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We refer to this unusual feature of Purple as the 6-20 split. The input plugboard permutation enables any six letters to be connected to the internal sixes in (2.4). If, after passing through the input plugboard, the resulting letter is permuted to one of the sixes, it then passes through a permutation, denoted by S in Figure 2.9, before being permuted by the output plugboard. The resulting letter is then sent to the output device. If, on the other hand, the plugboard permutation yields a twenties letter, the letter passes through three permutations, denoted L , M , and R in Figure 2.9, before being permuted by the output plugboard. Again, the resulting letter is then sent to the output device. As used by the Japanese, the output plugboard and input plugboard permutations were always the same. In fact, the Purple simulators built by the Allies used only a single physical plugboard, which could not have accurately modeled Purple if the input and output permutations were different. As can be seen in Figure 2.9, internally the sixes are permuted to sixes and the twenties are permuted to twenties. This was a major flaw that was carried over from a predecessor of Purple, a cipher known as Red. Why it was carried over is not clear, since there was no inherent limitation of Purple that necessitated such a split. In fact, two variants of Purple were used by the Japanese (Coral and Jade) that did not employ the 6-20 split. As discussed below, the 6-20 split was a crucial weakness that was exploited by the cryptanalysts who broke Purple. Each of S , L , M , and R cycle thorough a series of 25 fixed permutations, with S stepping once for each letter typed, and exactly one of L , M , or R stepping for each letter-which of these steps is determined by S as will be described shortly. Each of the S permutations is a permutation of the six vowels in (2.4), while each of t.he L , M , and R permutations is a permutation of the twenty consonants in (2.5). Like Enigma, Purple is a poly-alphabetic substitution cipher. However, the mechanisms employed by the two ciphers are completely different. Recall that Enigma is a rotor machine, where each rotor has a single hardwired permutation, and as a result of the rotor motion, the overall Enigma permutation changes with each letter. In contrast, Purple uses switches, where each step of a switch changes to a different permutation. That is, instead
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Figure 2.9: Purple encryption. of using permutations wired to rotors-which only requires one permutation per rotor-each step of Purple s S , L , M , and R switches t o a different, unrelated, hardwired permutation. Consequently, we refer to S , L , M , and R as switches and Purple as a stepping switch machine. While the distinction between rotors and switches might seem relatively minor, it is actually a major difference. For one thing, a stepping switch machine like Purple is inherently more complex and difficult to engineer than a rotor machine. From a cryptanlytic point of view, a rotor machine provides an elegant way to generate a large number of permutations, while a comparable stepping switch machine must be far more complex. There are also significant differences between the types of permutations that can be generated by Purple and Enigma, as discussed below and in Problem 16. The Purple encryption formula depending on whether the letter being encrypted corresponds to a six or twenty . Let x be the given input letter and let y be the corresponding output letter. Then we can denote the encryption by
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where PI is the input plugboard permutation (when going from the input keyboard to the switches), PO is the output plugboard permutation (from
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