Cryptography: A Fairy Tale for Mathematicians and Starring Mathematicians!
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1 Cryptography: A Fairy Tale for Mathematicians and Starring Mathematicians! University of California, Berkeley Mathematics Undergraduate Student Association October 27, 2014
2 Why Crypto? So why on earth was cryptography invented?
3 Why Crypto? So why on earth was cryptography invented? Most obvious: Keeping secrets!
4 Why Crypto? So why on earth was cryptography invented? Most obvious: Keeping secrets! but why keep secrets then?
5 Why Crypto? So why on earth was cryptography invented? Most obvious: Keeping secrets! but why keep secrets then? send messages to armies during war from Julius Caesar (and his silly shift cipher) to World War II Germany s [not so enigmatic] Enigma machine
6 Why Crypto? So why on earth was cryptography invented? Most obvious: Keeping secrets! but why keep secrets then? send messages to armies during war from 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 Scots plotted to assassinate Queen Elizabeth she was hanged because her cipher broke and proved her involvement.
7 Why Crypto? So why on earth was cryptography invented? Most obvious: Keeping secrets! but why keep secrets then? send messages to armies during war from 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 Scots plotted to assassinate Queen Elizabeth she was hanged because her cipher broke and proved her involvement. Etc.
8 Why Crypto? So why on earth was cryptography invented? Most obvious: Keeping secrets! but why keep secrets then? send messages to armies during war from 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 Scots plotted to assassinate Queen Elizabeth she was hanged because her cipher broke and proved her involvement. Etc. Less obvious: deciphering ancient languages, finding buried treasures, fame, glory, etc.
9 Why Crypto? So why on earth was cryptography invented? Most obvious: Keeping secrets! but why keep secrets then? send messages to armies during war from 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 Scots plotted to assassinate Queen Elizabeth she was hanged because her cipher broke and proved her involvement. Etc. Less obvious: deciphering ancient languages, finding buried treasures, fame, glory, etc. but most interesting: for fun!
10 When did math come in?
11 When did math come in? It didn t.
12 When did math come in? It didn t. Math has always been there.
13 When did math come in? Euclid (300 B.C.)
14 When did math come in? Euclid (300 B.C.) There are infinitely many primes...
15 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 after me!
16 When did math come in? Pierre de Fermat ( )
17 When did math come in? Pierre de Fermat ( ) Fermat s Little Theorem (1640): For any prime p and integer a, such that 1 a < p, then: a p 1 = 1 mod p
18 When did math come in? Leonhard Euler ( )
19 When did math come in? Leonhard Euler ( ) Eulers Theorem (1736): If gcd(a, n) = 1, then: a ϕ(n) = 1 mod n where ϕ(n) is number of integers x less than n such that gcd(x, n) = 1.
20 Some quick terminology
21 Some quick terminology Code: a word or phrase replaced with another, possibly shorter one.
22 Some quick terminology Code: a word or phrase replaced with another, possibly shorter one. Cipher: replacing each letter in a text by another based on a system
23 Some quick terminology Code: a word or phrase replaced with another, possibly shorter one. Cipher: replacing each letter in a text by another based on a system Plaintext: the message
24 Some quick terminology Code: a word or phrase replaced with another, possibly shorter one. Cipher: replacing each letter in a text by another based on a system Plaintext: the message Ciphertext: the encrypted message
25 ducing Characters (from wiki!)
26 ducing Characters (from wiki!) Alice Bob Carol or Charlie
27 ducing Characters (from wiki!) Alice Bob Carol or Charlie Eve, the passive eavesdropper
28 ducing 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.
29 Leaked photo of Evette in a beach
30 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
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