1. Cryptanalysis During World War II: The Allies
Succeeded while the Germans Struggled
Jillian (Maggie) Kasner
September 15, 2016
Abstract:
Prior to World War II, Germany was the center of the mathematical community: David
Hilbert, Emmy Noether, Oskar Bolza, Felix Hausdorff, Ludwig Bieberbach, and Albert Einstein
(amongst others) were significant mathematicians in the 1920’s and 1930’s. Despite Germany’s
prewar mathematical strength, it was the Allies who succeeded at mathematical cryptanalysis
during World War II. This success lead to the Allied victory. This paper will address the many
reasons for Germany’s struggle, involving mathematics and the government’s reaction to crypt-
analysis. Some of these reasons include the lack of government support, financially and politically,
in Germany. German mathematicians were also restricted on what they could research. The lack
of government support and restriction on what could be researched drove many mathematicians to
flee Germany. The only German cryptanalysis team Oberkommando der Wehrmacht (OKW) to
hire mathematicians was ostracized for trying something new. The German cryptanalysis teams
did not communicate with each other. Germany also did not trust their allies intelligence and
would not always share intelligence with them.
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2. Common Ground
The cryptologic process is used by individuals, and militaries, to send a message in secret. It is
important for militaries to send and receive secret messages in order to execute strategic commands
without the enemy’s knowledge. There are two branches of cryptography: transposition and
substitution. In transposition cryptography, the letters of a message are systematically rearranged.
In substitution cryptography, the letters of a message are replaced using some sort of algorithm.
When a message is encrypted using substitution, the encryption algorithm is called a cipher. This
algorithm creates the encryption and decryption key to encode and decode the message. This
paper focuses on substitution ciphers used by the Allies and Germans during World War II.
The following diagram provides an overview of the cryptologic process.
First, an individual writes a plaintext message that anyone can read. The individual selects an
established encryption key to encode the plaintext message into ciphertext. A ciphertext message
is supposed to be only readable with a decryption key. The individual sends the ciphertext over
a channel such as air radio, print, teleprinter, etc. The decryption key, which has already been
provided to the receiver, is applied to the ciphertext and the message is returned to plaintext.
Over multiple centuries, the cipher algorithms used to create encryption and decryption keys have
become increasingly complex and led to the creation of a subfield of cryptology, cryptanalysis.
Cryptanalysis is the process of analyzing (decrypting) an encrypted message intended for
someone else, even if the cryptographic key is unknown. A message in ciphertext is vulnerable to
interception as it passes through a channel. As the cipher algorithms have become more complex,
the techniques used by cryptanalysts to decrypt intercepted messages have become more complex,
as well. Prior to 1932, very little mathematics was used in cryptanalysis. Since 1854, when the
Vigen`ere cipher was cracked, the most common field of mathematics used in cryptanalysis was
probability and statistics. Poland began hiring mathematicians and scientists in 1932. Prior to this
change most cryptanalysts were classists and linguists. In World War II, the Allies’ cryptanalysts
believed that the Enigma, the German encryption machine, was unbreakable but Poland refused
to give up trying out of fear of being invaded by Germany. Over the course of the war the Enigma
had multiple variations creating a more complex encryption and decryption process. The famous
mathematicians Marian Rejewski, Jerzy R´o˙zycki, and Henryk Zygalski were able to crack the early
versions of the military Enigma. When Poland shared their information with the Allies in 1939,
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3. the Allies were impressed with their progress and began hiring mathematicians and scientists as
cryptanalysts. The German government did not support the implementation of mathematics and
science in cryptanalysis until approximately two years before the end of World War II when it was
too late to make any progress on the more complex Ally encryption systems. It took the Allies
many years to break the earlier, simpler, versions of the Enigma.
Issue
Prior to World War II, Germany was the center of the mathematical community. David Hilbert,
Emmy Noether, Oskar Bolza, Felix Hausdorff, Ludwig Bieberbach, and Albert Einstein were all
significant German mathematicians. Hilbert created twenty-three difficult meta-mathematical
problems, called Hilbert’s Problems, and was well known for his work in algebraist number theory.
