Duke University
Enigma
Yuanhao Guan
Math 89S: Mathematics of the Universe
Professor Hubert Bray
December 2016
1. Brief Introduction and Historical Background
The Enigma machine is one of the most famous cipher machines of all time. It was used
by Nazi Germany in World War II to send secret, coded messages. Enigma was invented by the
German engineer Arthur Scherbius at the end of World War I. Scherbius' Enigma, in a modified
and improved form, was later used widely throughout the German armed forces as the standard
method of encrypting messages prior to radio transmission. In fact, since it has so many possible
combinations of settings and each setting would produce a unique combination of encryption,
one German cryptanalyst stated that the Enigma's messages were unbreakable. Later, At the
Biuro Szyfrow, three brilliant cryptanalysts, Marian Rejewski, Henryk Zygalski and Jerzy
Rozicki, succeeded in breaking the Enigma in 1933. After hearing this groundbreaking news, the
Germans increased the sophistication of Enigma in 1939 which thwarted further Polish attempts
at cryptanalysis at that time. Actually, no one can invent a systematic way of cracking Enigma
messages efficiently which means lots of important Enigma messages remained encrypted until
Bombe was invented by Alan Turing with an important refinement devised in 1940 by Gordon
Welchman which can solve any Enigma message in less than 20 minutes. [1]
Though Enigma had some cryptographic weaknesses, in practice it was German
procedural flaws, operator mistakes, failure to systematically introduce changes in encipherment
procedures, and Allied capture of key tables and hardware that, during the war, enabled Allied
cryptologists to succeed and "turned the tide" in the Allies' favor. Enigma has already proved
itself to be one of the most successful electro-mechanical rotor cipher machines in 20th century.
2. Description of The Enigma Machine
(a less common version of military type Enigma Machine which has 4 rotors)
There were many variants of the Enigma, but the most common version, and the version
described here, is the standard Army and Luftwaffe machine. Its main components were: a
keyboard, a lampboard, a stecker or plugboard, a reflector, and a set of five rotors. (As the
picture illustrated below)
The keyboard is just a regular keyboard which only has 26 keys, each of which
corresponds to a letter of the alphabet.
The lampboard is consisted of 26 glow lamps, each of which, when illuminated, indicated
a letter.
The plugboard (Steckerbrett in German, or ‘Stecker’) is the unique feature that military
used to increase the sophistication of enigma. If you were a banker or businessman in Germany
at that time, you might buy a commercial enigma but you would not find a plugboard on your
enigma machine. The plugboard has 26 sockets, each has a letter printed next to it. There’s 10
pair of wires that can connect any pair of these sockets. When you use wire to connect two
sockets, the letter that are represented by these two sockets are now swapped. So if you connect
A to B, a letter that was originally encrypted into A would now become B and vice versa. The
enigma machine was carefully designed that if there’s no wire connecting the socket, that socket
makes an electrical connection with itself. (So if C socket was unoccupied, C is connected to C
itself.) Although there’s only 10 wires, it might surprise you that the number of permutations
afforded by this arrangement is greater than the number of permutations if all 26 letters are
connected with each other. The greatest number of permutations is attained when there are
eleven wires. It remains unknown why the Germans used only ten.
(a traditional plugboard used by military Enigma)
Each enigma has 5 distinct rotors, although it only requires 3 rotors to function. There’s 3
sockets in enigma that’s designed to contain rotors, and each rotor can be taken out and replaced
with ease. Each rotor had 26 contact pads on the left and 26 pins on the right. The pins and
contact pads are internally wired in an irregular and seemly random way. The pattern of wiring
was different in each of these 5 rotors and each rotor was stamped with a Roman numeral to
distinguish which rotor was which. Every time operator pressed a letter, the right motor moved a
bit, and it takes 26 times for the right motor did a full turn. And it would kick the middle rotor
one place after the right rotor rotated a full turn. Imaging these rotors as the hands on a clock and
you would get the idea.
There’s also a rotor on Enigma that can’t be removed or replaced, and it’s called a
‘reflector’. (German: Umkehrwalze, meaning 'reversal rotor') The reflector connected outputs of
the last rotor in pairs, redirecting current back into the rotors. The reflector gave Enigma a
unique and convenient property: self-reciprocal, which means encryption would be same as
decryption. (Actually, putting the reflector onto Enigma was not designer’s idea and it has been
proved that putting this reciprocal operation was a terrible idea since although it did increase the
sophistication for enigma, it also implements a serious flaw, otherwise Turing’s method would
not work, as we will discuss later.)
