WORLD W A RII CIPHERS

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permutation yields five pairs of outputs, one pair for each C, in Figure 2.22. These five output pairs correspond to five input pairs. Since the index permutation is a permutation, the five input pairs must include each of 0 through 9 exactly once, and all 26 letters from the output control rotors must appear. Count the number of valid sets of five pairs, using Table 2.10. b. How many distinct groupings are there in a., where groupings are considered distinct only if the numbers are different, not just the ordering

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28. Consider the Sigaba attack discussed in the text. There are = 45 choices for the pairs of index permutation inputs that get mapped to the C,. As discussed in the text and in Problem 27, the probability that C, is active (and, therefore, rotor i steps) is determined by the number of control rotor letters that feed into the pair of outputs that determine C,. The number of letters that can feed into a C, is in the range of 1 , 2 , 3 , .. . , 11. a. For each value k = 1 , 2 , 3 , .. . ,11, determine the stepping percentage for Ci when it is connected to exactly k control rotor letters. These percentages will sum to much more than one, since more than one rotor generally steps. Hint: Assume all outputs of the control rotors are equally likely. Generate all of these equally likely outputs, map these to the corresponding index perniutation inputs, and count the number of times that at least one element of each of the pairs in Table 2.10 occurs. Use this information to answer the question.

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b. Suppose that only one cipher rotor, say, i steps. What do you irnmediatcly know about the index permutation inputs that are combined to form Ci

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c. Suppose that exactly two cipher rotors, say, i and j step. What do you immediately know about the index permutation inputs that are combined to form Ci and Cj

29. For each letter encrypted by the Sigaba cipher machine, all five cipher rotors step and three of the five control rotors step. The two remaining control rotors and all five index rotors do not step. Since the cipher and control rotors each permute 26 letters, the maximum possible period for Sigaba is 268. However, in [146] it is claimed that the Sigaba cipher has a period of just 264, regardless of the initial settings. Write a program to determine the period of Sigaba for a given key. Use your program to verify that the Sigaba period is 264 for each of 100 randomly selected keys, or find a key that does not yield a period of 264.

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2.6 PROBLEMS

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30. Suppose that we create a new cipher, Sigaba Lite (SL), which is similar to Sigaba with the exception that SL uses only four cipher rotors. As with Sigaba, in SL from one to four (inclusive) of the cipher rotors steps with each letter encrypted. All other components of SL are the same as those of Sigaba. Also, SL is equipped with nine cipher and control rotors, that is, the number of rotors that will fit in the device (as is the case for Sigaba). Show that if sufficient known plaintext is available, then there is an attack on SL requiring work of about 2*' or less. Hint: Mimic the Sigaba attack outlined in this chapter.

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31. For the primary phase of the Sigaba attack:

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a. Determine the expected number of consistent paths (without merging) in the random case and the causal case.

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b. Determine the expected number of consistent paths (with merging) in the random case and the causal case.

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32. Consider the Sigaba attack discussed in this chapter. a. Using the results in Table 2.7, estimate the number of survivors from the primary phase, assuming that 40 know plaintext letters are available and paths are merged, but otherwise all surviving paths are saved.

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b. What is the work factor for the primary phase using the method in part a c. What is the total work, including the secondary phase, for the attack as outlined in this problem

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