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IntroResearch
Cryptography: A Fairy Tale for Mathematiciansand Starring Mathematicians!
Mahrud Sayrafi
University of California, BerkeleyMathematics Undergraduate Student Association
October 27, 2014
Mahrud Sayrafi Cryptography: FTMSM!
![Page 2: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/2.jpg)
IntroResearch
Why Crypto?
So why on earth was cryptography invented?
Most obvious: Keeping secrets!but why keep secrets then?
send messages to armies during warfrom Julius Caesar (and his silly shift cipher)to World War II Germany’s [not so enigmatic] Enigma machineorganize all sorts of conspiracies eg: Mary Queen of Scotsplotted to assassinate Queen Elizabethshe was hanged because her cipher broke and proved herinvolvement.Etc.
Less obvious: deciphering ancient languages, finding buriedtreasures, fame, glory, etc.
but most interesting: for fun!
Mahrud Sayrafi Cryptography: FTMSM!
![Page 3: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/3.jpg)
IntroResearch
Why Crypto?
So why on earth was cryptography invented?
Most obvious: Keeping secrets!
but why keep secrets then?
send messages to armies during warfrom Julius Caesar (and his silly shift cipher)to World War II Germany’s [not so enigmatic] Enigma machineorganize all sorts of conspiracies eg: Mary Queen of Scotsplotted to assassinate Queen Elizabethshe was hanged because her cipher broke and proved herinvolvement.Etc.
Less obvious: deciphering ancient languages, finding buriedtreasures, fame, glory, etc.
but most interesting: for fun!
Mahrud Sayrafi Cryptography: FTMSM!
![Page 4: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/4.jpg)
IntroResearch
Why Crypto?
So why on earth was cryptography invented?
Most obvious: Keeping secrets!but why keep secrets then?
send messages to armies during warfrom Julius Caesar (and his silly shift cipher)to World War II Germany’s [not so enigmatic] Enigma machineorganize all sorts of conspiracies eg: Mary Queen of Scotsplotted to assassinate Queen Elizabethshe was hanged because her cipher broke and proved herinvolvement.Etc.
Less obvious: deciphering ancient languages, finding buriedtreasures, fame, glory, etc.
but most interesting: for fun!
Mahrud Sayrafi Cryptography: FTMSM!
![Page 5: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/5.jpg)
IntroResearch
Why Crypto?
So why on earth was cryptography invented?
Most obvious: Keeping secrets!but why keep secrets then?
send messages to armies during warfrom Julius Caesar (and his silly shift cipher)to World War II Germany’s [not so enigmatic] Enigma machine
organize all sorts of conspiracies eg: Mary Queen of Scotsplotted to assassinate Queen Elizabethshe was hanged because her cipher broke and proved herinvolvement.Etc.
Less obvious: deciphering ancient languages, finding buriedtreasures, fame, glory, etc.
but most interesting: for fun!
Mahrud Sayrafi Cryptography: FTMSM!
![Page 6: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/6.jpg)
IntroResearch
Why Crypto?
So why on earth was cryptography invented?
Most obvious: Keeping secrets!but why keep secrets then?
send messages to armies during warfrom Julius Caesar (and his silly shift cipher)to World War II Germany’s [not so enigmatic] Enigma machineorganize all sorts of conspiracies eg: Mary Queen of Scotsplotted to assassinate Queen Elizabethshe was hanged because her cipher broke and proved herinvolvement.
Etc.
Less obvious: deciphering ancient languages, finding buriedtreasures, fame, glory, etc.
but most interesting: for fun!
Mahrud Sayrafi Cryptography: FTMSM!
![Page 7: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/7.jpg)
IntroResearch
Why Crypto?
So why on earth was cryptography invented?
Most obvious: Keeping secrets!but why keep secrets then?
send messages to armies during warfrom Julius Caesar (and his silly shift cipher)to World War II Germany’s [not so enigmatic] Enigma machineorganize all sorts of conspiracies eg: Mary Queen of Scotsplotted to assassinate Queen Elizabethshe was hanged because her cipher broke and proved herinvolvement.Etc.