(rotors and how they are wired internally)
Unlike a Caesar cipher, each letter would not be mapped to the same letter since each
time you press the letter the rotor rotates and maps it to a different letter. This makes it nearly
impossible to decipher an encrypted Enigma message by hand. Of course, it is pretty reasonable
to think that even though we can’t solve the encrypted messages, we would solve and decipher
every message Nazi sent if we can capture one functional Enigma machine. Unfortunately, it is
not the case. Even though you have the exact Enigma that Nazi used, you still don’t know how to
set the Enigma in order to decipher the code. So how many settings are there? Every Enigma is
equipped with 5 rotors, and you should pick 3 from 5 in order to get it to work. You have 5
choices for the first slot, 4 choices for the second slot, and 3 choices for the third slot which
gives you 5*4*3=60 ways that you can pick your rotors. You have 26 starting positions for each
rotor, which gives you 263=17576 different starting positions. Now things really get worse when
we move on to the plugboard. There are 26 letters in the alphabet and we want to know how
many ways we can arrange them, which gives us 26!. However, we only want 10 pairs, which
means there are 6 letters we don’t want, and we also don’t need to know the order of these 10
pairs, so we should divide 26! by 6! and 10!. And since AB and BA is the same pair, we divide 2
for each pair, so we divide the product above by 210 and we will have 150,738,274,937,250. If
we multiply all the numbers above together and we will have something around 159 quintillion.
[10] No wonder German cryptanalyst would say it is unbreakable. This number is so big that can
make any mathematician on the world to be desperate. So how did Nazi tell each other their
settings?
(an actual monthly sheet used in World War II)
German army and air force would use a monthly sheet which tells them how to set the
machine each day of that month. Since it is a monthly sheet and Nazi would change the settings
if they know it was stolen, it was not that valuable. Also, this was only a sheet of paper plus it
was written using soluble ink, it was really easy to destroy if soldiers get caught, which made it
even harder to obtain such a monthly sheet. No doubt, if you can steal one of these sheets
without Nazi noticing, which they did occasionally, you can use that sheet to decipher the code
until it runs out. However, without such monthly sheets, you would need mathematics to take
advantage of the flaw that Enigma has, otherwise you won’t decipher any Enigma message in
your life time. (As for German naval, they would use a separate part of code using another kind
of encryption just to tell receivers how to set the machine, which made the decryption “harder
than harder”.)
3. How to decipher
As we mentioned before, Enigma has a major flaw. In order for the reflector to work, the
electrical current has to be reciprocal, which means there has to be a loop inside the Enigma
machine that can take letter A to letter B and also take letter B to letter A, but here comes the
problem: it is possible A is mapped to any other letter in the alphabet, but it is not possible to
map A to A since it would create a single line connecting two points instead of a loop, which is
not allowed in the design of the Enigma. This flaw inspired Alan Turing to invent the famous
bombe machine (It get its name from the Polish decipher machine ‘bomba’, but they are similar
only in that both are rotary electro-mechanical machines). [5] However, contrary to the machine
described in the movie The Imitation Game, a real Bombe machine usually can’t decipher an
Enigma message. In the word of Gordon Welchman, "... the task of the bombe was simply to
reduce the assumptions of wheel order and scrambler positions that required 'further analysis' to
a manageable number". [6]
In order to decipher the ciphertext, the first step would be obtaining a crib—which is a
section of plaintext that was thought to correspond to the ciphertext. It was not that hard: German
military communications were often expressed in a stereotypical manner so that it was not
necessarily difficult to guess a particular phrase that might appear in the message, like “Weather
Report” or “Heil Hitler” in German. [11] Then, we can pin down the exact position of this crib in
the ciphertext by using Enigma’s property: it never encoded a letter as itself. Suppose we have
the following cirb: [7]
WETTERVORHERSAGEBISKAYA
