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Therefore, the theoret,ical keyspace for Purple is approximately of size 21g8, equivalent to a 198-bit key. However, all but a factor of 221.2 of this comes from the plugboard settings. In fact, the Japanese always used the same plugboard for both input and output, which immediately reduces the keyspace to 2109.6 The Purple plugboard is a very weak cryptographic element and, consequently, the effective keyspace is little more than 221. However, this presupposes that the switch permutations are known to the cryptanalyst, which was not the case when Rowlett and his team began their analysis of Purple. Consequently, the real cryptanalytic challenge for the Allies was to understand the inner workings of Purple and to recover the internal permutations-all without ever having seen the machine. Once this was accomplished, the actual decryption would not be difficult. In fact, the Japanese only used a very small fraction of the (already small) effective keyspace. Once the machine had been diagnosed, and a relatively simple message indicator (MI) system had been broken, the Allies could decrypt messages as quickly as-and sometimes faster than-the Japanese. In effect, maintaining the secret design of Purple was essential to maintain its security. It is hard to imagine a more striking violation of Kerckhoffs Principle. The fact that the Allies were able to break Purple without ever laying hands on an actual machine argues strongly for the wisdom of Kerckhoffs. In the next section we consider the diagnosis of Purple. This was the crucial cryptanalytic challenge in breaking Purple.

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9-3.4 Likelihood Ratio Approach to Development of Test Procedures (CD Only)

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Hypothesis testing is one of the most important techniques of statistical inference. Throughout this book we present many applications of hypothesis testing. While we have emphasized a heuristic development, many of these hypothesis-testing procedures can be developed using a general principle called the likelihood ratio principle. Tests developed by this method often turn out to be best test procedures in the sense that they minimize the type II error probability among all tests that have the same type I error probability . The likelihood ratio principle is easy to illustrate. Suppose that the random variable X has a probability distribution that is described by an unknown parameter , say, f (x, ). We wish to test the hypothesis H0: is in 0 versus H1: is in 1, where 0 and 1 are disjoint sets of values (such as H0: 0 versus H1: 0). Let X1, X2, p , Xn be the observations in a random sample. The joint distribution of these sample observations is f 1x1, x2, p , xn, 2 f 1x1, 2 f 1x2, 2 p f 1xn, 2

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No Purple cipher machine was available to Frank Rowlett, the American cryptanalyst most closely associated with the cryptanalysis of Purple. This meant that he first had to diagnose the machine before he could hope to break it. That is, he had to reconstruct the inner workings of the machine using the only available information, namely, intercepted ciphertext and knowledge of prior Japanese cryptosystems. In some cases, known plaintext was also available, and this would prove crucial to the diagnostic effort. Recall that ciphertext messages are said to be in depth if they are encrypted using the same key. If n, messages are all encrypted with the same key, then we refer to this as a depth of n legs. It is also possible to have an offset depth, where the messages do not begin on the same key, but from some point onward the messages go into depth. Suppose that the matched plaintext and ciphertext message snippets in Table 2.5 were generated by a cipher that uses a time-varying permutation of the alphabet. The Enigma cipher, for example, works in this manner. Purple is slightly more complicated due to the 6-20 split, but we ignore this issue for now.

Recall from our discussion of maximum likelihood estimation in 7 that the likelihood function, say L( ), is just this joint distribution considered as a function of the parameter . The likelihood ratio principle for test construction consists of the following steps: 1. Find the largest value of the likelihood for any in 0. This is done by nding the maximum likelihood estimator of restricted to values within 0 and by substituting this value of back into the likelihood function. This results in a value of the likelihood function that we will call L( 0). Find the largest value of the likelihood for any in lihood function L( 1). Form the ratio L1 L1

02 12

0 1 2 3 4 5 6 7 8 P E A R L H A R B J K F H N V P G P T 0 R A T 0 R A T K P L T H D W V J M I D W A Y I S L X H T E S A G N K

2. 3.

Call this the value of the like-

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