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# Attacking the Indicators

As is recounted in the several books and articles recounting the story of the Enigma, the Germans used two very dangerous indicator systems with it.

First, they started the Enigma, for a given day, with a fixed secret setting, called the ground setting or Grundstellung. At this setting, the actual starting rotor setting was repeated twice and enciphered.

Then, still before the war started, while only the Poles were breaking the Enigma, a change was made. Instead of a secret ground setting, a starting point was picked by the user at random, and started the message in the clear. Then the starting rotor setting used for the message itself was again repeated twice and enciphered.

In both cases, the relationship between the two repetitions of the rotor start position gave too much away.

### The Common Ground Setting

In the first case, since one had many messages for a given day, all with the same ground setting, one could look at the alphabet formed by combining that produced by the machine at the ground setting with that produced three letters later. This alphabet was visible in the clear: if a message began VBT RSQ, then V became R in that compound alphabet.

Essentially, tables could be made of the distinguishing features of the alphabets created at each starting position. And alphabets had such features, even after the plugboard was used. These were visible when an alphabet was reduced to 'cycle form': if V becomes R, then what does R become, and how many steps does it take to get back to V? Any scrambled alphabet basically divides the alphabet into pieces of various sizes, and those sizes aren't changed by the plugboard. The cyclometer, used to create these tables, will be described along with the Bombes in the next section.

The three alphabets derived from a day's intercepts (or three months' intercepts, actually) had, when reduced to cycle form, some very special properties. Although they were formed by comparing two encipherments at positions separated by 3, they could also be thought of, since decipherment and encipherment were the same, as the results of applying the encipherments at those two positions, one after the other.

This has the following curious result:

```A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

U V T X R W M I H Z Q P G S Y L K E N C A B F D O J
A B C D E F G H I J K L M N O P Q R S T U V W X Y Z

G K R P S Y A J M H B Q I U W D L C E V N T O Z F X
```

In the pair of reciprocal alphabets shown above, A becomes U, and then U becomes N. So, in the compound alphabet, A becomes N. Note that U also becomes A in the first alphabet, and N becomes U in the second, but these don't connect, so the compound alphabet is not reciprocal.

As noted, in the compound alphabet, A becomes N. Then N becomes (S, then) E, E becomes (R, then) C, C becomes (T, then) V, V becomes (B, then) K, K becomes (Q, then) L, L becomes (P, then) D, D becomes (X, then) Z, Z becomes (J, then) H, H becomes (I, then) M, and M becomes (G, then) A.

So, we have a cycle, involving the letters:

```(a n e c v k l d z h m)
```

What happens if we follow the fate of the letter U, which becomes A in the first alphabet, just as A became U?

U becomes A, which becomes G. G becomes M, which becomes I. I becomes H, which becomes J. In starting with the middle letter of each two-step substitution, we simply go through exactly the same cycle, but in reverse. So we have a second cycle of exactly the same length, with the letters:

```(u g i j x p q b t r s)
```

Because this sequence consists of the middle letters, taken in reverse order, of the previous cycle, the individual substitutions can be found at one of the positions these two sequences have when slid against each other after one is reversed:

```A N E C V K L D Z H M
U S R T B Q P X J I G
```

are all pairs in the first alphabet, and

```A N E C V K L D Z H M
G U S R T B Q P X J I
```

are all pairs in the second.

Because at that stage poorly chosen rotor starting positions, both consisting of adjacent letters on the keyboard, and also consisting of the same letter repeated three times, were common, it was possible to eliminate ambiguity in aligning the cycles and determine the rotor starting positions, without as yet knowing either the alphabet ring settings, the rotor order, or even the wiring of any rotors.

### The Slotted Sheets, or Grilles

Knowing the letters visible in the little windows on the enemy's Enigma was an accomplishment, but by itself it did not let you read any messages. The Poles did know the rotor wirings of the Enigma, though. They had not captured a machine, and doing so in peacetime would have made it obvious that one was missing. Instead, a spy had given the French, and the French had given them, keys for a period and some matching plain and enciphered messages, from which they laboriously reconstructed the rotor wirings and internal connections of the Enigma.

In addition to knowing the initial rotor settings, applying the pencil-and-paper method noted above provides you with six successive alphabets produced by the Enigma. At this stage, only a limited number of letters are modified by the plugboard, and the Enigma had only three rotors to insert in any of six different orders.

If, for the encipherment of the first six letters after the ground setting, the medium rotor did not move, these alphabets are produced from an inner alphabet, produced by the medium, slow, and reflecting rotors, at six consecutive positions of the fast rotor.

The grille or slotted sheet method involved sheets with the successive alphabets produced by one rotor printed on them, which were slid over a piece of paper on which these six alphabets were written.

