WOR,LDWARI 1 CIPHERS

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Figure 2.2: Enigma diagram [142] letter L is illuminated on the lightboard. We use the following notation for the various permutations in the Enigma:

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R, = rightrnost rotor R,, = middle rotor Re = leftmost rotor T = reflector

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S = stecker.

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If plaintext letter

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encrypts to ciphertext letter y, from Figure 2.2, we have

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=S-~R-IR-~R-~ =

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T m e TRFR,RTS(X) ( R! R, R, s) T (R!R, R ~ () . s X)

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(2.1)

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If that is all there were to the Enigma, it would be nothing more than a glorified siniple substitution (or mono-alphabetic substitution) cipher, with the initial settings determining the permutation. However, each time a keyboard letter is typed, the rightmost rotor steps one position, and the other rotors step in an odometer-like fashion --almost [26, 631 .l That is, the middle

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The alniost is due to the mechanical system used to step the rotors, which causes the iriiddle rotor to occasioriitlly step twice in succession. Whenever a rotor steps, it causes the rot,or to its right to also step. Suppose that the rniddle rotor just stepped to the position that engages the ratchet, rnech;tnisrn that will cause the leftniost rotor to step when the next letter is typed. Then when the next, letter is typed, the left rotor will step, and this will also

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2.2 ENIGMA

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rotor steps once for each 26 steps of the right rotor and the left rotor steps once for each 26 steps of the middle rotor. The reflector can be viewed as a fixed rotor since it permutes the letters, but it does not rotate. The net effect is that the overall permutation changes with each letter typed. Due to the odometer effect, the permutations R,, R,, and Re vary with time, but T and S do not. Figure 2.3 illustrates the stepping of a single Engima rotor. This example shows the direction that the rotors step. Note that to the operator, the letters appear in alphabetical order.

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R S T U

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Figure 2.3: Enigma rotor. The Enigma is a substitution cipher where each letter is encrypted based on a permutation of the alphabet. But the Enigma is far from simple since whenever a letter is encrypted (or decrypted), the permutation changes. That is, the Enigma is a poly-alphabetic substitution cipher, with an enormous number of possible alphabets.

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Enigma Keyspace

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The cryptographically significant components of the Enigma cipher are the stecker, the three rotors, and the reflector. The Enigma key consists of the configuration of the cipher used to encrypt and decrypt a particular message. The variable settings that comprise the key are the following: 1. The choice of rotors.

2. The position of a movable ring on each of the two rightmost rotors. This ring allows the outer part of the rotor (labeled with the 26 letters) to rotate with respect to the inner part of the ring (where the actual permutation is wired).2 Rotating this ring shifts the permutation and

cause the middle rotor to step again. The middle rotor thereby steps twice in succession, violating the odometer effect. Note that this same ratcheting mechanism causes the right rotor to step whenever the middle rotor steps, but since the right rotor already steps for each letter typed, there is no noticeable effect on the right rotor. 'This is analogous to rotating the position of a car tire relative to the rim.

WORLD WARI I CIPHERS

the point at which the odometer effect occurs relative to the letters on the rotors.

3 . The initial position of each rotor

4. The number and plugging of the wires in the stecker.

5. The choice of reflector

As mentioned above, each rotor implements a permutation of the 26 letters of the alphabet. The movable rings can be set t,o any of the 26 positions corresponding to the letters. Each rotor is initially set to one of the 26 positions on the rotor, which are labeled with A through Z. The stecker is similar to an old-fashioned telephone switchboard, with 26 holes, each labeled with a letter of the alphabet. The stecker can have from 0 to 13 cables, where each cable connects a pair of letters. The reflector implements a permutation of the 26 letters, with the restriction that no letter can be permuted to itself, since this would cause a short circuit. Consequently, the reflector is equivalent to a stecker with 13 cables. Since there are three rotors, each containing a permutation of the 26 lettcrs, there are 26! . 26! . 26! M 2265

ways to select and place rotors in the machine. In addition, the number of ways to set the two movable rings--which determine when the odometer-like effects occurs-is 26 . 26 M 2 .*. The initial position of each of these rotors can be set to any one of 26 positions. so there are 26.26.26 = 214.1ways to initialize the rotors. However, this mimber should not be included in our count, since the different initial positions are all equivalent to some other rotor in some standard position. That is, if we assume that each rotor is initially set to, say, A then setting a particular rotor to, say, B is equivalent to some other rotor initially set to A. Consequently, the factor of 2265 obtained in the previous paragraph includes all rotors in all possible init ial positions. Finally, we must consider the stecker. Let, F ( p ) be the number of ways to plug p cables in the stecker. From Problem 2, we have

The values of F ( p ) are tabulated in Table 2.1. Summing the entries in Table 2.1, we find that there arc more than 248 possible stecker configurations. Note that maximum occurs with 11 cables and that F ( 1 0 ) 247 As mentioned above. the Enigma reflector is equivalent