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Working principle of the Enigma
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This page is currently under construction.
Eventually, it will contains an accurate description of the working
principle of the Enigma machine.
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When studying the working principle of the Enigma, we have to consider
that there are in fact many different variants of this machine.
Some of the differences make it impossible to decrypt a message
that was encoded on another model. That does however not affect the working
principle as explained here. For this we study the circuit diagram
of an standard 3-wheel Army Enigma (Heeres Enigma).
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Simplified circuit diagram of a 3-wheel Service Enigma
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Letters are 'scrambled' by a set of rotatable wheels each with 26 contacts
on either side. Each contact on one side is connected (wired) to a
contact on the other side in some random fashion.
Some models, like the standard Service Enigma and the M3 have
3 such rotating wheels, but the M4 model, used later in the war
exclusively for the German U-Boats, has 4 wheels.
Each time a key is pressed, the right most wheel
is rotated by one step, resulting in a different mapping of the
internal wires. As a result, each new letter is encoded
differently.
Each wheel has one or more notches that may cause
the next wheel to be moved by one position too. If a wheel has only
one notch, it needs to complete a full revolution before the wheel
to the left of it is stepped by one position.
The keyboard consists of 26 keys, marked A-Z.
Whenever a key, say Q, is pressed the wheels will
be moved into a new position and a contact is closed.
As a result a current will flow.
The wires from the 26 keys are connected to a static wheel called
the Stator or Entrittswalze (ETW). The order in which
the keys are connected to the 26 contacts on the ETW varies
between the different Enigma models.
Leaving the ETW, the current enters the right most wheel (1) via one
of the contacts at its right hand side. The internal wiring of that wheel 'translates' this current to one of the contacts on the left side of
the wheel. From there the current is 'handed over' to the next wheel,
and so on. Left of the rotating wheels is the
Reflector, or Umkehrwalze (UKW). This wheel sends the current
back into the rotating wheels, but this time the current flows
from left to right, until it reaches the ETW again. From the
ETW the current goes to the lamp board where the corresponding
letter (E in the example) will be lit. It is inherent to this
design, that a letter can never be encoded into itself.
Before starting the ciphering process, the Enigma needs to be
setup in a known way at both sides of the communication link.
This means the wheel order (Walzenlage) needs to be known
as well as the starting position of each wheel (Grundstellung).
In order to further complicate things, each wheel has a settable
index ring that moves the contacts independant of the wheel's alphabet.
This is called the ring setting (Ringstellung).
To make life even more complex, the Army machines were all
equipped with a plug board, or patch panel (Steckerbrett),
that allows pairs of letters to be swapped.
Any number of cables from none to 13 may be connected to the
Steckerbrett, meaning that between 0 and 13 letter pairs may
be swapped. If a letter is not mapped (i.e. no stecker is used for
that letter), the letter is known to be Self-Steckered.
See below for more information.
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Wheel rotation in more detail
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Below each key of the keyboard, is a two-position switch.
The key has to be full depressed, before the switch is activated.
The key also controls the wheel stepping mechanism. Whenever a
key is pressed, the wheels are moved into a new position before
the switch is activated and a lamp is turned on.
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Each wheel has 26 positions that we will call A-Z. The index on the
wheels is engraved (either as A-Z or 1-26) along the side of the wheel.
When a key is pressed, the wheels are rotated counter clockwise,
when viewed from the ETW. If the letter A
was visible in the window, the letter B will be visible next time
the wheel is moved.
Each wheel has a ring that can be used to rotate
the wiring independantly of the index. This can be regarded as
creating an offset in the opposite direction.
The wheel-turnover notches are fixed to the index. Therefore the turnover
of the next wheel, will always happen at the same letter in the window.
Most Enigma models are equipped with stepping levers and notches
and will therefore exhibit a double stepping feature (see below).
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Exploded view of a wheel
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The Zählwerk Enigma and its successor the Enigma G however,
use a different wheel stepping mechanism based on cog wheels,
and does not suffer from the double stepping behaviour.
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The army variants of the Enigma (Service Enigma, M3 and M4) were
equipped with a Steckerbrett (plug board or patch panel) that would
allow any pair of letters to be swapped. If a patch cable was used beteen
G and
P, those two letters would be swapped.
As the Steckerbrett is connected between the keyboard and the ETW,
the encoded letter will go through stecker mappings twice. This
would still prevent a letter from being encoded into itself.
As we have 26 characters, a theoretical maximum of 13 cables could
be used. Due to the way the Steckerbrett is constructed, any number
of cables can be used, from none to 13.
The mathematical optimum however (i.e. the combination
that produces the highest number of permutations) is at 11 cables.
In practice, Enigma operating procedures dictated that 10 cables
had te be used at all times. This is remarkable, as it seriously
reduces the maximum number of permutations.
With most machines, 12 cables were supplied; 10 to be used on the
Steckerbrett and two spares that were stored inside the top lid of
the wooden case.
The picture shows a single patch cable with a plug at either side.
Each plug has two pins. To prevent a plug from
being inserted the wrong way around, one pin was thicker than the other.
Two wires were used to cross-connect the pins of both plugs.
