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Let X11, X12, p , X1n1 be a random sample from a normal population with mean 1 and variance 2, and let X21, X22, p , X2n2 be a random sample from a second normal pop1 ulation with mean 2 and variance 2. Assume that both normal populations are 2 2 independent. Let S 1 and S 2 be the sample variances. Then the ratio 2 F has an F distribution with n1 inator degrees of freedom.

Plaintext L I T T L E T I M E S 0 M U C H T 0 K N 0 W Ciphertext N B S M q T 9 Z I Y D D X K Y N E W J K Z R

1 denom-

This result is based on the fact that (n1 1)S 2/ 2 is a chi-square random variable with n1 1 1 1 degrees of freedom, that (n2 1)S 2 2 is a chi-square random variable with n2 1 degrees 2 2 of freedom, and that the two normal populations are independent. Clearly under the null 2 hypothesis H0: 2 S 2 S 2 has an Fn1 1,n2 1 distribution. This is the basis of 1 2 1 2 the ratio F0 the following test procedure.

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i-Contrast. Turns on i-Contrast to improve the tonal range and detail in the shadowed areas of the picture. AF Lock. Locks the focus in the Focus Frame. Display Off. Turns the LCD monitor display off quickly when its glow might be distracting to others.

Null hypothesis: Test statistic:

11. Suppose that the same Enigma rotors (in the same order) and reflector are used as in Problem 10, and the stecker has no cables connected. Solve for the initial rotor settings and recover the plaintext given the following ciphertext.

Alternative Hypotheses H1: H1: H1:

2 1 2 1 2 1 2 2 2 2 2 2

WWEZIZNBEOqDDDCJRZZTLRLGPIFYPHUSMBCAMNODVYSJWKTZEJCKP4YYN ZqKKJRQqHXLFCHHFRKDHHRTYILGGXXVBLTMPGCTUWPAIXOZOPKMNRXPMO AMSUTIFOWDFBNDNLWWLNRWMPWWGEZKJNH

EXAMPLE 10-11

Oxide layers on semiconductor wafers are etched in a mixture of gases to achieve the proper thickness. The variability in the thickness of these oxide layers is a critical characteristic of the wafer, and low variability is desirable for subsequent processing steps. Two different mixtures of gases are being studied to determine whether one is superior in reducing the variability of the oxide thickness. Twenty wafers are etched in each gas. The sample standard deviations of oxide thickness are s1 1.96 angstroms and s2 2.13 angstroms, respectively. Is there any evidence to indicate that either gas is preferable Use 0.05. The eight-step hypothesis-testing procedure may be applied to this problem as follows: 1. 2. 3. 4. 5. The parameters of interest are the variances of oxide thickness 2 and 2. We will 1 2 assume that oxide thickness is a normal random variable for both gas mixtures. H0: H1:

12. Develop a ciphertext only attack on the Enigma, assuming that all you know about the plaintext is that it is English. Analyze the work factor of your proposed attack and also estimate the minimum amount of ciphertext necessary for your attack to succeed. Assume that Enigma rotors, the rotor order, the movable ring positions, and the reflector are all known. Then you need to solve for the initial settings of the three rotors and the stecker. Hint: Since E is the most common letter in English, guess that the plaintext is EEEEEE.. . and use this noisy plaintext to solve for the rotor and stecker settings.

2 1 2 1 2 2 2 2

2 Since n1 n2 20, we will reject H0: 2 f0.025,19,19 2.53 or 1 2 if f0 if f0 f0.975,19,19 1 f0.025,19,19 1 2.53 0.40. 7. Computations: Since s2 (1.96)2 3.84 and s2 (2.13)2 4.54, the test statistic is 1 2

f0 8.

13. Suggest niodifications to the Enigma design that would make the attack discussed in Section 2.2 infeasible. Your objective is to make minor modifications to the design.

3.84 4.54

Conclusions: Since f0.975,19,19 0.40 f0 0.85 f0.025,19,19 2.53, we cannot 2 reject the null hypothesis H0: 2 1 2 at the 0.05 level of signi cance. Therefore, there is no strong evidence to indicate that either gas results in a smaller variance of oxide thickness.

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