Some of the letters in the second list are duplications from the first list, but others are not, terminal G being one example. As a consequence of this, via terminal G, cycle C2, will give rise to voltages on the other terminals B, E, H, N, V and X.
Again as a consequence of the voltage on terminal B, cycle D2 will give rise to voltages on terminals D and Y, and at this stage the terminals of all the relays in the indicator unit, with the exception of terminal I, will have a voltage on them.
As the following table shows, beginning with the false hypothesis S/A, all of the 25 possible false conclusions are arrived at. The reader may wish to test other false initial hypotheses, and should find that they will all produce the same final outcome on the indicator unit where, with the exception of relay I, all of the others will receive the test voltage.
In the way that has already been explained, if the true hypothesis (S/I) were to be tested, then only the terminal I would receive the test voltage.
The required discrimination between the correct stecker hypothesis and all the false ones has been achieved, and the general behaviour of the bombe characterized by the above example, is summarized in the following statements:
(i)
At each possible rotor configuration, the feedback circuit, in effect, tests simultaneously all of the 26 possible stecker hypotheses for a chosen letter on the menu, and if the rotor configuration is correct then the true hypothesis can be identified from a distinctive pattern of voltages appearing on the relay terminals of the indicator unit.
(ii)
The distinctive pattern can take two forms: either the test voltage will reach 25 of the 26 relay terminals of the indicator unit, or alternatively only one of them.
The 26 relays in the indicator unit, one for each letter of the alphabet, were used to provide the required logical control of the machine, so that when it was operating with the rotors progressively rotating through all their possible positions, the system would be halted automatically whenever the test voltage failed to reach all of the 26 relays (known as a ‘stop’).
The introduction of the concept of a feedback loop was of huge importance; providing that a suitable menu was used it enabled simultaneous scanning to be achieved in a remarkably simple way. This enabled part of an Enigma key to be found in a way that avoided having to confront at the same time the severe complications arising from the unknown connections on the Enigma plugboard (the steckers).
As the process of finding the true stecker hypothesis does not depend on which switch is used to input the test voltage, it was a common practice at the time to use switch A. The indicator unit was connected by a 26-way cable to the ‘best’ position in the circuit and this was usually taken to be the one corresponding to the letter on the menu that had the most links attached to it.
Eliminating All the Wrong Rotor Configurations
Clearly a matter of great importance is to consider how this circuit will behave when the rotor configuration is wrong as this will be by far the most common occurrence. In all such cases both sequences S1 and S2 will produce random permutations and a ‘random stop’ will occur when by chance both of these permutations happen to include a unit cycle of the same letter. The probability of this happening is 1/(26×26) implying that the expected number of random stops will be about 26 for every rotor order tested.
However, as a consequence of the feedback processes previously described, a much more likely outcome is that the test voltage will reach the terminals of all the relays in the indicator unit, no matter which input switch is used to ‘inject’ the test voltage. As all the stops had to be subsequently checked by hand the expected number of random ones given by a menu with 2 loops was inconveniently large. When a menu with three loops was available, the expected number of random stops would be reduced to only one for each rotor order that was tested.
The First Prototype
In the previous circuit diagram, there are duplications of some of the scramblers used at certain positions, and it is possible to eliminate these by making some simple changes to the electrical configuration. The resulting circuit, shown below, has an improved structure that is more suitable for setting up on the bombe, and corresponds to the circuit configurations that were used in the prototype.
The first machine arrived at Bletchley Park in March 1940, and was rather optimistically called ‘Victory’; however, the operational performance of the machine was somewhat below expectations. The effectiveness of the prototype bombe was critically dependent on the use of menus containing multiple loops and these were not available as often as had been hoped.
An additional reason for the limited success of the machine was that it was mainly used in attempts to break German Navy ciphers, and this meant that many more possible rotor orders had to be tested than was the case for the messages transmitted by the German Air Force and Army. Consequently a considerable amount of ‘bombe time’ was expended in attempting to find Navy Enigma keys, using relatively weak menus. German Navy Enigma operators had eight rotors available for use compared to the five for the operators in the other two services. With eight rotors in use there were 336 possible rotor orders to consider!
The Diagonal Board
The invention of the diagonal board by Gordon Welchman was of vital importance in improving the performance of the bombe. It had the valuable effect of hugely increasing the number of electrical connections between the contacts in the circuits that represented valid logical deductions, and enormously increased the likelihood that with a given menu the process of simultaneous scanning would be successfully completed.
This meant that after the addition of the diagonal board to the bombe it became possible to achieve simultaneous scanning using much weaker menus than before. The diagonal board also greatly reduced the number of random stops obtained from a given menu. It enabled menus that covered spans of less than 13 consecutive Enigma positions to be more easily formulated, thus reducing the chances of a middle-rotor turnover (explained below) occurring to less than 50%. In some extreme circumstances the diagonal board even made it possible to achieve simultaneous scanning when using a menu without any loops at all.