If it weren't for the plugboard, negating the fast rotor would produce six identical alphabets: even with the plugboard, the right position produced six alphabets with many letters in common, and from the similarities and differences, both the plugboard settings and the inner permutation could be determined.

The initial rotor settings are known in terms of where the alphabet rings are; now the ground setting is known in terms of the positions of the rotors themselves. Still more work needs to be done to read messages: the original Polish Bombe automated that next step.

### Arbitrary Ground Settings

Since the ratchet that allowed one rotor to move the next slower rotor and the alphabet ring which was used to refer to rotor positions were fixed together, in the second system it was often possible to determine, since the Enigma's original five rotors all had their one carry in different positions, which rotor was the fast rotor.

Among the many indicators for a day, there would be some with repeated letters. And there would be some with the starting positions given in the clear possibly adjacent to each other. Which positions are adjacent depends on when carries occur. And if the same letter becomes first N and then V three letters later, and then fails to change from N to V when it is apparently in the same two positions, then one is mistaken about when the medium rotor moved.

Combinations where the same letter becomes the same letter in the same message, in two positions three places different could not always occur, and these indicators, called "females", eliminated possible indicator settings.

Also useful to Bletchley Park at this stage were "Herivel tips", which were the result of Enigma operators choosing, after setting up their machine for the day, to use as their first setting the position of the rotors as they stood, or possibly moving them only a few places, from the position after setup which meant the rotors were near their initial positions, and the setting sent in the clear would give away the alphabet ring settings (either by itself, or in combination with the arbitrary ground settings of other operators making the same mistake).

### The Perforated Sheets

The perforated sheets were one set of 26 square sheets, with punched holes, for each possible rotor order. They were used when the ground setting was sent in the clear, followed by two encipherments of the current setting. Each sheet corresponded to the position of one of the wired rotors (in Poland, apparently the slow one) and the rows and columns on the sheets corresponded to the positions of the other two rotors. The rows and columns, except the first one, were repeated twice. A hole was punched in a sheet whenever a letter would be enciphered to itself by an Enigma with that position of the three rotors plus an Enigma with the fast rotor three places further ahead.

Since a rotor order is assumed by one's choice of perforated sheet set to use, one can exclude all indicators in which the medium rotor moved. This is because the ratchet wheel and the alphabet ring are attached to each other. In practice, unless there is other information available, one needs to try all 60 rotor orders. And, yes, one also needs to try 26 stacks of sheets for each rotor order. However, 1560 trials is not an unmanageable brute-force search, even by manual methods.

One finds numerous indicators with repeated letters in the same position, and places, one on top of each other, alphabet sheets so staggered as to match the relative displacements of the ground settings for the different messages. Then, a hole through the stack of alphabet sheets indicates which position of the wired rotors corresponds to each of the ground settings.

### The Final Period

During the war, the double encipherment of the indicator was eventually abolished. This increased the reliance of cryptanalysts on a large piece of electrical machinery, the Bombe. But for a period of time before it became available messages were still deciphered, as the result of the continuation of the practice of choosing adjacent keyboard letters, then called "cillies" by the British, (well, they were silly) even if settings consisting of three identical letters were by then suppressed.

With the Naval Enigma, instead of switching to a single encipherment of the initial rotor position, the use of a common ground setting was revived; however, the indicators recieved an additional layer of encryption; the two repeated encipherments of the initial rotor setting were enciphered by means of a table of digraphs.

If the enciphered starting setting was ABC DEF, then the digraphs that would be enciphered in the table would be AY BD CE XF where X and Y represent two additional letters chosen by the encipherer at random; that is, the starting setting was staggered like this:

```A B C X
Y D E F
```

and the digraphs were taken off by columns.

This would have made the indicators useless for cryptanalysts, except that the digraph table was not itself part of the daily key. Instead, for one period, a set of nine digraph tables were used, and only which one was to be used was part of the key that changed each day. As copies of the digraph tables (which were, for convenience, reciprocal) were captured, the result was merely that all nine possibilities had to be checked. With a common ground setting, as noted above if letter 1 was A and letter 4 was D once, then letter 4 had to be D whenever letter 1 was A, and so one would normally be able, with 27 messages, to swiftly determine the right table to use.

Had this method been used with the later indicator method used by the other services, an arbitrary ground setting and a single encipherment, considerably more work would have been multiplied by a factor of nine, even using the digraph tables that had been captured. Had the digraph tables been changed more often, that too would have diminished the usefulness of the indicators, but by then the advanced Bombes in use eliminated the need to break the indicator system; it would have meant more work, but their Engima messages would not have become inviolate.

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