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					                  Cryptograpy
                  O




By Roya Furmuly
                          What Is It?

Enables two people (Alice and Bob) to
communicate over an insecure channel in such
a way so that an opponent (Oscar) cannot
understand what is being said.
              How Does It Work?

   Alice encrypts the information (Plaintext),
    using a predetermined key, then sends the
    result (Ciphertext) to Bob.
   Oscar cannot determine the plaintext because
    he doesn’t know the key.
    Bob, who knows the encryption key, decrypts
    the ciphertext and reconstructs the plaintext.
                          Formal Definition
A Cryptosystem is a five-tuple (P,C,K,E,D )
P = finite set of plaintexts
C = finite set of ciphertexts
K = finite set of keys (keyspace)
For each K K  eK E and a corresponding dK
                                            
                       
  D. Each eK:P C and dK:C P are functions such
  that dK(eK(x))=x  x P.
                              
                               Observations
   The encryption function eK must be injective to
    avoid ambiguity.
i.e. if y= eK(x1)= eK(x2) where x1 not equal x2
Bob doesn’t know whether y= x1 or y= x2

   If P = C , then the encryption function is a
    permutation.
                                       Protocol

   Choose random key K in K (when Oscar not present
    or through a secure channel).
   Alice
    Message: x=x1x2...xn where i in (1,n), xi in P
    encrypts each xi using encryption rule yi= eK(xi)
                      y=y1y2…yn
   Bob uses decryption function dK(yi)=xi
                    x=x1x2...xn
                                         Diagram

                    Oscar

        x                y               x
Alice       encrypter        decrypter       Bob


             K


            key source
What makes a Cryptosystem practical?


1. Encryption and Decryption functions
  should be efficiently computable.

2. Upon seeing ciphertext y, the opponent
  should be unable to determine the key K
  used (“security”).
                              Shift Cipher
Let P =C =K = Z26.
        eK(x)=x+K mod 26
and
        dK(y)=y-K mod 26            (x,y in Z26)

      cool fact: for K=3, cryptosystem is called the
      Caesar Cipher.
                Shift Cipher (cont’d)

We encrypt English text, by the following
 correspondence:
A 0, B 1, …, Z 25,

ABCDEFGHIJ KLMNOPQRSTUVW
0 1 2 3 4 5 6 7 8 9 101112 13 14 15161718192021 22
XY Z
23 24 25
                   Let’s Encrypt!
Let the key be K=7, encrypt: UCLA BRUINS
convert letters to integers using chart:
20 2 11 0 1 17 20 8 13 18
add 7 to each value, reduce mod 26:
1 9 18 7 8 24 1 15 20 25
convert to sequence of integers:
      BJSHIYBPUZ
                    Let’s Decrypt!
BJSHIYBPUZ
convert letters to integers:
1 9 18 7 8 24 1 15 20 25
subtract 7, reduce mod 26:
20 2 11 0 1 17 20 8 13 18
convert to letters:
     UCLA BRUINS
      Shift Cipher, any Good?
 Nope! Fails security property.
 Keyspace is very small, only 25 possible
  keys.
 Can easily be deciphered by an exhaustive
  key search.
 Try K=1…25, until get a text that makes
  sense.
                          Vigenere Cipher
Let m>0 be fixed. Let P =C =K = (Z26)m
For a key K=(k1,k2,…km) define

        eK(x1,x2,…,xm)=(x1+k1, x2+k2,…,xm+km)
and
       dK(y1,y2,…,ym)=(y1-k1, y2-k2,…,ym-km)

*all operations done in Z26
                            Let’s Encrypt!
Let key=hot=(7,14,19), encrypt: SUMMER IS
                                    HERE
convert to integers & “add” the keyword mod
  26:
18 20 12 12 4 17 8 18 7 4 18 4
 7 14 19 7 14 19 7 14 19 7 14 19
----------------------------------------------------
25 8      5 19 18 10 15 6 0 11 6 23
      ZIFTSKPGALGX
                              Let’s Decrypt!
ZIFTSKPGALGX
convert to integers and “subtract” the keyword
  hot=(7,14,19) mod 26:
 25 8 5 19 18 10 15 6 0 11 6 23
  7 14 19 7 14 19 7 14 19 7 14 19
--------------------------------------------------------
 18 20 12 12 4 17 8 18 7 4 18 4

      SUMMER IS HERE
Vigenere Cipher, any Good?
 Better than Shift Cipher
 Possible number of keys of length m is
            (26)m
 Say m=5, then keyspace size is
      (26)5 approx 1.1x107
 So, exhaustive key search not feasible by
  hand (but OK by computer).
            Other Cryptosystems
 Data Encryption Standard (DES)
  Based on permutaion of 64 bits at a time.
 RSA
  Based on difficulty of factoring large
  integers into primes.
 Enigma
   Machine with rotors that shifted letters in
  complicated manner.
                            Summary
 Cryptography allows us to communicate
  through insecure channels.
 Shift Cipher…insecure (small keyspace)
 Vigenere Cipher…less insecure
 Complicated Cryptosystems
      DES, RSA, ENIGMA
WKH HQG

				
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posted:6/27/2012
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