Transcript Slide 1

Introduction to
Cryptography
Lecture 5
Shift Cipher
As we saw in class, shift cipher is not
secure.
 There are only 26 possible shifts for
English alphabet
 The letters are not mixed enough

Affine Cipher
Definition: The affine cipher is the cipher
which encrypts message using the
formula y  (kx  b) modm.
 m is number of letter in alphabet
 k and b are chosen with some restrictions
from 0,1,…,m-1
 x is numerical equivalent of
corresponding letter
Modular Arithmetics
Claim: If a  b mod m and c  d mod m ,
then ac  bd mod m .
Modular Arithmetics
Claim: If a  b mod m and c  d mod m ,
then ac  bd mod m .
Proof: a  b  k1m and c  d  k2 m .
ac  (b  k1m)(d  k2 m) 
 bd  (k1d  k2b)m  k1k2 m 
2
 bd  (k1d  k2b  k1k2 m)m
ac  bd  (k1d  k2b  k1k2 m)m
Decimation Cipher
Definition: The decimation cipher is the
cipher which encrypts message using the
formula y  kx modm .
 m is number of letter in alphabet
 k is chosen with some restrictions from
0,1,…,m-1
 x is numerical equivalent of corresponding
letter

Decimation Cipher
Examples: When k = 3 we get the table:
A B C D E F G H
I
J K L M N O P Q R S T U V W X Y Z
A D G J M P S V Y B E H K N Q T W Z C F
I
L O R U X
When k = 4 we get the table:
A B C D E F G H
A E
I
J K L M
N
O P Q R S T U V W X Y Z
I M Q U Y C G K O S W
A
E
I M Q U Y C G K O S W
Decimation Cipher
k must be one of the: 1, 3, 5, 7, 9, 11, 15,
17, 19, 21, 23, 25.
 In other words, we must have
gcd(k,26)=1.
 How secure this cipher is?

Decimation Cipher
Example: Decrypt message:
CFQGEKAZEMFZMAGVMCNMOVYSV.
 We know A is A.
 Lets guess that E is M.
 Then k  4  12 mod 26 , has two solution:
k = 3 and k =16.
 STOCK MARKET REACHES NEW HIGH
Affine Cipher
Definition: The affine cipher is the cipher
which encrypts message using the
formula y  (kx  b) modm.
 m is number of letter in alphabet
 k and b are chosen with some restrictions
from 0,1,…,m-1
 x is numerical equivalent of
corresponding letter
Affine Cipher
Using simple modular arithmetic we can
1
get: x  (k ( y  b)) mod26.
 We don’t have A encrypted to A
anymore.
 How secure this cipher is?

Affine Cipher
Example: Decrypt message:
SUYMCPTTWCMLCPUFRRUGKCPWIT
JQMAO.
 Lets use letter frequencies to make a
guess for two letters.
 Assume C is E and T is U.
Affine Cipher
Then we get two equations:
2  (4k  b) mod26, 20  (19k  b) mod26
 So k = 22, not relatively prime 26.
 Bad guess make another.
 The correct answer k = 5 and b = 2.
 YOU CAN TEACH AN OLD DOGMA NEW
TRICKS.

Homework
Read pg.69-80.
 Exercises: 8, 9, 10 on pg.81- 82.
 Those questions will be a part of your
collected homework.