Less obvious: deciphering ancient languages, finding buriedtreasures, fame, glory, etc.
but most interesting: for fun!
Mahrud Sayrafi Cryptography: FTMSM!
![Page 8: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/8.jpg)
IntroResearch
Why Crypto?
So why on earth was cryptography invented?
Most obvious: Keeping secrets!but why keep secrets then?
send messages to armies during warfrom Julius Caesar (and his silly shift cipher)to World War II Germany’s [not so enigmatic] Enigma machineorganize all sorts of conspiracies eg: Mary Queen of Scotsplotted to assassinate Queen Elizabethshe was hanged because her cipher broke and proved herinvolvement.Etc.
Less obvious: deciphering ancient languages, finding buriedtreasures, fame, glory, etc.
but most interesting: for fun!
Mahrud Sayrafi Cryptography: FTMSM!
![Page 9: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/9.jpg)
IntroResearch
Why Crypto?
So why on earth was cryptography invented?
Most obvious: Keeping secrets!but why keep secrets then?
send messages to armies during warfrom Julius Caesar (and his silly shift cipher)to World War II Germany’s [not so enigmatic] Enigma machineorganize all sorts of conspiracies eg: Mary Queen of Scotsplotted to assassinate Queen Elizabethshe was hanged because her cipher broke and proved herinvolvement.Etc.
Less obvious: deciphering ancient languages, finding buriedtreasures, fame, glory, etc.
but most interesting: for fun!
Mahrud Sayrafi Cryptography: FTMSM!
![Page 10: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/10.jpg)
IntroResearch
When did math come in?
It didn’t. Math has always been there.
Mahrud Sayrafi Cryptography: FTMSM!
![Page 11: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/11.jpg)
IntroResearch
When did math come in?
It didn’t.
Math has always been there.
Mahrud Sayrafi Cryptography: FTMSM!
![Page 12: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/12.jpg)
IntroResearch
When did math come in?
It didn’t. Math has always been there.
Mahrud Sayrafi Cryptography: FTMSM!
![Page 13: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/13.jpg)
IntroResearch
When did math come in?
Euclid (300 B.C.)
”There are infinitely many primes ...””... also there is this algorithm for finding GCD. Let’s name it afterme!”
Mahrud Sayrafi Cryptography: FTMSM!
![Page 14: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/14.jpg)
IntroResearch
When did math come in?
Euclid (300 B.C.)”There are infinitely many primes ...”
”... also there is this algorithm for finding GCD. Let’s name it afterme!”
Mahrud Sayrafi Cryptography: FTMSM!
![Page 15: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/15.jpg)
IntroResearch
When did math come in?
Euclid (300 B.C.)”There are infinitely many primes ...””... also there is this algorithm for finding GCD. Let’s name it afterme!”
Mahrud Sayrafi Cryptography: FTMSM!
![Page 16: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/16.jpg)
IntroResearch
When did math come in?
Pierre de Fermat (1601-1665)
Fermat’s Little Theorem (1640):For any prime p and integer a, such that 1 ≤ a < p, then:
ap−1 = 1 mod p
Mahrud Sayrafi Cryptography: FTMSM!
![Page 17: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/17.jpg)
IntroResearch
When did math come in?
Pierre de Fermat (1601-1665)Fermat’s Little Theorem (1640):For any prime p and integer a, such that 1 ≤ a < p, then:
ap−1 = 1 mod p
Mahrud Sayrafi Cryptography: FTMSM!
![Page 18: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/18.jpg)
IntroResearch
When did math come in?
Leonhard Euler (1707-1783)
Eulers Theorem (1736):If gcd(a, n) = 1, then:
aϕ(n) = 1 mod n
where ϕ(n) is number of integers x less than n such thatgcd(x , n) = 1.
Mahrud Sayrafi Cryptography: FTMSM!
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IntroResearch
When did math come in?
Leonhard Euler (1707-1783)Eulers Theorem (1736):If gcd(a, n) = 1, then:
aϕ(n) = 1 mod n
where ϕ(n) is number of integers x less than n such thatgcd(x , n) = 1.
