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

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216 appendix So, for example, the points (1,�5) are integer points of the curve Y 2 �X 3 �2X�3 (mod 19). From any one or two points on the curve another may be constructed by using the tangent at the single point or the chord joining the two points. This tangent or chord meets the curve in a third point which must have rational co-ordinates and these rationals are convertible into integers in GF( p), the Galois field (mod p). So, for example, for the curve above with p�19, the equation of the tangent at the point (1, 5) is 2Y�X�9 and we find that this tangent meets the curve again at the point where X��7/4. This is equivalent to an integer value in GF(19); since 4 is the denominator of this fraction we must first find the integer n such that 4n�1 (mod 19). This gives n�5 since 20�1�19�1; hence �7/4�(�7)�5��35�3 (mod 19) and so the fraction �7/4 is equivalent to the integer 3 in GF(19). This gives 3 as the integer value of X and the corresponding value of Y, obtained from the tangent above, is 6. Since Y 2 �36 and X 3 �2X�3�27�6�3�36 we have verified that the points (3,�6) lie on the curve above. (We only need to show that they lie on the curve in GF(19); in fact they lie on the curve (mod p) for all p, but that is a fluke; this will not normally be the case.) Thus another integer point is found on the curve. Since all arithmetic is (mod p) there are only a finite number of possible points (X, Y) with integer values. It follows therefore that the construction method that gives new points must eventually terminate. If we start with a particular (integer) point Q(X, Y) on the curve we can generate a finite set, �Q�, of points which we denote by 2Q, 3Q, 4Q,... etc. (these are not to be confused with the points (2X, 2Y) etc). For example, starting with the point Q (1, 5) on the curve above we have just found the point, 2Q, generated from the tangent at Q. Continuing in this way we find that we are led to the points 2Q�(3,�6), 4Q�(10,�4), 8Q�(12,�8) and so on (for another example see [13.9]). If we are given a point R(X�, Y�) and asked to find if there is an integer n such that R�nQ, a point within the set �Q�, we will have a very difficult
Mathematical aspects 217 problem unless the prime, p, is not too large. If R is not in the set �Q� no such value of n will be found. The values of p that are used are likely to exceed 10 50 and the number of trials that will have to be made (except in some rare cases) is of the order of the square root of p, which makes the computational task beyond the power of even the most powerful computers. The way in which Q, R and n are used to provide a signature to a message is somewhat involved and will not be described here. A reasonably brief and readable account will be found in [13.9]. Anyone wishing to know more about this particular aspect of Galois theory should consult books on finite fields. Galois was killed in a duel at the age of 20 in 1832. Knowing that he was almost certain to be killed he stayed awake during the night before the duel and wrote a paper, which he hoped would be published, explaining his ideas. The paper was eventually published, in 1846. Further details of his life and work will be found in books on the history of mathematics, such as [13.13].
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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
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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
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
Page 185 and 186: number of tests increased by a fact
Page 187 and 188: In this case the remainder is 4, no
Page 189 and 190: key, d, we use the Euclidean Algori
Page 191 and 192: the decipher key, d, is used instea
Page 193 and 194: and and, since Table 13.1 n N�2 n
Page 195 and 196: The Data Encryption Standard (DES)
Page 197 and 198: the message were known to relate to
Page 199 and 200: whereas in block cipher systems, su
Page 201 and 202: (1) X precedes M with information w
Page 203 and 204: ways, since choosing to pair, say,
Page 205 and 206: For example; if someone claims that
Page 207 and 208: depth, mentioned in Chapter 3. If w
Page 209 and 210: M9 Combining two biased streams of
Page 211 and 212: and so 2�4�6�12 �(4095)�4
Page 213 and 214: Verification Let the recurrence be
Page 215 and 216: the lettersatsetting2ofthewheel,and
Page 217 and 218: M16 Probability of a ‘depth’ in
Page 219 and 220: (2) In how many ways can N be repre
Page 221 and 222: Chapter 13 M21 (Rate of increase of
Page 223 and 224: Multiply each of these by M: M, 2M,
Page 225 and 226: If x 3 �0 divide x 2 by x 3 to gi
Page 227: then k�[ log 2 n], where [z] deno
Page 231 and 232: RHAPSODY and SYMPHONY agree in posi
Page 233 and 234: 4.2 (Number of possible transpositi
Page 235 and 236: Table S.5 R H A P S O D Y B C E F G
Page 237 and 238: Chapter 8 8.1 (Recurrences of order
Page 239 and 240: columns in each case, are full of c
Page 241 and 242: Chapter 11 11.1 (Pin-setting errors
Page 243 and 244: [2.4] Moroney, M.J.: Facts from Fig
Page 245 and 246: [10.3] Almost any elementary book o
Page 247 and 248: Name index Adelman, L. 171, 234 And
Page 249 and 250: Subject index Abwehr Enigma 124, 13
Page 251 and 252: active pin 136 cage:‘good’ 141;
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