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Differences in Enigma models
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It has been stated before that there are many
different versions
of the Enigma. The various models may differ in:
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- Steckerbrett
Some models have a plug panel and some don't. The theoretical
maximum number of patch cables is 13 (as we have 26 letters),
but the number of cables supplied with the unit varies. The
highest number of permutations is achieved with 11 patch cables.
- ETW mapping
The Eintrittswalze (ETW) can be mapped in a linear fashion: ABCDEFGH... etc, but also
in the order of the keys on the keyboard: QWERTZUIO...
On the Japanese Enigma machine, the Tirpitz, the contacts of the ETW
are organised in a random fashion: KZROUQHY...
- Numbers or letters
Some wheels have numbers (01-26) on their perimeter, whilst others
carry letters (A-Z). Initially all Enigma machines used letters (A-Z)
on their wheels. This is definitely the case for all commercial Enigma
machines produced prior to WWII. When the German Army adopted the machine
for military use, they added a Steckerbrett (see above) and changed the
lettering of the wheels into numbering (01-26). The (later) Naval machines
however (M3 and M4), had letters again.
- Number of different of wheels
Some models have 3 rotatable wheels, but the M4 has 4 wheels.
Also some models have a range of wheels (e.g. 8) to choose from.
The wheels may be placed in the machine in any particular order.
On an Enigma M4 (a 4 wheel machine), the extra wheel is not moved
automatically, but can be set manually to an initial position.
Furthermore the extra wheel cannot be exchanged with the other
three wheels as it is a 'thin' one.
The 4th wheel was supplied as a pair with an UKW.
For UKWs B and C, the extra wheels Beta and Gamma where supplied,
hence the name Griechenwalze (Greek wheel). They may be used however in any combination.
The 4th wheel on an Abwehr Enigma (G-series) is moved by the other wheels, due to
the mechanical difference of this model.
- UKW mapping and setting
Some models have more than one UKW available.
On most models the UKW is fixed, but on some the UKW can be given a start position.
Additionally, the G models have a movable UKW, which means that the wheel can
be moved by the notches of the wheel next to it.
- Number of notches on each wheel
In the basic situation, each wheel has one notch which, after a full revolution,
causes the next wheel to be stepped by one position. Some versions have two
or even more notches on each wheel, causing more frequent changeovers of the next wheel.
The three wheels of the Enigma-G have 11, 15 and 17 notches respectively.
- Single or double stepping
As a result of the mechanical principle of the stepping mechanism, the middle
rotor 'suffers' from a double stepping anomaly, described in a paper by David
Hamer
[1]
. The Enigma-G, which use a gear box instead, does not suffer
from this anomaly.
- Manufacturer
Before and during WWII, the Enigma machines were built by various manufacturers.
Although these machines were mathematically compatible, there are a few cosmetic
differences. Additionally there are physical differences between the thin wheels
from some manufacturers.
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The basic Enigma design has a number of waknesses that helped the
allied codebreakers during WWII to some extend.
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- A letter can never be encoded into itself
One of the key properties of the Enigma design is the fact that
a letter can never be encoded into itself. In other words: when
the letter A is pressed, every lamp on the lamp panel can be
lit, except for the letter A itself.
This property is caused by the fact that a reflector (UKW) is used.
- Regular stepping of the wheels
In most Enigma machines, the righmost wheel needs to complete a
full revolution before the wheel next to it is moved by one
position. As a result, the 2nd wheel is only stepped once every
26 characters and the 3rd wheel hardly ever moves. This makes
the Enigma more predictable.
Some Enigma variants (such as the Enigma T) had
multiple turnover notches and
the Zählwerk Enigma even featured a cog
wheel mechanism to cause irregular stepping.
- Double stepping of the middle rotor
Under certain circumstances, the middle rotor can make two
steps on two subsequent key presses. This effectively halves
the cipher period. The double stepping feature is described
in a paper by David Hamer
[1]
.
- 4th wheel not moving
In the Naval Enigma M4, the extra wheel (Zusatswalze) can be set
to any of 26 position at the start of a message. During
encipherment, however, the wheel never moves. Together with the
UKW, this wheel can be regarded as a selection between 26
different UKWs.
- 2 Notches on the extra Naval wheels
The three extra Naval wheels (VI, VII and VIII) each have two
notches to cause a more frequent wheel turnover. However,
because 2 is a relative prime of
26 and because the two notches are positioned oposite each
other, the cipher period is effectively halved.
- Mandatory use of extra Navel wheels
If the operator could pick any three wheels from the available 8
on any given day, there would have been 336 possible different
wheel orders. In practice however, the Navy was instructed to
use at least one extra Naval wheel each day (VI, VII or VIII)
and that the selected wheel could not be used on two adjacent
days.
- Fixed number of cables on the Steckerbrett
The Steckerbrett has 26 sockets, one for each letter of the
alphabet. Cables were used to swap pairs of letters. If
a cable was omitted, that letter would not be swapped.
In theory, any number of cables between 0 and 13 would thus
be possible. In practice, the procedures commanded the use
of exactly 10 cables at all times, which greatly reduces the
maximum number of possibilities.
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