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Relatives of the Enigma

Relatives of the Enigma

Despite the fact that the Enigma never enciphered a letter as itself, thereby making it easy to align probable plaintext with an intercept, several other rotor machines were based on the Enigma.

W. F. Friedman devised Converter M-325. It was similar to the Enigma, except that the rotor nearest the reflecting rotor, rather than the rotor farthest from it, was the fast rotor. This produced a machine that did not need a plugboard to be reasonably secure.

The British cipher machine during World War II was the Typex. Its rotors were notched several times for frequent and irregular motion. Three rotors moved, and the farthest of them from the reflecting rotor was still the fast rotor. But there were two other rotors beyond the fast rotor, and they acted as a plugboard replacement.

Some models of the Typex had two concentric sets of 26 contacts. According to those who have had the opportunity to examine surviving specimens of this machine, they were not used for the obvious purpose of providing a return path for electrical current, thus allowing the unique weaknesses of the Enigma design to be avoided.

Despite this, I still wonder; there is a simple way, using dummy rotors connected by cables, to turn a SIGABA into a plain Hebern machine. As the SIGABA and the Typex were made to interoperate by means of accessories which turned them into the "Combined Cipher Machine", it makes sense to conclude that the second set of contacts was used as a return path on the Typex, since then suitably wired rotors with some sets of contacts wired without scrambling could allow Hebern type operation of such a machine as well, thus facilitating this particular method of interoperation. Although the fact that the fast rotor is on the entrance side (like an Enigma, unlike an M-325) makes it more difficult to turn a Typex into a Hebern machine with the fast rotor in the middle, it is not impossible: an auxilliary wiring harness, with dummy rotors, is also required as with the SIGABA, and the reflecting rotor is one of those that has to be replaced. The following diagram shows a principle that may have been used:

As the details of the actual CCM are available, no doubt I will soon be hearing if my reconstruction is near the mark or not. As only three rotors aren't much, the actual CCM may have been somewhat more complicated, for example by including entry and exit plugboards to replace the two stators in a real Hebern five-rotor machine.

As the Typex is said to have very closely resembled the Enigma, I had assumed that it had a "650 break" as well, and this would have further complicated interoperation with the SIGABA. However, since the Typex used heavily multinotched wheels, such a gearing arrangement would have shortened its period considerably, so perhaps that problem did not arise. (It's even possible there was a lever somewhere to choose with!)

On some later models of the Typex, the two entrance rotors also moved. This is merely a speculation on my part, but one simple way in which they could have been caused to move is this: the entrance rotor would be the slow rotor, the next one the medium, and the same notches in the wheels that caused the regular carry would also cause movement of the wheels on the entrance side, but the ratchet wheels would be displaced. So, if the positions B, G, J, M, O, R, T, V, and X caused the original medium wheel to move when the fast wheel reached them, and the original slow wheel to move when the medium wheel reached them, then perhaps the positions Z, E, H, K, M, P, R, T, and V, each two positions earlier, would cause movement on the other side of the fast rotor. Or another row of notches could be present on the wheels, and the number of notches in that row could very well be even, since the ordinary rotor motion would already be providing the maximum period. No such motion, however, unlike the original motion with those two rotors as stators, or better yet a motion resembling that of the Converter M-325, would allow easy interoperation with a SIGABA using dummy rotors by emulating a Hebern machine, basically the lowest common denominator of the two machine types.

Some time after World War II, Switzerland, which had been using a version of the Enigma for its own communications, designed its own significantly improved cipher machine based on the Enigma, the NEMA (for Neue Maschine, not National Electrical Manufacturers Association). Recently, a computer program simulating its operation, accompanied by a description of the machine, became available. One of the program's authors, Frode Weierud, has a forthcoming article in Cryptologia about the machine.

It used lamps to indicate the letters it produced as output, and physically it looked like a metal box with a sloped front in which a typewriter keyboard was sunk. Recently, examples have become available on the open market, at prices considerably below those of World War II Enigmas.

