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Figure 2.22: Control and index permutations. In Figure 2.22, the control permutation changes with each letter typed, but the index permutation is fixed for the entire message. Since the control permutations are changing, we model their output as uniformly random, that is, we assume that each of the combinations of output letters is equally likely at each step. Then, due to the way that the control rotor outputs are ORed together, the inputs to the index permutation are not uniform. For example, input 8 will be active much more often than inputs 1, 2, or 9 and input 0 is never active. The outputs of the index rotors are ORed in pairs, and the results determine the cipher rotor stepping. Therefore, if we have sufficient information on the frequency of the stepping of individual cipher rotors-which is available from known plaintext using the putative cipher rotor settings obtained in the primary phase of the attack-we can assign probabilities to the index permutations. For example, suppose that the index permutation is 5479381026, that is, input 0 is mapped to output 5, input 1 is mapped to output 4, and so on. Then by considering the pairs of outputs that determine the cipher rotor stepping, we can see that some cipher rotors will step more often than others. The data in Table 2.9 makes this more precise. For example, cipher rotor C d steps if either output 1 or 2 (or both) of the index permutation is active. For the index permutation in Table 2.9, inputs 6 and 8 are mapped to outputs 1 and 2, respectively. Inputs 6 and 8 of the index permutation correspond to outputs 6 and 8 of the current control permutation, and at

(b) What is the P-value for this test (c) Find a 95% con dence interval on the difference in the two means 10-25 The melting points of two alloys used in formulating solder were investigated by melting 21 samples of each material The sample mean and standard deviation for alloy 1 was x1 420 F and s1 4 F, while for alloy 2 they were.

2.9 The Custom Display menu is where you set the amount of information that is displayed when reviewing the picture in Playback mode. This gure shows the default display view.

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x2 426 F, and s2 3 F Do the sample data support the claim that both alloys have the same melting point Use 005 and assume that both populations are normally distributed and have the same standard deviation Find the P-value for the test 10-26 Referring to the melting point experiment in Exercise 10-25, suppose that the true mean difference in melting points is 3 F How large a sample would be required to detect this difference using an 005 level test with probability at least 09 Use 1 4 as an initial esti2 mate of the common standard deviation 10-27 Two companies manufacture a rubber material intended for use in an automotive application The part will be subjected to abrasive wear in the eld application, so we decide to compare the material produced by each company in a test.

(246)

Twenty- ve samples of material from each company are tested in an abrasion test, and the amount of wear after 1000 cycles is observed For company 1, the sample mean and standard deviation of wear are x1 20 milligrams /1000 cycles and s1 2 milligrams/1000 cycles, while for company 2 we obtain x2 15 milligrams/1000 cycles and s2 8 milligrams/1000 cycles (a) Do the data support the claim that the two companies produce material with different mean wear Use 005, and assume each population is normally distributed but that their variances are not equal (b) What is the P-value for this test (c) Do the data support a claim that the material from company 1 has higher mean wear than the material from company 2 Use the same assumptions as in part (a) 10-28.

The thickness of a plastic lm (in mils) on a substrate material is thought to be in uenced by the temperature at which the coating is applied A completely randomized experiment is carried out Eleven substrates are coated at 125 F, resulting in a sample mean coating thickness of x1 1035 and a sample standard deviation of s1 102 Another 13 substrates are coated at 150 F, for which x2 997 and s2 201 are observed It was originally suspected that raising the process temperature would reduce mean coating thickness Do the data support this claim Use 001 and assume that the two population standard deviations are not equal Calculate an approximate P-value for this test 10-29 Reconsider the coating thickness experiment in Exercise 10-28.

How could you have answered the question posed regarding the effect of temperature on coating thickness by using a con dence interval Explain your answer 10-30 Reconsider the abrasive wear test in Exercise 10-27 Construct a con dence interval that will address the questions in parts (a) and (c) in that exercise 10-31 The overall distance traveled by a golf ball is tested by hitting the ball with Iron Byron, a mechanical golfer with a swing that is said to emulate the legendary champion, Byron Nelson Ten randomly selected balls of two different brands are tested and the overall distance measured The data follow:.

least one of these outputs is active if one or more of the 10 letters L, M, N, 0, U, V, W, X, Y, or Z, which are connected t o these outputs, are active. On the other hand, CZ steps only when output A of the control permutation is active. As a result, C will step much more often than Cz. 4 Table 2.9: Index Permutation 5479381026 Cipher Rotor

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