Noether worked in multiple subjects including abstract algebra and physics. Bieberbach worked on
in the field of complex analysis. These are just a few of their noteworthy accomplishments. Even
though the mathematical strength of these and other German mathematicians in the 1920’s and
1930’s was significant, it was the Allies’ who succeeded at mathematical cryptanalysis during World
War II. The struggle the German’s faced in mathematical cryptanalysis during the war occurred
for multiple reasons, many of which revolved around mathematics. For example, restrictions were
placed on what mathematicians could research and the application of the research. Many German
mathematicians either fled Germany or attempted to leave because of the hostile environment in
their country and in the mathematical community. Other reasons involved the lack of government
financial and political support.
Reasons
Allies Cryptanalytic Successes
The Allies are famous for cracking the German military’s high level security Enigma encryption
machine. Multiple versions of the Enigma were developed throughout the war. Some divisions of
the German military had specialized versions of the Enigma. One example is the famous Naval
Enigma which was barely compromised by the Allies who struggled with decrypting its messages.
This version of the Enigma used ten plugboard cables instead of the traditional three. The
Naval Enigma was very important because it was this Enigma that encrypted U-Boat instructions
to other nearby U-Boats. The codename for military messages encrypted by the Enigma that
were later decrypted by British cryptanalysts was Ultra. The Allies built different versions of
the “bombe” (machine designed to mechanize the decipherment process) to crack the different
versions of the Enigma over the course of the war.
Another high level security system cracked by the Allies was the German teleprinter ciphers.
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4. The information gathered by German teleprinter ciphers was given the codename Fish. The most
difficult of the Fish ciphers was the Lorenz cipher, codename Tunny. The machine built to crack
the Lorenz cipher was called the “Tunny Machine.” The German Navy and Luftwaffe’s (air force)
used the Siemens Halske T-52 (Geheimfernschreiber or SFM, a machine used to send messages
faster in place of the German field machine, Enigma. It was the equivalent of the German Army’s
Lorenz cipher. The codename Sturgeon was given to the information collected from messages
encrypted by the T-52.
The Allies were successful at breaking the low security systems, as well, but these systems will
not be discussed since no new mathematical techniques were created to crack those systems [4].
Germans Cryptanalytic Successes
After World War II, cryptanalysts at the National Security Agency (NSA) interviewed German
cryptanalysts about the German’s success in cracking the Allies encryption systems. The Germans
conveyed that they were very successful at cracking the Allies’ low and medium security systems,
including the pre-war British Naval Code, British and Allied Merchants Ships (BAMS) code, U.S.
Navy codes prior to 1942, British ”Naval Cypher No. 3” which was used for radio communication,
various low grade British Naval and Air Codes, the M-209 (U.S. Navy) and M-94 (U.S. Army)
field cipher machine, U.S. strip ciphers, and the ”Black” codes used by U.S. diplomats [5].
The Germans created two machines to decrypt the U.S. strip cipher and the M-94 cipher.
The machine crafted to crack the U.S. strip cipher was called the “statistical depth-increaser.”
The Germans used a similar cryptanalytic approach on the M-94 cipher. Some Germans thought
they were intercepting and decrypting U.S. strip ciphers but the messages were actually encrypted
using the M-94. The machine created to decrypt the M-94 was called “automation” [1].
The Germans were not successful at cracking the Allies’ high security systems, like the SIGABA
and TypeX. The SIGABA, also referred to as the ECM Mark II, (U.S. encryption machine) and
TypeX (British encryption machine) were advanced versions of the Enigma. After the Allies
cracked the Enigma, they transformed the Enigma into the SIGABA and TypeX in order to
assure the ciphers could not be broken using the cryptanalytic techniques created by the Allies;
thus, making the SIGABA and Type X more difficult to crack than the Enigma. The TypeX was
easier to crack than the SIGABA and harder to crack than the Enigma, although the TypeX was
never broken. The TypeX was easier to crack because it was made to be portable, which means
the designers were unable to put all the security measures used in the SIGABA which is why
the SIGABA was extremely large. The TypeX was created to facilitate secure communication
between Americans and the British on the battlefield.
