Mathematics of
Cryptology: Decoding
Nazi Secrets
MATH150 Documentary Project – Jared Ennico
Introduction
Cryptology is the study of codes, ciphers, and hidden
messages.
The NOVA documentary Decoding Nazi Secrets shows
how math helped break Nazi codes.
How probability, logic, and pattern recognition exposed
the limits of "randomness" in the Enigma machine.
Before Enigma: A
Short History of
Cryptology
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Early substitution ciphers (Caesar
cipher → shift by 3)
Modular arithmetic origins (wrapping
around alphabet)
Monoalphabetic → polyalphabetic
ciphers
Human pattern-breaking (frequency
analysis)
Why countries needed stronger,
machine-based encryption by WW2
Overview of the Documentary
Focus on Allied
codebreaking
during WWII
Alan Turing’s
breakthroughs
Work at Bletchley
Park
Polish
mathematicians
who first broke
early Enigma
Development of
the Bombe and
Colossus
Ending with the
impact of
cryptology on the
war
The Enigma and
the Illusion of
Randomness
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The Enigma machine encrypted
messages using rotors and
plugboards.
Despite appearing random, it was
deterministic
The same input always produced
the same output.
Mathematicians used
combinatorics and logical
reasoning to narrow trillions of
settings.
Randomness was only an illusion;
machines can’t be truly random.
Probability and Pattern Recognition
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Codebreakers relied on probability to find patterns in German messages.
Repeated words like London, "Heil Hitler" or weather reports were key clues
(cribs).
They eliminated impossible settings instead of guessing the correct one directly.
This process was mathematical elimination, not brute force.
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Mathematical
Foundations in
Cryptology
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Combinatorics: Calculating total Enigma
configurations.
Modular Arithmetic: Letter shifting, mod
26 arithmetic.
Mod 2 Arithmetic: Used in binary and
Lorenz cipher (0s and 1s).
Morse and Binary Code: Information
encoded through math.
These show how math governs both
communication and secrecy.
Logic, Algorithms, and
Turing
Alan Turing applied mathematical logic
to create algorithms for decoding.
His Bombe machine automated logical
testing of possible settings.
Early computers used Boolean logic and
binary arithmetic (mod 2).
Showed how human logic + machines =
powerful cryptanalysis.
Randomness vs.
Predictability in
Cryptology
True randomness
vs mechanical
randomness
Enigma was
deterministic, NOT
random
Probability used to
spot non-random
patterns
German habits
(weather reports,
greetings) ruined
randomness
The “first 3 letters”
pattern from
operators entering
same key twice
Logical elimination:
reject wrong
settings until one
remains
Modern
Connections
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All modern encryption uses math
(modular arithmetic, prime
numbers, randomness)
Turing’s algorithms → basis of
computing
Randomness still the hardest part
of cryptography
Lessons from Enigma: predictable
systems eventually break
Modern Connections
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Modern cryptography still uses
modular arithmetic and probability.
True randomness remains nearimpossible, computers use pseudorandomness.
The foundation of digital security today
traces back to Enigma and Turing.
Cryptology is where math, logic, and
technology intersect.
Conclusion
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Mathematics exposed structure inside
what appeared to be chaos.
Probability, logic, and modular
arithmetic made codebreaking
possible.
The story of Enigma shows that even
'randomness' has rules.
Cryptology remains a living example of
math in action.