Mahrud Sayrafi Cryptography: FTMSM!
![Page 20: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/20.jpg)
IntroResearch
Some quick terminology
Code: a word or phrase replaced with another, possiblyshorter one.
Cipher: replacing each letter in a text by another based on asystem
Plaintext: the message
Ciphertext: the encrypted message
Mahrud Sayrafi Cryptography: FTMSM!
![Page 21: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/21.jpg)
IntroResearch
Some quick terminology
Code: a word or phrase replaced with another, possiblyshorter one.
Cipher: replacing each letter in a text by another based on asystem
Plaintext: the message
Ciphertext: the encrypted message
Mahrud Sayrafi Cryptography: FTMSM!
![Page 22: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/22.jpg)
IntroResearch
Some quick terminology
Code: a word or phrase replaced with another, possiblyshorter one.
Cipher: replacing each letter in a text by another based on asystem
Plaintext: the message
Ciphertext: the encrypted message
Mahrud Sayrafi Cryptography: FTMSM!
![Page 23: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/23.jpg)
IntroResearch
Some quick terminology
Code: a word or phrase replaced with another, possiblyshorter one.
Cipher: replacing each letter in a text by another based on asystem
Plaintext: the message
Ciphertext: the encrypted message
Mahrud Sayrafi Cryptography: FTMSM!
![Page 24: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/24.jpg)
IntroResearch
Some quick terminology
Code: a word or phrase replaced with another, possiblyshorter one.
Cipher: replacing each letter in a text by another based on asystem
Plaintext: the message
Ciphertext: the encrypted message
Mahrud Sayrafi Cryptography: FTMSM!
![Page 25: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/25.jpg)
IntroResearch
Introducing Characters (from wiki!)
Alice
Bob
Carol or Charlie
Eve, the passive eavesdropper
Craig, the password cracker
Mallet or Mallory, the malicious attackers
Trudy, the intruder
etc.
Mahrud Sayrafi Cryptography: FTMSM!
![Page 26: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/26.jpg)
IntroResearch
Introducing Characters (from wiki!)
Alice
Bob
Carol or Charlie
Eve, the passive eavesdropper
Craig, the password cracker
Mallet or Mallory, the malicious attackers
Trudy, the intruder
etc.
Mahrud Sayrafi Cryptography: FTMSM!
![Page 27: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/27.jpg)
IntroResearch
Introducing Characters (from wiki!)
Alice
Bob
Carol or Charlie
Eve, the passive eavesdropper
Craig, the password cracker
Mallet or Mallory, the malicious attackers
Trudy, the intruder
etc.
Mahrud Sayrafi Cryptography: FTMSM!
![Page 28: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/28.jpg)
IntroResearch
Introducing Characters (from wiki!)
Alice
Bob
Carol or Charlie
Eve, the passive eavesdropper
Craig, the password cracker
Mallet or Mallory, the malicious attackers
Trudy, the intruder
etc.
Mahrud Sayrafi Cryptography: FTMSM!
![Page 29: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/29.jpg)
IntroResearch
Leaked photo of Evette in a beach
Mahrud Sayrafi Cryptography: FTMSM!
![Page 30: Cryptography: A Fairy Tale for Mathematicians and Starring ......For any prime p and integer a, such that 1 a < p, then: ap 1 = 1 mod p Mahrud Sayra Cryptography: FTMSM! Intro Research](https://reader035.vdocuments.site/reader035/viewer/2022071105/5fdfad6b0435d10432635cd6/html5/thumbnails/30.jpg)
IntroResearch
New research problems and directions
More math
elliptic curve-based crypto
lattice-based crypto
braid-based crypto
secret-sharing
random oracles
quantum cryptography
zero-knowledge proofs
More EECS or applied math
multi-party protocols
homomorphic encryption
private information retrieval
anonymity
bit commitment
oblivious transfer
secure voting systems
crypto hardware
public-key infrastructure
tweakable encryption
Mahrud Sayrafi Cryptography: FTMSM!