Allies’ Government Reaction to Cryptanalysis
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5. The Allied governments were very supportive of cryptanalysts and provided them with plentiful
resources. The Polish government did not have many cryptanalysts when they started intercepting
messages from the Germans. The government turned to mathematicians because of the political
environment and fear of German invasion. In order to boost recruitment, the government cre-
ated a cryptology course in Pozn`an to increase mathematician’s interest in cryptanalysis. Three
famous mathemeticians, Marian Rejewski, Jerzy R´o˙zycki, and Henryk Zygalski were trained as
cryptanalysts in Pozn`an courses [7].
During World War I, Britain expanded the Government Code and Cypher School (GCCS).
Bletchley Park was a famous unit of GCCS. It was a very small cryptanalysis bureau in 1919 that
grew during World War II to a peak of almost 9,000 employees [6]. The cryptanalysts worked with
the employees at GCCS to crack the Enigma. The British government provided Bletchley Park
more resources to train mathematicians in cryptanalysis in order to accelerate progress and crack
the Enigma. The British government spent £100,000 per bombe (machine designed to mechanize
the decipherment process). The bombes were approximately 2.1 x 2.0 x 0.6 meters and weighed
approximately one ton. The British government used resources to build these bombes instead of
building new ships to replace the ones sunk by German U-boats. Approximately 210 bombes were
created and sent throughout Britain [6]. Even though finances were tight during the war, the
British government provided cryptanalysts with as many resources as possible.
Winston Churchill was so impressed with the members of Bletchley Park that on September
6th, 1941 he visited Bletchley Park’s codebreakers. After Churchill’s visit, Alan Turing, a famous
mathematician cryptanalyst, and his colleagues sent Churchill a letter asking for more staff to
improve the speed of the bombes decryption process. Previously the Director of Bletchley Park,
Commander Edward Travis, had blocked Turing’s request because he could not justify hiring
more cryptanalysts. Churchill issued a memorandum to his staff officer that stated Bletchley Park
is an “extreme priority” and should be given the necessary resources to succeed [8]. Churchill
realized that to end the war he needed the upper hand when making strategic military decisions;
for example, being able to decipher German military messages to ensure British ships were able
to avoid the U-boats who were lying in wait. This advantage saved countless lives and resources.
German’s Government Reaction to Cryptanalysis
During World War II, Germany was a very hostile environment. This hostility was not only
directed towards Jews, but anyone who opposed what the government associated with being
“Jewish.” The Deutsche Mathematiker-Vereinigung (DMV), the German Mathematical Society,
declared that pure mathematics was considered Jewish and therefore, would not allow or support
applied research in the mathematics field, only theoretical mathematics was allowed. Mathematics
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6. professors who did not follow this rule would be visited by Vahlen, Beiberbach, and other members
of the Nazi Party and DMV who dressed in Nazi uniforms [2].
The Hitler Regime passed multiple laws that restricted who could work in certain professional
fields. One law was the “Law for the Restoration of the Professional Civil Service” (passed on
April 7th, 1933) which stated that “civil servants who are not of Aryan descent are to be retired; if
they are honorary officials, they are to be dismissed from their official status” and that this “does
not apply to civil servants in office from August 1, 1914, who fought at the Front for the German
Reich or its Allies in the World War [I] or whose fathers or sons fell in the World War [I]” [3].
These hostilities led many mathematicians to flee or attempt to flee from Germany. Math-
ematicians also fled because they had lost their job, had opposing political views, or supported
Jews. Some of these mathematicians worked for the Allied governments in cryptanalysis and other
mathematics fields that helped build a stronger Allied military. Once the Germans realized Ger-
man mathematicians were working with the Allied forces they stopped allowing mathematicians
and scientists to leave the country. Some mathematicians who tried to flee after this point were
sent to concentration camps and others were publicly executed to create deterrence.
The Hitler Regime allocated a lot of resources to scientists studying the toxic chemicals used at
concentration camps to kill large portions of the Jewish population. Scientists who worked for the
government largely worked in military equipment and chemical warfare divisions [2]. The Hitler
Regime also focused their material resources on building and maintaining military equipment.
This left very few resources to build machines to decrypt Allied messages. The Regime also lacked
monetary resources to hire and train cryptanalysts to tackle the higher security systems used by
the Allies. Towards the end of the war, the Hitler Regime realized more resources should have
been dedicated to cryptanalysis. Hitler became very open about his support of cryptanalysis and
mathematicians’ working in the field, but it was too late to make any headway on the complex
systems used by the Allies (SIGABA and TypeX). It took the Allies almost a decade to crack the
Enigma and there was only about two years left in the war [5].