The device had four rotors, plus a reflecting rotor, all of which (including the reflecting rotor) moved during encipherment. However, it had the appearance of having ten (rather thin) rotors. This was because each rotor was manually adjusted by means of a flange on its left with 26 segments labelled with the letters of the alphabet, and bore on its right a ring of appearance similar to that flange, but which contained a notched gear. Each rotor's movement was controlled by the ring to its right; when a notch on the ring came into position, its corresponding rotor did not advance for the current letter being enciphered. In addition, the ring on the entrance rotor, towards the right of the rotor bank, was distinctively colored (it was red) and had a second notched gear on it. This gear, when a notch on it came into position, prevented the second and fourth rotors, and their corresponding rings, from moving. Usually, all the rings would move with every character enciphered.

Thus, the NEMA had a period of 17,576 letters, each ring-rotor pair having a period of 676. If the red ring had stopped only two other rings, and not their corresponding rotors as well, the period could have been 26 times longer. In any case, it has a large number of initial settings, and a very irregular rotor movement, bettered only by that of the SIGABA (although the Enigma A and Enigma B also had a rotor movement that is more irregular than usual). Note that the diagram, for simplicity, shows the mechanics of the rotor movement by lines which may suggest electrical wires; actually, the result was achieved mechanically. Also note that the wheels with notched gears, although concentric on the same shaft as the rotors, are shown in the diagram in a row below the rotors, using the usual diagram symbol for pinwheels.

And, during the sixties, an Italian firm, O.M.I., offered a version of the Enigma with seven rotors. But it embodied an ingenious idea for providing security in a machine sold openly without requiring customers to rewire their rotors. The rotors came in two pieces, both with scrambled wiring, that the customer could interchange and rotate to form the rotor set to use.

It would be a little harder to do this with rotors of the Hebern type, if one wished to take full advantage of the fact that both sides of these rotors are identical, but it could still be done using androgynous connectors positioned in alignment with the centers of the contacts, as shown below:

An interesting web site in Germany describes several cipher machines from behind the Iron Curtain. Three of them were described in enough detail (which I tried to make out as best I could, not understanding German) that I found their mention of interest.

The T217, or ELBRUS, is an electronic cipher machine, having at least a superficial external resemblance to the early electronic cipher machines used by the United States (as those machines are, as far as I know, still classified, I have no knowledge of whether the resemblance is more than superficial). A linear-feedback shift-register generator feeds a longer shift register, and bits from that are selected under the control a key cassette to go into combinatorial logic to produce bits for stream cipher use.

The M105 was claimed to use a principle that seems hard to understand: 5-level code characters are converted to a 12-bit form, then XORed with a key tape, then converted back down to 5-bit form. It doesn't seem clear at first how any transformation more involved than a simple XOR could actually be accomplished by this principle, particularly when decoding is also performed by precisely the same method.

It would seem to make far more sense to do something like this: XOR the plaintext with five bits from the tape, perform a nonlinear transformation chosen with two bits from the tape, and then XOR the result with the next five bits from the tape. Something like that would clearly make it possible to scramble 5-bit inputs according to 4,096 different schemes.

But it might be possible to make such a machine work through methods that seem like cheating; for example, the 5-bit character


could be converted to


to be enciphered, and then the 12-bit result of being XORed with the key tape could be converted to 5 bits by taking the first 5 bits, performing a nonlinear operation controlled by the last 2 bits, and then XORing with the second 5 bits. To decipher using the same key tape, the 5-bit ciphertext character


would simply be converted to


before the XOR with the key tape. The machine would, though, in that case need a "decipher" setting, both to change the conversion, and to reverse the order of the steps taken. Even something like that, particularly if a plugboard can be used to scramble the 12 bits from the key tape, might not be entirely silly.

As well, although these conversions are obviously trivial, their nature could be concealed by applying an invertible modulo-2 matrix multiplication to the two possible "converted" 12-bit forms of a 5-bit input value. That would just lead to the one-time tape acting like a linearly-transformed version of itself; a similar principle is used in implementations of Rijndael decipherment.

