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Generally, if there are k factors and they each have two levels, a factorial experimental design will require 2k runs. For example, with k 4, the 24 design in Fig. S1-3 requires 16 tests. Clearly, as the number of factors increases, the number of trials required in a factorial experiment increases rapidly; for instance, eight factors each at two levels would require 256 trials. This quickly becomes unfeasible from the viewpoint of time and other resources. Fortunately, when there are four to ve or more factors, it is usually unnecessary to test all possible combinations of factor levels. A fractional factorial experiment is a variation of the basic factorial arrangement in which only a subset of the factor combinations are actually tested. Figure S1-4 shows a fractional factorial experimental design for the four-factor version of the connector experiment. The circled test combinations in this gure are the only test combinations that need to be run. This experimental design requires only 8 runs instead of the original 16; consequently it would be called a one-half fraction. This is an excellent experimental design in which to study all four factors. It will provide good information about the individual effects of the four factors and some information about how these factors interact. Factorial and fractional factorial experiments are used extensively by engineers and scientists in industrial research and development, where new technology, products, and processes are designed and developed and where existing products and processes are improved. Since so much engineering work involves testing and experimentation, it is essential that all engineers understand the basic principles of planning ef cient and effective experiments. We discuss these principles in 13. 14 concentrates on the factorial and fractional factorials that we have introduced here.

That is, at each step, the number of possible paths increases, which seems to be the opposite of what we would like to see occur Nevertheless, we can obtain useful information from this process, as outlined below, but first we consider a simple example Suppose we have selected five of the ten available candidate rotors as cipher rotors, and we have placed them in a specified order and selected their orientations This, together with the initial positions of the selected cipher rotors constitutes a putative setting Consider, for example, the case where these cipher rotors are set to AAAAA, that is, each of the five cipher rotors is initialized to A Then we know the putative encryption pcrmutation and if it does riot encrypt the first known plaintext to the first known ciphertext, this cipher rotor setting is not causal and we can discard it.

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Minitab will also perform power and sample size calculations for the two-sample t-test (equal variances). The output from Example 10-7 is as follows:

Power and Sample Size 2-Sample t Test Testing mean 1 mean 2 (versus not ) Calculating power for mean 1 mean 2 difference Alpha 0.05 Sigma 2.7 Sample Target Actual Difference Size Power Power 4 10 0.8500 0.8793

This immediately reduces the number of candidates by a factor of 26, since there is only a 1/26 chance of a letter matching at random Suppose that the first letter does match Then we must try all 30 possible steps of the five cipher rotors arid save any of these that encrypt the second plaintcxt letter to the second ciphertext letter Since we make 30 comparisons, The expected number of matches that occur at random is, as mentioned above, 30/26 M 1154 An example of this process is illustrated in Figure 220 In this example, the first letter is consistent with the initial setting AAAAA, and the first three letters are consistent with each of the given paths Note that at the third plaintext letter in Figure 220 we have two paths ending at BBBBA.

10-3.4

Detailed. You can choose to display the last image taken with shooting information shown alongside the image. Focus Check. This setting keeps the image displayed on the LCD until you change modes or go back to shooting. Leaving the image displayed definitely taxes the battery life, so be sure to use the Focus Check option sparingly.

2 2 Case 1: 2 1 2 To develop the con dence interval for the difference in means are equal, note that the distribution of the statistic

1 n2

Since the next step depends only on the current cipher rotor settings, and since we are only interested in the initial setting (not the entire path), we can merge these paths as illustrated in Figure 221 This merging is useful since it effectively reduces the number of paths under consideration, while riot degrading the success of this phase of the attack In the random case, the analysis above holds, so that at each step we expect an increase by a factor of 1154 (before merging) In contrast, the causal case provides a slightly higher increase on average, since we are assured one causal match, with the remaining elements matching as in the random case; see Problem 31 This gives us a method to distinguish random from.

(10-18)

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