Now suppose we know the following ciphertext includes the sequence:
QFZWRWIVTYRESXBFOGKUHQBAISEZ
To determine the location of crib inside the ciphertext, we slide the crib alongside the
ciphertext until we find a position that same letter doesn’t appear in the same position. [9]
For example, we will find the first S in ciphertext would be in the same position as in the
crib, so we would silde the crib along until we find a position such that there would be no
contradiction: [7]
Now we know the crib is correctly matched with the ciphertext. We can number the crib
and ciphertext as follows:
Now we can generate a diagram using the chart above: [8]
We proceed by guessing the pairing on the plugboard. Let’s assume A is paired with B
and use the diagram above to see if it has any contradictions. We would do 26 possible pairs for
A (remember A can also be paired with A itself on the plugboard by simply leaving the socket of
A unoccupied.) If none of them is possible, we would move the right rotor by one place until we
figure out all the possible settings of the Enigma. [8] It still sounds unmanageable, so Alan
Turing came with this clever idea: Human needs at least 5 minutes to examine if a combination
of setting is possible or not, but current can run through the circuit and find if the circuit is
opened or closed by less than a second. Why not we build a machine that use current in circuits
to do this work for us? That’s exactly what bombe did. Each drum in the picture below has
identical interconnections to those of the relevant Enigma rotor. Each ‘Bombe’ would contain 36
Enigma equivalents, which means it can eliminate 36 ways of wrong settings in less than a
second. During 1940, 178 messages were broken on ‘bombe’ machines, nearly all successfully.
[2]
(Real view of the rebuilt Bombe at Blechley Park)
4. Inspiration of Typex
Since Enigma has so many amazing features and is incredibly hard to solve without
taking advantage of its major flaw, British army wanted an improved version of Enigma, which
can encrypt a letter into itself sometimes. That’s where Typex appeared on the scene. Basically
Typex was built upon the commercial Enigma machine, but it has additional features that can
improve its security drastically. The follow features are also advantages Typex has over Enigma.

Most versions of the Enigma required two operators to operate
effectively—one operator to input text into the Enigma and the other to copy down the
enciphered or deciphered characters—Typex required just one operator.

Typex avoided operator copying errors, as the enciphered or deciphered
text was automatically printed on paper tape.

Unlike Enigma, Typex I machines were linked to teleprinters while Typex
II machines could be if required.

Enigma messages had to be written, enciphered, transmitted (by Morse),
received, deciphered, and written again, while Typex messages were typed and
automatically enciphered and transmitted all in one step, with the reverse also true. [3]
A Typex machine was captured later by German forces during the Battle of France and
more than one German cryptanalytic section proposed attempting to crack Typex; however, the
B-Dienst codebreaking organization gave up on it after six weeks, when further time and
personnel for such attempts was refused. One German cryptanalyst stated that the Typex was
more secure than the Enigma since it had seven rotors, therefore no major effort was made to
crack Typex messages as they believed that even the Enigma's messages were unbreakable. [4]
Reference:
1. "The Enigma Machine." Historical Background. N.p., n.d. Web. 03 Dec. 2016.
2. "Outstations from the Park", Bletchley Park Jewels, retrieved 16 April 2010
3. Ralph Erskine, "The Development of Typex", The Enigma Bulletin 2 (1997): pp. 69–
86
4. Cipher A. Deavours and Louis Kruh, "Machine Cryptography and Modern
Cryptanalysis", Artech House, 1985, pp. 144–145; 148–150.
5. Rijmenants, Dirk; Enigma message procedures Cipher Machines & Cryptology
6. Miller, A. Ray (2001). "The Cryptographic Mathematics of Enigma" . National
Security Agency.
7. "The Enigma Machine." Description of the Enigma. N.p., n.d. Web. 04 Dec. 2016.
8. "The Turing Bombe." Description of the Bombe. N.p., n.d. Web. 04 Dec. 2016.
9. "The Turing Bombe." The Turing Bombe - Cribs and Menus. N.p., n.d. Web. 04 Dec.
2016.
10. Numberphile. "Flaw in the Enigma Code - Numberphile." YouTube. YouTube, 14
Jan. 2013. Web. 04 Dec. 2016.
11. Numberphile. "158,962,555,217,826,360,000 (Enigma Machine) " YouTube.
YouTube, 10 Jan. 2013. Web. 04 Dec. 2016.