However, as a diagram of the scheme makes clear,

one is just using extra circuitry to accomplish an encipherment that could be performed just as well by using the twelve bits from the key tape to encrypt the five-bit plaintext directly. Still, even if this is not how the actual M105 worked, the principle may be of some use for other purposes, even if I don't think it could serve as a basis, say, for a form of the Shamir three-pass protocol that doesn't rely, like the Massey-Omura cryptosystem, on functions also useful for public-key cryptography as a basis.

Note that the positions of the matrix multiplication and its inverse are not reversed for decipherment; this is not an error in the diagram; the same transformed 12-bit keystream values need to be recoverable for both functions.

It may also be noted that if the nonlinear transformation included a memory for preceding values of its input or output, or if the device had some other form of internal state, then its encipherment could be made arbitrarily complicated.

And, finally, the M125, or the Fialka, is described on his site, which is why I am mentioning his site on this page: it is a variation of the Enigma. My original description of that machine, based on his site, contained several inaccuracies, and even a first attempt at an improved version was seriously in error on many points. An accurate, detailed, and illustrated, description of the Fialka is now available at this site.

This machine has several features of interest. When two adjacent rotors both move, they move in opposite directions, rather than in step. As well, there are two independent paths by which one rotor causes subsequent ones to advance; thus, this rotor machine has fast rotors near both ends, which thoroughly defeats an attack by the isomorph method.

An extended manual for the Fialka is available from this site, and that has enabled me to further revise my description of the machine, to the point where I believe it is now completely accurate.

Four wires come out of the reflecting rotor.

One group of three wires goes to an electronic circuit designed so that, if the three wires are called A, B, and C, during encipherment, current to wire A is reflected back along wire B, current to wire B is reflected back along wire C, and current to wire C is reflected to wire A. In decipherment, this is reversed; current to wire C is reflected back along wire B, current to wire B is reflected back along wire A, and current to wire A is reflected back along wire C. In this way, the cipher produced by the machine is no longer fully reciprocal, removing one of the potential weaknesses of an Enigma-based design.

Originally, I had suspected that the circuit showing three transistors in a circle in the diagram on the site where I first heard of the FIALKA had some such purpose, but I had thought that the rotors had 31 contacts instead of 30, and so the circuit was needed to make an Enigma-like design work with an odd number of contacts on the wheels.

In a way, that was true, because the number of letters handled by the reflecting rotor was reduced from 30 to 29 by the other wire also coming out of the reflecting rotor. That wire was used to deal with an even more serious weakness of Enigma-like machines; when a signal came to that wire, the plaintext letter was enciphered as itself. This reversed nicely during decryption, since the same letter would, by the same rotor arrangment, also go to that contact on the reflecting rotor.

Like the Typex, we have an Enigma with both multinotched rotors and a plugboard; British cryptanalysts working on the Abwehr Enigma on the one hand, and the Army and Navy Enigmas on the other saw that to be an effective combination. This may not be terribly surprising, as there was Soviet penetration of the British Ultra effort during World War II. The patch panel, like the Uhr Box, avoids the problem of the plugboard substitution being reciprocal in itself.

In addition, a later version of the M-125 had rotors that, like those of the O.M.I. machine, were in two parts, so that the rotor wirings could be varied by changing how the two parts were combined. In addition, like rotors on the SIGABA, the small supplementary part of a rotor that modified its wiring could be inserted upside-down, which meant that it could be inserted in sixty, rather than thirty, positions.

Although the machine is connected to 5-level tape equipment, instead of enciphering normal 5-level code (leaving, perhaps, the all-zeroes character unenciphered) the German-language version of this device omits the letters W, X, and Y, and uses J to represent the space (thus printing a space when deciphering, and a J when enciphering), leaving room for the digits from 2 through 9 in the character set. As Russian has a 32-letter alphabet, presumably the original Russian-language version did not include digits in its character set.

My initial attempt to reconstruct the Fialka from the information I had found resulted in a somewhat different machine, with its own features of interest.