The effect of the diagonal board was to enable additional logical inferences to be made about the stecker hypotheses that potentially existed in the original bombe circuits during the course of an operational ‘run’ but which could not have been foreseen in advance from the menu.
Basically the device consisted of an array of 26 × 26 ( = 676) electrical terminals that were connected to each other and to the scramblers in the bombe in a particular way that made it possible to exploit the symmetrical properties of the electrical connections on the Enigma plugboard.
As an illustrative example the diagram overleaf shows the sequence of encipherments S2 previously described, but now set up in the form of a circuit as used on the bombe with the indicator unit connected to position S. The diagram shows an additional row of 26 terminals (A–Z), all connected to position S on the menu by means of a 26-way cable, so that they will have the same voltages on them as the corresponding terminals of the relays in the indicator unit.
As has previously been explained, when used with the correct rotor configuration, the voltages on the relays represent the false conclusions for the stecker partner of letter S that follow as logical consequences from an initial false hypothesis. It should be clear that the voltages on the row of terminals could be used instead to represent the false conclusions for the stecker partners of letter S, instead of the relays in the indicator unit.
Additional rows of 26 terminals are also shown connected to the other six scrambler positions in the menu circuit, and likewise the voltages on the terminals in these rows could be used to represent the false conclusions for the stecker partners of these other letters (Q, I, F, E, H and U) also occurring on the menu. (For this particular menu an array of 7 × 26 of terminals is sufficient for the purpose.)
Welchman realized that permanent connections could be made between s
ome of the pairs of the terminals in this array, and that electrically they would significantly increase the number of false conclusions that could be made from any false initial hypothesis that might have been used. His reasons for adding these permanent connections to the circuits in the bombe were based upon the reciprocal relationship existing between the pairs of stecker letters on the Enigma plugboard. For example if say the conclusion S/H is deduced by the electrical circuits in the bombe then as a consequence of this reciprocal stecker relationship, the additional conclusion H/S can also be made.
In electrical terms, this means that if a voltage happens to appear on the terminal H in row S of the array, (corresponding to the conclusion H/S), then there should also be a voltage on terminal S in row H (corresponding to the conclusion S/H). This can easily be arranged by simply making a permanent electrical connection between these two terminals to create a new electrical path in the scrambler circuit. If the conclusion H/S is false then so will the conclusion S/H and this in turn is highly likely to result in the generation of further false conclusions (as voltages on other terminals). The same argument can be made for all the other reciprocal stecker letter pairs represented in the array of terminals.
The combined effect of all the connections in the array is to increase hugely the number of false conclusions arrived at from any false initial hypothesis, to the extent that even for some menus without any loops at all, it remains highly probable that all of the possible false conclusions will be obtained, thus maintaining the desired simultaneous scanning that was essential for the correct functioning of the bombes. (The diagonal board had the same powerful effect when the rotor order was wrong.)
A much larger array of 26 × 26 ( = 676) terminals is required to provide for all the possible situations that might arise, consequently the wiring of the complete diagonal board is too complex to be given in the form of a diagram, but for purposes of illustration a ‘mini’ 3 × 3 version of the board for just three letters, is shown below.
To avoid confusion lower case letters have been used in this diagram to denote the individual terminals in the array and upper case letters to denote the multiple cable connectors used to link the rows of terminals on the board to the corresponding positions between the scramblers.
This ‘mini’ board would require three 3-way external connecting cables at the positions indicated by the capital letters to make the necessary electrical connections to the other circuits in the bombe.
As previously explained, the wiring of the diagonal board is based on the symmetrical property of the reciprocal stecker relation between pairs of letters on the plugboard. This means that with the wiring of the diagonal board, if the test voltage appeared on wire c, at position A it would cause the test voltage also to appear on wire a at position C. The arrows in the diagram show how the board provides the required electrical connections corresponding to the following stecker relationships:
A/c implying C/a and C/b implying B/c
(Note: the ‘self steckers’, for example B/b, do not require any interconnections on the board.)
When the diagonal board is extended to cover all 26 letters, each of the necessary 26 cable connectors, represented by capital letters in the diagram, will have 26 individual contacts, so that the internal electrical connections between the diagonal board and the other circuits in the bombe were made by means of 26-way cables.
The way in which the diagonal board functioned is not obvious, and it is on record that when Welchman first described it to Turing, the latter was ‘incredulous’ (for a moment or so)!
In order to provide some idea of how it functioned, the following simplified example is given that is based upon an alphabet restricted to the five letters A, B, C, D, E. As a consequence there will only be five connections through the scramblers and only a ‘5 × 5’ diagonal board will be needed.
