Cryptology

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Cryptology

• Harriet Muncey

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Encryption

• What? Why?
• Classical Cryptography
• Enigma
• Internet Encryption Public Key
• RSA Algorithm
• Cryptology in the Classroom

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Why?
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Caesar Cipher

• shift each letter in the text cyclically k
  places in the alphabet
• HELLO becomes MJQQT
• c E(m) (m k) (mod 26), 0 k 26
• D(c) (m - k) (mod 26) m

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Frequency Analysis
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Vigenère

• keyword (of length r)
• apply r separate Caesar ciphers periodically
• E (mi) mi ki (mod r) (mod 26)

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Enigma
Keyboard
Indicator Light
Rotor 1
Rotor 2
Rotor 3
Reflector
Key (i) Choice and order of rotors
(ii) Their initial position (iii) Fixed
initial permutation of the alphabet
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Internet Encryption Public Key

• Mathematically linked keys generated from a large
  random number
• Asymmetric Cryptosystem solves problem of key
  distribution
• Security relies on adversary having limited
  computational power
• Hybrid System

• Alice

Public Key
Private Key
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Sending a Message
Bob encrypts the message
Plain Text
Alices Public Key
Cipher Text
Plain Text
Alices Private Key
Alice decrypts the message
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RSA Algorithm

• Two distinct large primes p, q
• Public modulus n pq
• Public exponent e
• coprime to f(n) ( p 1)(
  q 1)
• Exists unique 1 lt d lt f(n)
• satisfying ed 1 mod
  f(n)

Private Key d
Public Key ( n , e )
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Encryption

• Alices message M1, M2,,Mt Ci Mie
  mod n
• Bob decrypts the message by Mi Cid
  mod n

ed 1 mod f(n) gt ed 1 t f(n) Cid
Mied Mi 1 t f(n) Mi mod p Fermats
Little Thm gt Mip-1 1 mod
p Similarly Cid
Mi mod q Chinese Remainder Thm gt
Cid Mi mod n
WHY?
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Cracking RSA

• Need exponent d
• Euclids Algorithm ed 1 mod f(n)
• Factorisation of n
• Proof
• Requires enormous computational power

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Cryptology in the Classroom

• Interesting AND Relevant
• Classical Cryptography
• Frequency Analysis Statistics
• Combinations / Permutations
• Modular Arithmetic
• Logic / Problem solving
• Extensions
• Asymmetric encryption
• Primes
• Hashing Functions
• Authentication
• Error Correction

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Any Questions?