Ally Cryptanalysts and Techniques
Understanding the Enigma
The Polish were the first to successfully crack the Enigma. In September of 1932, Biuro
Szyfr´ow of the Polish Cipher Bureau in Warsaw, Poland hired Marian Rejewski, Jerzy R´o˙zycki,
and Henryk Zygalski. After Rejewski’s work became unclassified he wrote multiple papers on his
work cracking the early versions of the Enigma. The Polish recognized that the application of
mathematics to cryptanalysis was needed to crack the new cipher machines, like the Enigma. It
is important to first know how the Enigma works in order to understand how the Polish cracked
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7. the Enigma. The Enigma had a twenty-six letter keyboard. Behind the keyboard, twenty-six
letters were illuminated by lamps. The Enigma had three rotors, a fourth stationary reflector,
a plugboard, and a stationary entry ring which was the link between the commutator and the
right rotor. The three rotors had twenty-six letters each [7]. The figure below shows the path of
a plaintext letter turning into the ciphertext.
Retrieved from Francisco R. Villatoro’s blog [9]
When enciphering a message using the Enigma, the German’s made multiple mistakes which
helped the Allies’ crack the Enigma. These mistakes included, but were not limited to, multiple
transmissions of the same message, repetition in messages, starting with the same words or phrases,
ended transmissions with Heil Hitler!, sending early morning weather reports promptly at 6am,
often sending the message “Nothing to Report,” and the initiator sending new rotor locations for
each message twice, like “BGRBGR.”
Polish Mathematical Cryptanalysis Techniques
The encipherment process creates permutations. We know that any permutation can be written
as a set of cycles. Let
dmq vpn
von puy
puc fmq.
This means that d → v, v → p, and p → f. Continuing this pattern we find that
AD = (dvpfkxgzyo)(eijmunqlht)(bc)(rw)(a)(s).
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8. Multiple sets like this one are collected and called “characteristic sets.”
Rejewski, R´o˙zycki, and Zygalski created many permutations-based theorems to crack the
Enigma. Three of these theorems were mentioned in Rakus-Andersson’s paper The Brains be-
hind the Enigma Code Breaking before the Second World War:
Theorem 1: If two permutations X and Y of the same degree comprise disjunc-
tive transpositions, then their product XY will include disjunctive cycles of the same
lengths in even numbers. The converse is true as well.
Theorem 2: If a permutation includes disjunctive cycles of the same lengths in even
numbers, then the permutation may be regarded as a product XY of two permutations
X and Y , composed of disjunctive transpositions.
Theorem 3: Letters entering into one and the same transposition of permutation X
and Y , always enter into two different cycles of the same length, which belong to the
permutation XY .
These theorems helped determine the connections between the plaintext and ciphertext. They
denoted the permutation caused by the commutator as the letter S, permutations caused by the
rotors as the letters N, M, L, and the permutations caused by the reflector as the letter R. Let
H be associated with the permutation created by the entry ring, this is to be considered as the
identity permutation. The path a letter takes is represented as
SNMLRL−1
M−1
N−1
S−1
.
The last permutation needed is P, which is P = (abcd...xyz). This information led Rejewski,
R´o˙zycki, and Zygalski to find the following:
AD = SPNP−1
MLRL−1
M−1
PL−1
P3
NP−4
MLRL−1
M−1
P4
N−1
P−4
S−1
BE = SP2
NP−2
MLRL−1
M−1
P2
N−1
P3
NP−5
MLRL−1
M−1
P5
N−1
P−5
S−1
CF = SP3
NP−3
MLRL−1
M−1
P3
N−1
P3
NP−6
MLRL−1
M−1
P6
N−1
P−6
S−1
This reconstruction of the permutations created by the above method only works on deciphering
a single day’s cipher [7].
In summary, Rejewski, R´o˙zycki, and Zygalski used cyclic notation to analyze the relationship
between the 1st
and 4th
, 2nd
and 5th
, and 3rd
and 6th
letters which was called a “cyclometer.”
