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?
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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
 C and d
D. Each eK:P
P are
K:C
functions such thatdK(eK(x))=x
x P.
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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
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
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
Alice
x
y
encrypter
K
key source
x
decrypter
Bob
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