The Uhr box was a replacement for the patch cords that were plugged into the Enigma's plugboard. Each day, the relationship between the alphabet and the contacts on the Enigma's rotors was changed in accordance with the daily key by plugging in the patch cords as it specified.
I am indebted to some USENET posts, in reply to questions of my own concerning the Uhr box, by Frode Weierud from Switzerland in clearing up some of the details about how it worked. The facts he provided allowed me to propose a tentative reconstruction of how the Uhr box might have been wired internally.
By using the cords coming out of the Uhr box instead, it became convenient to change the plugboard setting more often, because it could be changed just by turning the knob on top of the Uhr box. The knob had forty settings, numbered from 0 to 39. These settings were indicated by a two letter code, given by a sticker on the lid of the Uhr box.
When set to setting 0, the Uhr box acted like a set of ordinary patch cords. Of the twenty plugs coming out of the Uhr box, ten were painted white, and ten were painted red. The plugs in the two groups were numbered from 1 to 10, and the pairs of one red plug and one white plug both bearing the same number each behaved like a patch cord in setting 0.
Even in other settings, although the connections between the input of one plug and the output of another plug were now scrambled, it was still true that the input of any red plug was always connected to the output of a white plug, and the input of any white plug was always connected to the output of a red plug.
Because patch cords always connected their input and output to the output and input on the other end, they produced a reciprocal permutation of the alphabet. The Uhr box allowed this to be avoided. This didn't change the fact that the Enigma's cipher was reciprocal, since the permutations of the Enigma's individual rotors weren't reciprocal either, but the cipher of the Enigma as a whole was still reciprocal, since the reflecting rotor caused the electrical signal from the keyboard to the glowlamps to go through each rotor twice, first one way, and then the other. The same thing was true of the Uhr box.
But the fact that ordinary patch cords produced a permutation which was reciprocal on, as it were, a second level did prove helpful to British cryptanalysts. For one thing, it permitted an adjunct to the Bombe called the 'diagonal board' which allowed it to be both more versatile and more effective.
Based on these facts concerning the Uhr box, I had suspected that it may have been wired as follows:
The knob on the Uhr box would have turned what was, essentially, a reflecting rotor with 40 contacts.
These 40 contacts could be thought of as being labeled with the letters ABCDABCDABCD... over and over again.
The contacts marked A would all have wires connecting them to the contacts marked C, but in scrambled order. Similarly, the contacts marked B would all be connected to the contacts marked D, again in a scrambled order.
The contacts in the box which connected to this movable reflecting rotor would also be divided into four groups, which we can also think of as being labelled ABCDABCDABCD... and so on.
The input contacts from the red plugs would be connected to the contacts marked A, and since these were always to be connected to the output contacts of the white plugs, those would be connected to the contacts marked C.
Similarly, the input contacts from the white plugs could go to the contacts marked B, and the output contacts from the red plugs to the contacts marked D.
The order in which the plugs were connected to the contacts would match the scrambled wiring in the 40-contact reflecting rotor, so that when it was set to position 0, the desired objective of emulating plain patch cords would be achieved.
It would be possible, although unnecessary, for all the input contacts, for example, from the patch cord plugs to be wired in numerical order around the circle.
The following diagram may make the description of my tentative reconstruction of the Uhr box clearer:
However, a paper in the July 1999 issue of Cryptologia has now explained the actual workings of the Uhr box, and it differed somewhat from my tentative reconstruction.
Instead of having 40 contacts, the reflecting rotor that was the heart of the Uhr box had 80 contacts, of which only 40 were used at any one time.
The contacts on the reflecting rotor were in two consecutive rings of 40 contacts, and either the even or odd ones were used in any position. Thus, the contacts on the Uhr box were in two rings of 20.
The outer ring was wired to the red plugs, and the inner ring was wired to the white plugs. In both rings, the even-numbered contacts of the Uhr box were wired to the thick pin of the two pins on the plugs. The wires from the red plugs were wired to these contacts in order, but those from the white plugs were wired in a scrambled fashion.
Essentially, therefore, the Uhr box worked somewhat like my hypothetical reconstruction, except that it had two sets of scrambled wirings, each one of which could be rotated to only half as many positions.
However, there was another peculiarity of the wiring of the Uhr box that led to a weakness.
Although the wires from the white plugs were not wired to the Uhr box contacts in order, they were wired to those contacts in pairs. This meant that of the two wirings in the reflecting rotor, one, in order to allow the zero setting to emulate ordinary plugboard wires, had to take pairs of contacts (with a contact in between belonging to the other wiring) to pairs of contacts in the other ring, reversing the two elements in the pair (so as to take a large pin from a red plug to a small pin on a white plug and vice versa). This meant that every fourth setting of the Uhr box behaved like a set of conventional plugboard wires.
A diagram of the actual Uhr box may help to make its design clearer:
in this diagram, only the wires in the rotor are shown that are in the set used in the zero position, which reflect the flaw in the device that makes every fourth setting (every second setting using that set of wires) reciprocal. The Uhr box is shown here set in the zero position. Because of the extra complexity of inner and outer contacts, a more schematic diagram, rather than one showing the rotor contacts in a circle, is given.
While the Uhr box did not do much by itself, only providing a fixed substitution that did not change during a message, this kind of design illustrates how one could, for example, build an interesting type of rotor machine for a 26-letter alphabet using especially-wired 52-contact rotors. The idea of wiring a rotor with a number of contacts that is a multiple of the size of the alphabet used, so that it acts like two different rotors that are used alternately, also will surface in the Hagelin B-21, which we will meet later.
Another enhancement used on some Enigmas late in the war was a reflecting rotor that could be completely rewired by the user based on a daily key.
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