Then, starting with the false stecker hypothesis A/d, the additional connections through the diagonal board will lead to the following sequence of outcomes:
The voltages on the ‘live’ contacts, representing the false steckers (A/d, B/e, C/d, D/b and E/c), will be passed to the diagonal board and returned from it to other contacts on the scramblers that represent the reciprocal steckers D/a, E/b, D/c, B/d, C/e, so that these contacts become ‘live’.
Each of the five ‘live’ contacts representing these reciprocal steckers will be on a ‘line of electrical continuity’ through the scramblers and consequently as well as the contacts on the scramblers these lines will also be made ‘live’.
As the diagram shows, two additional lines of continuity marked (2) & (3) are made ‘live’ by the wiring in the diagonal board. Consequently through these additional lines, more contacts on other scramblers will also be made ‘live’. These ‘live’ contacts will also indicate false steckers, and the voltage on them will be carried back again to the diagonal board to be returned as reciprocal false steckers. This virtually instantaneous process will continue until a stable state is reached.
There are three special circumstances that usually occur that will reduce the number of scrambler contacts involved:
(i)
‘Repeats’. It is possible that the voltage representing a reciprocal stecker may be returned from the diagonal board to a contact that is already ‘live’.
(ii)
‘Self steckers’ (A/a, B/b, C/c, D/d, E/e). These have no effect on the process and the ‘voltages’ are not returned from the diagonal board.
(iii)
‘Letter not on the Menu’. In general (but not in this example) a returned reciprocal stecker may involve a letter that is not on the menu. Consequently the voltage from the diagonal board cannot be returned to any of the scramblers.
The complex sequences of events carried out by means of the diagonal board are not easy to quantify. However, the following approximate analysis based on probability considerations may help to provide a better understanding of the role of the diagonal board than is possible from a purely descriptive account.
Consider for example a linear menu (one without loops) consisting of 12 letters, and suppose that the rotor order and current drum settings are both correct. If a false initial hypothesis for the partner of a chosen letter on the menu is used, then in electrical terms the test voltage is applied to a chosen contact on a particular scrambler so that it becomes ‘live’. The voltage will be conveyed by means of a line of electrical continuity through the scramblers so that one contact on each of the scramblers has become ‘live’. These 12 ‘live’ contacts each represent a stecker partner for the corresponding letter on the menu.
Then, by means of the electrical connections to and from the diagonal board, the voltage on some of these ‘live’ contacts will be returned and will make other scrambler contacts ‘live’, these corresponding to the reciprocal steckers. For each of these reciprocal steckers the probability that it will involve one of the 12 letters on the menu = 12/26. So assuming there are no ‘repeats’ the expected number of reciprocal steckers returned via the diagonal board that will result in additional ‘live’ contacts = 12 × (12/26).
This, however, is still a slight overestimate, because ‘self-steckers’ are not returned. The probability that a letter on the menu is ‘self-steckered’ = 1/26, hence the expected number of self-steckers = (12 × 1/26), and so a better estimate for the expected number of reciprocal steckers that will be returned from the diagonal board to contacts on the scramblers = 12 × (12/26) – 12/26 (≈ 5).
The processes involving the diagonal board can be considered in a number of stages. In the first stage, as explained above, simplified theory shows that a voltage on one line of continuity will cause one contact on each of the 12 rotors to become ‘live’ and via the diagonal board an estimate of five other contacts on the scramblers will be made ‘live’. Hence, assuming there are no ‘repeats’, the total number of contacts to become activated during the first stage = 12 + 5 ( = 17).
However, the possibility of the occurrence of one or
more ‘repeats’ must be taken into account. The diagonal board consists of 26 rows of contacts, one for each letter of the alphabet, and each row has 26 contacts in it (in all 26 × 26 = 676 contacts). For a menu with 12 letters, the total number of operational contacts on the diagonal board is 12 × 26 ( = 312).
Suppose that at any stage, T represents the total number of terminals on the board that are ‘live’. Then the probability that a given terminal is not ‘live’ = 1 – T/312 or alternatively = (312 – T)/312. This expression can be used at the next stage as an approximation for the fraction of the predicted number of new ‘live’ contacts that are not ‘repeats’ of ones previously counted.
Hence an approximation for the expected total number of new contacts made ‘live’ by the end of the first stage is: 5 × (312 – 12)/312] (≈ 5) (so at this stage repeats are unlikely). By the end of the first stage a total of 12 + 5 = 17 contacts will be ‘live’, (the same approximate value as before).
During the second stage the five additional scrambler contacts that became ‘live’ during the first stage will have two effects:
(i)
As each of them is on a ‘line of electrical continuity’ through the 12 scramblers this will result in a contact on each of the 12 scramblers becoming ‘live’, so in all 5 × 12 = 60 contacts will be made ‘live’ in this way. However, taking into account the probable occurrence of ‘repeats’ this number is reduced to about: 60 × (312 – 17)/312 (≈ 57). So the current cumulative total number of ‘live’ contacts = 17 + 57 ( = 74)
Gordon Welchman Page 30