In the a block diagram I saw, there were three signals going from the reflecting rotor to a transistor symbol. I thought that perhaps this meant that three contacts were wired in a loop, with diodes going from one to the next, in rotation. As a DC current is used to go from the keyboard to the rotors and back to the output mechanism, this can mean that A becomes B, B becomes C, and C becomes A. (However, there is nothing to prevent the current from going through two diodes in a row. This can be avoided by the use of resistors or multiple diodes, and a suitable choice of voltage threshold, or by a simple transistor circuit.)

That would mean the cipher of the machine as a whole would not be quite reciprocal, but no letter could represent itself. The easiest way to reverse the action of such a circuit during decipherment would be simply to reverse the polarity of DC current through the machine.

The fact that the punched-card "plugboard" affected only 30 of the machine's 31 characters was baffling to me, and implied some sort of external constraint. As I assumed a simple crossbar arrangement for card sensing, to minimize parts count, the fact that a standard 80-column card has over 900 holes in it meant that it could indeed control a 30 by 30 crossbar patch panel. (One needs 961 hole positions for a 31 by 31 crossbar patch panel!) The diagram shows a possible arrangement for the crossbar circuitry that uses the first 75 columns of the card, and which, by taking into account the zone and digit positions of the card, allows key cards to be produced with conventional punched card equipment, using only &, -, /, and the digits and the upper-case letters.

Perhaps the construction I reached through speculation will be helpful to amateur builders intending to build a replica of the Fialka at home, but with the use of more standard parts, even if it departs from the historical reality.

The following diagram is provided for those who aren't familiar with the code used on the 80-column punched card:

or, if that diagram is difficult to understand, here is a more explicit one:

Gordon Welchman, the inventor of the diagonal board, has commented that it would have been possible, but difficult, to design an Enigma that did not have the feature of being unable to encipher a letter as itself.

Obviously, the simplest method to avoid this problem with the Enigma would be to revert to sending the signal through the rotors in only one direction, as with the Enigma A and Enigma B, or as with the original Hebern machine. But if one wanted to avoid having to switch a large number of wires when changing from encipherment to decipherment, or if one felt it useful to have the signal going through the rotors twice instead of just once, how would an Enigma not having this problem be designed?

One can begin by merely changing the labels which are over the indicating light bulbs. If the light bulb that is on the circuit with the Q key is now labelled B instead of Q, then it is B that the letter Q will not be enciphered as.

But then one would have to have one wiring for encipherment, and a different wiring, either selected by a large and complex switch, or by constructing enciphering Enigmas and deciphering Enigmas.

One way to avoid the problem would be to replace the plugboard of the Enigma with two switchboards, one connecting the keys to the machine, and one connecting the lamps to the machine. Then one could select any wiring configuration for encipherment, and its counterpart, with lamp and key wirings exchanged, for decipherment.

Could a suitable choice of new labels for the light bulbs allow a simpler solution, one that involved a lesser change to the way the Enigma worked? One possibility would be, given the keys are connected to the rotor contacts in alphabetical order, to label the light bulb connnected to the A key with N, the light bulb connected to the B key with O, and so on. Then, if the setting rings are labelled with two alphabets similarly displaced, in contrasting colors, and the gear teeth by which one ring advances the next are so arranged that if a tooth is at one position, there is also a tooth at the position diametrically opposite (shortening the period, because all the rotors must now have an even number of teeth), one could indeed have a machine that behaved almost exactly like the Enigma.

However, the fact that the letter A can now never be represented by N, instead of never by A, is just a fixed substitution that would soon be discovered, and then it would cause no real difficulties for the cryptanalyst. A normal Enigma plugboard would not change this: and the sockets on it would also have to be labelled in two colors, as the plugboard wiring would have to be displaced 13 positions to decipher.

Another thought would be, instead of having two switchboard-type plugboards, where a set of 26 sockets is matched by 26 plugs from within the Enigma, to have a second plugboard of the regular Enigma type in a single Enigma. In addition to the one between the rotors and the lamps and keys both, the additional plugboard would be between the lamps and the keys. The reciprocal nature of the plugboard substitution would not be a problem, and would in fact allow the same wiring to serve for encryption and decryption. This arrangement is illustrated below:

However, it wouldn't quite work as illustrated. Although the rotor substitution and the plugboard substitution are both reciprocal, that does not mean that they commute. But a more complicated wiring based on this principle would be possible, such as placing the rotors on the lamp side for decipherment; this would require a bank of 26 relays to simulate the effect of the switches in the keyboard.

