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Sigaba was developed by American cryptographers-including Friedman and Rowlett--prior to World War II.6 As far as is known, no successful attack on Sigaba was every conducted during its service lifetime. During WWII, the Germans are said to have quit collecting Sigaba intercepts since they deemed the problem hopeless [ 2 3 ] . In this section we first give a detailed description of the cipher. Sigaba is a rotor machine, but its inner workings are far more complex than thc Enigma. Then we consider the size of the Sigaba keyspace in some detail, followed by an outline of an attack on the machine. As we describe it, the attack is impractical-and it would have been even more so using WWII technology. Some of the problems at the end of this chapter point to improvements in the attack, which make it far more practical, but still beyond the realm of a realistic WWII-era attack. This attack highlights the crucial features of the Sigaba that make it so much more secure than Enigma or Purple.

(10-1)

This result will be used to form tests of hypotheses and con dence intervals on 1 2. Essentially, we may think of 1 as a parameter , and its estimator is X1 X2 2 2 2 with variance 2 1 n1 2 n2. If 0 is the null hypothesis value speci ed for , the test . Notice how similar this is to the test statistic for a single mean statistic will be 1 02 used in Equation 9-8 of 9.

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There were several variants of the basic Sigaba design, and to further muddy the water, different branches of the military used different names for the same machine. The Sigaba machine in Figure 2.13 is thought to be equivalent to the CSP-889 (used by the Navy) and the Converter M-134C or Sigaba (different mines, but the same device, used by the Army). In addition, the name ECM Mark I1 wa.s used during the development of the machine that would become Sigaba. Here, we stick with the name Sigaba. The Sigaba cipher includes a typewriter keyboard for entering the plaintext (or ciphertext), and a.n output device for printing t.he corresponding

10-2.1

We now consider hypothesis testing on the difference in the means 1 2 of two normal populations. Suppose that we are interested in testing that the difference in means 1 2 is equal to a speci ed value 0. Thus, the null hypothesis will be stated as H0: 1 2 0. Obviously, in many cases, we will specify 0 0 so that we are testing the equality of two means (i.e., H0: 1 2). The appropriate test statistic would be found by replacing 1 2 in Equation 10-1 by 0, and this test statistic would have a standard normal distribution under H0. That is, the standard normal distribution is the reference distribution for the test statistic. Suppose that the alternative hypothesis is H1: 1 x2 2 0. Now, a sample value of x1 that is considerably different from 0 is evidence that H1 is true. Because Z0 has the N(0, 1)

distribution when H0 is true, we would take z 2 and z 2 as the boundaries of the critical region just as we did in the single-sample hypothesis-testing problem of Section 9-2.1. This would give a test with level of signi cance . Critical regions for the one-sided alternatives would be located similarly. Formally, we summarize these results below.

"231.

Null hypothesis: Test statistic:

0 0 2 2

Figure 2.13: A Sigaba machine [105]. ciphertext (or plaintext). Like the Enigma, Sigaba is a rotor machine, but there arc several important differences between the two. Cryptographically, the most significant differences are that whereas Enigma uses three rotors, Sigaba employs five rotors to permute the letters, and whereas Enigma rotors step like an odometer, the Sigaba cipher rotor motion is controlled by a set of ten additional rotors, for a total of 15 rotors. In effect, it is as if the motion of the Sigaba encryption rotors is controlled by another rotor cipher machine. This causes the Sigaba rotors to step irregularly, which is a major improvement over the Enigma and other regularly stepping rotor machines. Sigaba also lacks the Enigma s reflector and stecker. The use of irregularly stepping rotors and the lack of a reflector and Enigma-like stecker make Sigaba a stronger cipher than Enigma (the attack in Section 2.2 points to some of the weaknesses of the Enigma design). The Sigaba rotors are illustrated in Figure 2.14.

(10-2)

Alternative Hypotheses H 1: H 1: H 1:

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Figure 2.14: Sigaba rotors [126]. The fifteen Sigaba rotors consist of five cipher rotors, five control rotors, and five in,dez rotors, where the cipher rotors permute the input letters and the othcr two banks of rotors drive the cipher rotors. The cipher and control rotors are interchangeable, and these rotors are also designed so they can

1 1 1

2 2 2

0 0 0

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