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To reiterate, the crucial observation here is that once we specify the rotor settings, all permutations Po, P I ,Pl,. . . and P i 1 ,P;', P;', . . . are known. Then if we substitute a putative value for S ( E ) , we can imniediately check the validity of both cycle equations. For an incorrect guess of S ( E ) (or incorrect rotor settings) there is a 1/26 chance any given cycle will hold true. But with two cycles, there is only a ( 1/26)2 chance that both cycle equations will hold true. Consequently, with two cycles involving S ( E ) , we can reduce the number
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2.2 ENIGMA
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of possible initial rotor settings by a factor of 26. Since there are about 230 rotor settings, after completing the attack in Table 2.3, we expect to have about 230/26 M 225.3 putative rotor settings remaining. The attack in Table 2.3 can be extended to more than two cycles, in which case we obtain a proportionally greater reduction in the number of surviving keys. With a sufficient number of cycles, we can uniquely identify the initial rotor settings. In fact, with n pairs of cycles we expect to reduce the number of possible keys by a factor of 26 . Therefore, with a sufficient number of cycles, we can recover the key. Amazingly, by recovering the initial rotor settings in this manner, stecker values are also recovered-essentially for free. However, any stecker values that do not contribute to a cycle will remain unknown, but once the rotor settings have been determined, the remaining unknown stecker settings are easy to determine; see Problem 9. It is interesting to note that in spite of an enormous number of possible settings, the stecker contributes little to the security of the Enigma.
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More Secure Enigma
Several of the design features of the Enigma conspired to create weaknesses that were exploited by the Allies. For example, the fact that the right rotor is the fast rotor (i.e., the right rotor steps with each letter typed) was said to be crucial in one particular attack. If instead, the left rotor had stepped with each letter-and the designers of the Enigma could just as easily have chosen any of the rotors as the fast rotor--this particular attack would not have succeeded [23]. The attack described in this section would still work, regardless of which rotors are fast, medium, and slow. However, in spite of it being extremely efficient by modern standards, the attack presented here would have been impractical using 1940s technology. The practical attacks of World War I1 required that the cryptanalyst reduce the number of cases to be tested to a small number. Many clever techniques were developed to squeeze as much information as possible from the messages before attempting an attack. In addition, much effort was expended finding suitable cribs (known plaintext) since all of the practical attacks required known plaintext.
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Is there any relatively simple modification to the Enigma that would prevent the attack discussed in this section We leave this as an exercise (Problem 13). It is important to note that our attack exploits the fact that the rotors can, in a sense, be isolated from the stecker. Any modifications designed to prevent this attack must take this fact into account.
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WORLDWAR, 1 CIPHERS 1
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The .Japanese Government regrets to have to notify hereby the American Government that in view of the attitude of the Arnerican Government it cannot hut consider that it is impossible to reach ari agreement through further negotiations. - From the 14-part message [72]
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The World War I1 era Japanese cipher machine Angooki Tuipu B was known to Allied cryptanalysts as Purple--due to the color of the binders used to hold information on the cipher. The Japanese used Purple to encrypt diplomatic traffic and it was in use from the late 1930s until the end of the war. Contrary to some reports, Purple was not used to encrypt tactical Naval commuriications-that was the role of the JN-25 cipher [ 2 3 ] . In particular, it was JN-25 decrypts (not Purple decrypts, as is sometimes claimed [75]) that provided the information enabling American pilots to shoot down Admiral Yamamoto s airplane in 1943.
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