All rights reserved

Figure 12: Tree Before Branch 

Collapsing and Pruning [1]

Figure 13: Tree After Branch 

Collapsing and Pruning [1]

The results in Table 4 are derived using the following method. Given a known 

plaintext/ciphertext pair, we ran a certain number of tests on it. A random setting for the 

order and initial positions of the rotors was generated. Next, the first plaintext letter is 

encrypted before any rotors are stepped. Here, a random setting will survive the first step 

with a 1/26 probability. If the setting survives the first letter, the rotors can be stepped in 30 

different ways. Any stepping that is consistent for the second letter is saved. This is 

repeated for all the letters until one of two things happens. If the last letter is reached, then 

the path is valid and will be considered for Phase 2. The other option is that if at any letter, 

none of the 30 possible steppings for the cipher rotors yields a consistent path, that path is 

eliminated since it is a random path. Table 5 was generated using the same method. 

However, instead of a random setting, the causal setting was used. 

From Table 4, we can see that as the number of letters (steps) increases, the number of 

surviving settings decreases. This shows that the number of random settings can be reduced 

using a small amount of known plaintext and ciphertext. However, we need to note that 

although the number of surviving settings decreases, the number of surviving paths (non-

zero settings * average per non-zero) increases. For example, in Table 4, for a 30-letter 

message, we expect that 0.427% of the random settings (Non-Zero Settings) will survive, 

with each survivor expanding to an average of about 16.5 paths and a maximum of 84 

paths. For the causal case, Table 5 shows that for a 30-letter message, we expect 29.6 paths 

with the 10,000 cases tests. We also expect a minimum of one and a maximum of 151 

consistent paths.

21

---

Steps 

(Letters)

Tests

Non-Zero 

Settings

Average Per 

Non-Zero

Maximum

10

10

5

763

6.5

27

20

10

5

516

11.8

56

30

10

5

427

16.5

84

40

10

5

324

20.8

105

50

10

5

290

28.4

194

60

10

5

275

38.8

163

70

10

5

269

47.1

415

80

10

5

212

71.3

524

90

10

5

216 

77.6

486

100

10

5

203

100.5

1005