five binary digits each time. Binary digits are commonly known as bits and are frequently referred to in that way. Binary keys are frequently used in cryptography. Not only have they the great merit that non-carrying addition (mod 2) is particularly simple and identical to non-carrying subtraction (mod 2), which makes encipherment and decipherment the same, but also (mod 2) arithmetic is very easy to simulate electronically, and so is particularly suitable both for cipher machines and for simulators on computers. Lottery type draws The system used to draw lottery (or bingo) numbers could be used provided it was modified so that a number which has been drawn is immediately returned to the pool. Thus 100 balls numbered 00 to 99 are spun in a barrel and selected one by one, each number selected is noted and provides two decimal digits for the table of random numbers. The selected ball must be put back in the barrel for otherwise it couldn’t be drawn again and each page of 100 two-digit decimal numbers would contain each number once, and only once, and so would not be random. A typical page of 100 two-digit random numbers would be expected to contain some numbers three, or even four, times whilst between 30 and 40 numbers might not occur at all. (For an explanation see M8.) Cosmic rays Cosmic rays are produced when particles from the Sun enter the Earth’s atmosphere and generate cascades of other particles by collisions and so provide a ‘natural’ source of (presumably) random events. If we were to install ten detectors, such as Geiger counters, numbered 0 to 9, in a room and record the order in which the detectors ‘fire’ we would obtain a genuinely unpredictable decimal sequence. Care would have to be taken that when a detector has ‘fired’ no other event is recorded until that detector has had time to ‘recover’, for otherwise there is likely to be a deficiency of ‘doublets’ such as 00, 11 etc. in the resultant sequence. Amplifier noise Producing random numbers and letters 97 Noise in electrical circuits is usually regarded as a problem, but it can also be turned to good use in cryptography. The noise can be converted into a
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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
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
Page 63 and 64: The plaintext digraphs are now sepa
Page 65 and 66: ox increases. By enumerating the po
Page 67 and 68: Example 5.1 HAPPY BIRTHDAY encipher
Page 69 and 70: Encryption The ‘plaintext’ is A
Page 71 and 72: plaintext digraphs according to som
Page 73 and 74: Cryptanalytic aspects of Playfair P
Page 75 and 76: OURXSITUATI ONXISXDESPE RATEXSENDXS
Page 77 and 78: the alphabet are represented by up
Page 79 and 80: From a cryptographic point of view
Page 81 and 82: 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: going into the mathematical criteri
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 and 134: first day of usage and so the asses
Page 135 and 136: The Enigma cipher machine 123 The m
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
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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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