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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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