Then they created a list, called the Cryptologic Card Catalog, of all the possible scrambler settings
associated with the number of links within a set of permutations. They designed the “bombe”
to decipher messages once the link length had been determined. In 1939 when Germany invaded
Poland, Rejewski, R´o˙zycki, and Zygalski were forced to flee to Romania, then France, and lastly
England. In 1946, Rejewski returned to Poland [7].
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9. English Mathematical Cryptanalysis
Right before Germany invaded Poland, the Polish government shared Rejewski, R´o˙zycki, and
Zygalski’s findings with the Allies. Alan Turing, Gordon Welchman, and Hugh Alexander, while
at Bletchley Park, expanded on the initial research to create a second mechanical bombe. The
second bombe mechanized breaking the cipher in addition to deciphering messages [4]. Sadly, the
Naval Enigma could only be cracked by hand. Turing used sequential analysis which required a lot
of data or messages. This technique is called “Turingery.” He solved the German naval indicator
system which was more complex than the indicator systems used by the other services. The
indicator system consists of the rotors, pluboard, and reflector previously described. The Naval
Enigma had five rotors that could be moved around and ten plugboard cables. The complexity
of the indicator system is why the messages had to be deciphered by hand. The handmade
cryptanalysis technique developed for the Lorenz cipher used on the German Army Enigma was
similar to the technique Turingery [4].
German Cryptanalysts and Techniques
The Germans did not create any new cryptanalytic techniques during World War II for many
reasons. One key reason was the German’s failure to use applied mathematics. The Oberkom-
mando der Wehrmacht (OKW) cryptanalysis organization was the youngest of the German orga-
nizations and the only organization to hire mathematicians. The team was ostracized by other
cryptanalysis organizations for hiring mathematicians and researching applied mathematics. Karl
Stein was a mathematician who worked for OKW. He was the head of division IVa at the Cipher
Department. Since his success was not appreciated by the Germans it is difficult to find exam-
ples of his work. Another famous cryptanalysis team was located at B-Dienst. Wilhelm Tranow
worked at B-Dienst as a linguist and is well known for his success in low level cryptanalysis using
probability and statistics and trial and error techniques. He is known for cracking the British
Naval Cipher No. 3, but not the updated Cipher No. 5, by comparing the encoded messages with
merchant ship movements [5].
Conclusion
After reviewing how Marian Rejewski, Jerzy R´o˙zycki, and Henryk Zygalski cracked the Enigma
it becomes obvious that the use of applied mathematics was essential. These mathematicians used
permutations and cycle notation to create a machine that could decipher high security German
messages, whereas the Germans relied on probability and statistics, such as frequency analysis, to
crack the Allies’ low security systems.
This paper has discussed the many reasons why the Germans struggled with cryptanalysis, es-
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10. pecially mathematical cryptanalysis. The German’s struggle with mathematical cryptanalysis was
not due to the lack of brilliant mathematicians in Germany. Many brilliant mathematicians and
scientists, but not all, fled Germany because of the hostile environment created by the government
and the German mathematical society. The hostile environment for mathematicians was created
from the mathematical society’s belief that pure mathematics was Jewish and led to the prohibi-
tion of research in the field of applied mathematics. This belief blinded the German government
to the value of mathematicians and mathematical cryptanalysis. If the environment in Germany
had been less hostile, then perhaps these brilliant mathematicians would have contributed to the
German cryptanalysis effort, rather than fleeing Germany and helping the Allies. Another key
reason the Germans struggled with cryptanalysis is that the government did not provide adequate
financial and material resources to cryptanalysts. This meant that cryptanalysts had a difficult
time designing and making mechanized deciphering machines like the ones the Allies used to
crack the Enigma. Also, the German cryptanalysis organizations ostracized the one team that
hired mathematicians and used applied mathematics. This action would have dissuaded other
teams from following in their footsteps.
It was of equal importance to the outcome of World War II that the Germans struggled with
mathematical cryptanalysis while the Allies succeeded. If Germany had had the ability to decrypt
high security ciphers, the Allies may not have gained the upper hand, the Germans may have
been able to counter the Allies avoidance of the U-boats, and the war may have continued for
several more years. The Germans would have also been able to strategically plan where and when
field combat occurred. If the Germans used their knowledge gained from German scientist study
of chemical warfare, then they would be able to kill a large number of Ally troops. It is possible
that if the Germans had been more supportive of mathematicians in the field of cryptanalysis and
applied mathematics, World War II may have had a different ending.
