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twenties. For most random selections of six letters from the alphabet, the selected letters will occur at either a higher or lower average frequency than the remaining letters (see Problems 14 and 15). For example, if the plaintext letter E is among the sixes, and the other sixes are all letters of average frequency, then the expected frequency of each of the ciphertext sixes will be higher than the expected frequency of each of the ciphertext twenties. Therefore, the ciphertext sixes can usually be determined simply from a frequency count of individual ciphertext letters. That is, the six highest-frequency or six lowest-frequency letters are most likely the sixes. Once the sixes have been isolated, it is relatively easy to determine the sixes permutations from a small amount of known plaintext, since in Purple there are only 25 sixes permutations. Using their knowledge of the Red cipher, Rowlett s team was quick to realize that Purple also employed a 6-20 split. They were then able to reconstruct the sixes permutations. But the twenties proved far more difficult to crack. The output permutations generated by a switch-based cipher, such as Purple, have certain identifiable characteristics. For example, consider the permutations in Table 2.6, which were generated by a process analogous to that used by Purple to generate its twenties permutation. In this example, there are three banks of permutations, as with the twenties permutations in Purple, but here each of the switches contains just three permutations (as opposed to 25 for Purple), giving a cycle length of 33 = 27. Table 2.6: Successive Permutations

2 n2

x2 2 1

Consider permutations P and Ps in Table 2.6. The element in position 0 4 is 4, while the 4 is in position 2 of P5. That is, the first element

02 2

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The AF Frame options control how the G11 achieves autofocus. The AF Frame offers two choices: Face AiAF FlexiZone When FlexiZone is selected, you can move the focus frame around after pressing the AF Frame/Erase button or hold this button down for a few seconds to return the AF Frame to its normal center position. Face AiAF is an intelligent autofocus system that detects and prioritizes faces in the scene. It automatically switches the G11 between a nine-area AF system and face detection. It s great to have the two most common AF modes on a single option rather than having to manually select face detection as users of the earlier G10 had to do.

d t 2 is the square of the value of a random variable that has the s2 n t-distribution with n 1 degrees of freedom when the null hypothesis H0 : 0 is true. So we may write the value of the likelihood ratio as c 1 2 3t 1n 12 4 s

It is easy to nd the value for the constant k that would lead to rejection of the null hypothesis H0. Since we reject H0 if k, this implies that small values of support the alternative hypothesis. Clearly, will be small when t 2 is large. So instead of specifying k we can spec2 ify a constant c and reject H0: c. The critical values of t would be the extreme 0 if t values, either positive or negative, and if we wish to control the type I error probability at , the critical region in terms of t would be t t

4 3 3 3 4 4 4 2 2 5

2 or, equivalently, we would reject H0: c t 2 2,n 1. Therefore, the likelihood 0 if t ratio test for H0: 0 versus H1: 0 is the familiar single-sample t-test. The procedure employed in this example to nd the critical region for the likelihood ratio is used often. That is, typically, we can manipulate to produce a condition that is equivalent to k, but one that is simpler to use. The likelihood ratio principle is a very general procedure. Most of the tests presented in this book that utilize the t, chi-square, and F-distributions for testing means and variances of normal distributions are likelihood ratio tests. The principle can also be used in cases where the observations are dependent, or even in cases where their distributions are different. However, the likelihood function can be very complicated in some of these situations. To use the likelihood principle we must specify the form of the distribution. Without such a speci cation, it is impossible to write the likelihood function, and so if we are unwilling to assume a particular probability distribution, the likelihood ratio principle cannot be used. This could lead to the use of the nonparametric test procedures discussed in 15.

9-5.2

of P is offset by 2 in Ps The element in position 1 of P is a 2 and the 2 4 4 appears in position 4 of Ps, so these have an offset of 3 Continuing in this manner, we see that the offsets, or differences, between the elements of P 4 and the corresponding elements of Ps are ( 2 , 3 , 1 , 4 , 1 ,3,O) As explained in Section 223, consecutive permutations from a rotor all share the same difference sequence The difference sequence for Ps and Ps: is different than between P and P5, so we can be sure that these three permutations were not 4 generated by consecutive shifts of a single rotor Now if we compute the difference sequence for permutations P and PCJ, 8 we see that it is the same as for P and Ps.

Small-Sample Tests on a Proportion (CD Only)

Tests on a proportion when the sample size n is small are based on the binomial distribution, not the normal approximation to the binomial. To illustrate, suppose we wish to test H0: p H1: p p0 p0

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