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Table 2.4: Medium Switch Steps and Slow Switch Steps
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ple stepping method is t h a t the period length of the twenties is maximized, whereas in Enigma, the period length is reduced slightly, since more than one rotor steps a t the rollover points. However, the additional complexity of the Purple stepping more t h a n offsets any possible advantage due to the greater period length.
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Now we consider the size of the Purple keyspace. Suppose for a moment that the permutations on the switches were selectable. Then the Purple keyspace would be enormous-just the selection of the S, L , M , and R permutations would give 2237. 26628 = 26865 ( 6 ! ) 2 5 . (26!)7~5
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However, givcn the design of Purple, it was not possible t o change the hardwired Permutations, so we compute the keyspace assuming that the permutations are fixed. Under this restriction, the Purple key consists of the following: 1. Initial settings of the switches S. L , M , and R: There are 254 ways t o initialize thesc switches.
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2. Choose fast, medium and slow switches from L , M , and R: Since these can tie selected in any order, there are 6 = 22.6 combinations.
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3 . Select input and output plugboard permutations: If the input and outjput plugboards can be chosen independently, there arc (26!) = 21763.8 combinations.
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Therefore, the theoret,ical keyspace for Purple is approximately of size 21g8, equivalent to a 198-bit key. However, all but a factor of 221.2 of this comes from the plugboard settings. In fact, the Japanese always used the same plugboard for both input and output, which immediately reduces the keyspace to 2109.6 The Purple plugboard is a very weak cryptographic element and, consequently, the effective keyspace is little more than 221. However, this presupposes that the switch permutations are known to the cryptanalyst, which was not the case when Rowlett and his team began their analysis of Purple. Consequently, the real cryptanalytic challenge for the Allies was to understand the inner workings of Purple and to recover the internal permutations-all without ever having seen the machine. Once this was accomplished, the actual decryption would not be difficult. In fact, the Japanese only used a very small fraction of the (already small) effective keyspace. Once the machine had been diagnosed, and a relatively simple message indicator (MI) system had been broken, the Allies could decrypt messages as quickly as-and sometimes faster than-the Japanese. In effect, maintaining the secret design of Purple was essential to maintain its security. It is hard to imagine a more striking violation of Kerckhoffs Principle. The fact that the Allies were able to break Purple without ever laying hands on an actual machine argues strongly for the wisdom of Kerckhoffs. In the next section we consider the diagnosis of Purple. This was the crucial cryptanalytic challenge in breaking Purple.
Purple Diagnosis
No Purple cipher machine was available to Frank Rowlett, the American cryptanalyst most closely associated with the cryptanalysis of Purple. This meant that he first had to diagnose the machine before he could hope to break it. That is, he had to reconstruct the inner workings of the machine using the only available information, namely, intercepted ciphertext and knowledge of prior Japanese cryptosystems. In some cases, known plaintext was also available, and this would prove crucial to the diagnostic effort. Recall that ciphertext messages are said to be in depth if they are encrypted using the same key. If n, messages are all encrypted with the same key, then we refer to this as a depth of n legs. It is also possible to have an offset depth, where the messages do not begin on the same key, but from some point onward the messages go into depth. Suppose that the matched plaintext and ciphertext message snippets in Table 2.5 were generated by a cipher that uses a time-varying permutation of the alphabet. The Enigma cipher, for example, works in this manner. Purple is slightly more complicated due to the 6-20 split, but we ignore this issue for now.