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I know how an Enigma works and that the reflector means a current flow must return by a different route but my question is this. After returning from the reflector and through the rotors, it passes through the plugboard before lighting up a different letter than was entered. Now imagine the ‘a’ key is pressed and returns to the plugboard as, say, a ‘w’. But what then happens if, on the plugboard, the ‘w’ is paired with ‘a’? Then isn’t the current then directed to ‘a’, the key that was pressed.

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What you missed is that the signal goes through the plugboard twice.

Quoting Wikipedia:

Current flows from the battery (1) through a depressed bi-directional keyboard switch (2) to the plugboard (3). Next, it passes through the (unused in this instance, so shown closed) plug "A" (3) via the entry wheel (4), through the wiring of the three (Wehrmacht Enigma) or four (Kriegsmarine M4 and Abwehr variants) installed rotors (5), and enters the reflector (6). The reflector returns the current, via an entirely different path, back through the rotors (5) and entry wheel (4), proceeding through plug "S" (7) connected with a cable (8) to plug "D", and another bi-directional switch (9) to light the appropriate lamp.[21]

So in your example, if A and W are connected in the plugboard, then when "A" is pressed, the plugboard changes it to "W". It then heads to the rotors (which we assume change it back to "A"), and back to the plugboard, which changes it back to "W". I will note each plugboard cable has two wires to allow this to happen.

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Short answer: Enigma, with or without a plugboard, is self-inverse and never encode to itself. If $\mathtt{A}$ maps to $\mathtt{W}$, for the same machine state $\mathtt{W}$ maps to $\mathtt{A}$.

Longer answer: Instead of a hypertheoretical example, trying-it-out using one of the many excellent Enigma simulators, Cryptii Enigma Simulator, out there could be far more fruitful. Let's use a training message, Frank Carter (2010) The Turing Bombe, The Rutherford Journal 3, used at the Bletchley Park. Typing in plaintext $\mathtt{WETTE\ RVORH\ ERSAG\ E}$ at a particular initial position returns ciphertext $\mathtt{SNMKG\ GSTZZ\ UGARL\ V}$. With the same initial position, typing in ciphertext $\mathtt{SNMKG\ GSTZZ\ UGARL\ V}$ returns plaintext $\mathtt{WETTE\ RVORH\ ERSAG\ E}$. The self-inverse property operates at a character-by-character level, for example, typing in $\mathtt{WEMKE\ GSTZZ\ EGAAG\ E}$ returns $\mathtt{SNTTG\ RVORH\ URSRL\ V}$.

One of the 6760 possible initial settings is Reflector UKW-B, Rotors II V III, Rings 26 26 26, Indicators U L A and Plugboard AD ET FU GQ HM JL NV ZP IK XO.

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Here is a partial Enigma schematic showing only 4 keys and light-bulbs, and 6 contacts per reflector (UKW), rotor side, and plug-board (Stecker) side (out of 26 due to space limitation). Simplified circuit diagram of a 3-wheel Service Enigma Simplified circuit diagram of a 3-wheel Service Enigma. Source: Cryptomuseum.com, used with permission.

Looking at the switch of a key (red-tinted rectangle for the letter Q), we see that when a key is pressed, the corresponding light-bulb is disconnected, thus it can't light up.

The current out (in red) from the keys and light-bulbs on the right goes thru a permutation defined by plug-board and rotors. The reflector routes the in-going current reaching one of it's connection to another of it's connections. The current returns (blue) to some connection of the keys and light-bulbs block. That can't be the connection of the originating key, because rotors and plug-board perform a permutation. Therefore the returning current goes thru a connection of another key and light-bulb. That other key is not pressed thus the other light-bulb is connected and lights up.

Thus, for any fixed setting of the plug-board, rotors (wiring, order, rotation), and reflector, the key-to-light-bulb function has two properties:

  • $\forall x\in[\mathtt A,\mathtt Z],\ f(x)\ne x$; that is, it's non-stationary. Pressing key $x$ never lights $x$.
  • $\forall x\in[\mathtt A,\mathtt Z],\ f(f(x))=x$; that is, it's an involution. If pressing key $x$ lights $y$, then pressing key $y$ would have lit $x$ for that position of the rotors.

That follows from each key and light-bulb pair having a single connection to the plug-board, and use of wires and contacts which conduct in both directions between these 26 connections. That limits to $25\times23\times21\ldots3=2^{\approx42.8}$ possible key-to-light-bulb functions; however the particular function changes each time a key is pressed, thanks to the rotors.

The above stands for all variants and settings of all Enigmas, even those (Mark 1 and Mark 2) not using pairs of wires for the plug-board, something that's specific to Mark 3 devices (including those deployed during WW2). The three linked Cryptomuseum pages show the plug-board differences between the models. It's explained how the Mark 3 design restricts the plug-board to involutions ($2^{\approx47.6}$, with $2^{\approx47.0}$ using the 10 wires common during WW2), rather than arbitrary permutations ($26!$ that is $>2^{88}$) in Mark 2.

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Why not? I'll rephrase your question as "Now imagine the $\mathtt{G}$ key is pressed and returns to the plugboard as, say, a $\mathtt{Q}$. But what then happens if, on the plugboard, the $\mathtt{Q}$ is paired with $\mathtt{G}$? Then isn’t the current then directed to $\mathtt{G}$, the key that was pressed." @Eugene Styer said, "I will note each plugboard cable has two wires to allow this to happen." Having said/done that, I don't see how this exercise brings any insight on how Enigma works.

The Steckerbrett

Image source: Crypto Museum (2009) How does an Enigma machine work?

                               Q  G
                               ^  |
                               |  |
                               |  v
plugboard        (AD)(ET)(FU)(GQ)(GQ)(HM)(IK)(JL)(NV)(OX)(PZ)(B)(C)(R)(S)(W)(Y)
                              ^    |
                              |    |
                              |    +---+
                              +-------+|
                                      ||
                                      |v
Letchworth       (AF)(BO)(CK)(DW)(EI)(GQ)(HV)(JZ)(LN)(MS)(PX)(RY)(TU)
scrambler
pos   4  U L E - 

One of many possible Enigma machine states is Reflector UKW-B, Rotors II V III, Rings 26 26 26, Indicators U L E and Plugboard AD ET FU GQ HM JL NV ZP IK XO. Note: this setting is after an input key is pressed. If one enters this setting into a typical Enigma simulator, s/he needs to set initial Indicators U L D.

Further note: a Letchworth scrambler is the combined effect of three rotors, a reflector and return thru' three rotors in reverse order, i.e. from entry wheel ETW back to entry wheel.

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