One simple way to make this work, with some limitations, is this: for encipherment, use only the plugboard between the lamps and keyboard. For decipherment, also use the regular plugboard, and wire it in the same way as the plugboard between the lamps and keyboard. This works because plugboard wirings are reciprocal.

The basic problem of this sort of arrangement is that the rotors are connected to the keyboard and not the lamps in an Enigma, hence a permutation between the keyboard and lamps disrupts the relationship between the rotor contacts and the lamps and keys that makes decipherment possible when it is reversed for decipherment. Could an Enigma be modified so that the rotors were connected to the lamps? Yes, if each lamp were shorted with a diode:

This diagram shows the flow of electricity in an Enigma. On the left, a simplified diagram of the Enigma is shown. Wires at a positive potential are red, wires at a negative potential are blue, and wires carrying current are shown thicker with yellow symbols indicating the direction of current.

On the right, the arrangement involving diodes needed to allow the rotors to be connected to the lamp side of a modified Enigma is illustrated. Current that pressing a key allows to flow goes first through the shorting diode belonging to the lamp with the same letter, then through the rotors, and on its return trip lights the correct lamp, since it is going the other way, and cannot be shorted by that lamp's diode.

Suitable switching circuitry, allowing either arrangement to be selected, and reversing other connections, would allow a two-plugboard Enigma of the type shown above to even have an Uhr box connected at either or both plugboards.

Another possible way to modify the Enigma so that a letter could represent itself would be to modify the reflecting rotor so that two of its positions did not connect back to another position, but instead connected out to one of two concentric contact rings on the other side of the rotor. If either one were live, some action could be taken to bypass the wiring that normally prevented the light for the letter itself from lighting up. But because an even number of letters are necessarily involved, this makes the chance of a letter representing itself twice what the chance of any other letter representing it would be, and this creates a new, different weakness. Switching to rotors with an odd number of contacts could solve this. The illustration of the Fialka may show an attempt to do something of a similar nature.

Another machine similar to the Enigma of interest was the SG-39. It was designed by Ostwin Fritz Menzer, working for the Abwehr. This machine had four rotors, but instead of a reflecting rotor, the signal was sent back through the four rotors by a plugboard; thus, the rotors made contact with stationary contacts at both ends as in a conventional Hebern rotor machine.

This machine was designed to be used as a telecipher machine, having circuits to convert the output of the rotors to 5-level code.

However, it has an additional very important feature of interest, which makes it almost comparable to that pinnacle of rotor machine security, the SIGABA - of course, this is also true of the NEMA, described above; and perhaps a better comparison would be to the ECM Mark III.

It had three pinwheels, with 21, 23, and 25 positions. These pinwheels, like those in a Hagelin machine or in the Lorenz Schlusselzusatz, advanced one step with every letter enciphered.

Each pinwheel corresponded to one rotor, and an active pin would make that rotor advance.

In addition, the rotors had pins on them so that one rotor would cause another rotor to advance. These worked like the pins on the pinwheels, rather than like gear teeth, and so if a rotor was stationary, an active pin on it would, as long as it remained in the position where that active pin was in the appropriate position, cause the appropriate other rotor to advance repeatedly.

The machine had four rotors, 1, 2, 3, and 4. Pinwheel 1 made rotor 1 advance, pinwheel 2 made rotor 2 advance, and pinwheel 3 made rotor 3 advance.

The pins on the rotors themselves worked as follows: rotor 1 caused rotor 2 to move; rotor 2 caused both rotors 1 and 3 to move; and rotor 3 caused rotor 2 to move.

Rotor 4 was a stator, and did not move at all during encipherment.

This diagram shows how it worked, although it shows a lampboard instead of an assembly to convert to teleprinter code.

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