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11. References
[1] National Security Agency. European axis signal intelligence in world war ii as revealed by
”ticom” investigations and by other prisoner of war interrogations and captured material,
principally german: Volume 2-notes on german high level cryptology and cryptanalysis, May
1946.
[2] John Cornwell. Hitler’s Scientists: Science, War, and the Devil’s Pact. Penguin Group, 2003.
[3] German History Documents. Law for the restoration of the professional civil service of 1933,
2015.
[4] A. Hinsley, F. H. Stripp. Code Breakers: The inside story of Bletchley Park. Oxford University
Press, 1994.
[5] John Jackson, editor. Hitler’s Codebreakers: German Signals Intelligence in World War 2.
BookTower Publishing, 2012.
[6] Martin Oberzalek. Breaking the enigma, 2000.
[7] Elisabeth Rakus-Andersson, editor. The Brains behing the Enigma Code Breaking before the
Second World War. Bleckinge Institute of Technology, Department of Health, Science, and
Mathematics, 2003.
[8] Simon Singh. The code book: the science of secrecy from ancient Egypt to quantum cryptogra-
phy. Anchor, 2000.
[9] Francisco R. Villatoro. Nota dominical: Los matem´aticos polacos, alan turing y el secreto de
la m´aquina enigma., 2012.
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12. [1] National Security Agency:
National Security Agency (NSA) declassified Alan Turing’s research on cracking the
Enigma, as well as the TICOM (Target Intelligence Committee) interrogations of Ger-
man cryptanalysts and prisoners of war back in 2009. John Jackson edited this release
into a book [5]. Jackson’s book is easier to follow because he reorganized it into more
understandable subsections. NSA is one of the few places where you can see this
information since Alan Turing’s work was burned after the war.
[2] John Cornwell:
Cornwell talks about the scientist that worked for Hitler. These scientist mostly worked
on creating chemical warfare to exterminate Jews. Hitler’s plan to exterminate the
Jews was called the ”Final Solution”. Although this book mostly focuses on the ”evil”
scientists it also mentions the German mathematicians and the German Mathematical
Society.
[3] German History Documents:
The German History Documents website shows a scanned copy of the Law for the
Restoration of the Professional Civil Service of 1933 with Hitler’s signature. It also
has brief descriptions of many other laws similar to the one mentioned in this paper.
[4] Hinsley and Stripp:
Hinsley and Stripp compiled interviews from individuals who worked at Bletchley Park
during World War II. They organized the book by what these individuals were working
on, such as individuals that focused on the Naval Enigma went in the same section.
They also added descriptions of what encryption systems these individuals were trying
to crack. It gave a good description of the Lorenz cipher amongst the other types of
ciphers briefly mentioned in this paper.
[5] John Jackson:
Jackson’s book was already described in the annotation of the NSA declassified file.
This book did a good job at being less technical than the NSA file and gave better
descriptions of the organizations and individuals being talked about.
[6] Martin Oberzalek:
Oberzalek’s website provides a lot of information regarding the Engima and the break-
ing of the Engima. This author has published multiple webpages related to the Enigma.
This paper only uses this reference for talking about the size and cost of the bombes.
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13. [7] Elisabeth Rakus-Andersson:
Rakus-Andersson’s paper is an edited version of Rejewski’s paper on cracking the
Engima. She added a more historical context about Poland and their programs for
cryptanalysis. She had a great example of how the Enigma works and how Rejewski
cracked the Enigma.
[8] Simon Singh:
Simon Singh has written multiple books on a variation of mathematical topics. This
specific book focuses on the evolution of cryptography and cryptanalysis and the inter-
action between the two fields. He provides many examples for all the ciphers mentioned
in the book that are easy to follow for individuals with no higher mathematical edu-
cation.
[9] Fransisco Villatoro:
This paper only uses the picture describing how the Enigma works. It was a very good
picture that made the more complicated machine easy to understand. The rest of the
website is in Spanish. Since this website is a blog instead of published paper, it was
not used very much in the paper.
I pledge that I have neither given nor received unauthorized aid on this paper.
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