1. Introduction to Cryptography
Lecture 5

2. 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

3. Affine Cipher
Definition: The affine cipher is the cipher which encrypts message using the formula
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

4. Modular Arithmetics
Claim: If and , then

5. Modular Arithmetics
Claim: If and , then
Proof: and

6. Decimation Cipher
Definition: The decimation cipher is the cipher which encrypts message using the formula
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

7. Decimation Cipher Examples: When k = 3 we get the table:F G H I J K L M N O P Q R S T U V W X Y Z 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

8. 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?

9. Decimation Cipher Example: Decrypt message: CFQGEKAZEMFZMAGVMCNMOVYSV.
We know A is A.
Lets guess that E is M.
Then , has two solution: k = 3 and k =16.
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10. Affine Cipher
Definition: The affine cipher is the cipher which encrypts message using the formula
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

11. Affine Cipher Using simple modular arithmetic we can get: .
We don’t have A encrypted to A anymore.
How secure this cipher is?

12. Affine Cipher
Example: Decrypt message: SUYMCPTTWCMLCPUFRRUGKCPWITJQMAO.
Lets use letter frequencies to make a guess for two letters.
Assume C is E and T is U.

13. Affine Cipher Then we get two equations:
So k = 22, not relatively prime 26.
Bad guess make another.
The correct answer k = 5 and b = 2.
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14. Homework Read pg.69-80. Exercises: 8, 9, 10 on pg.81- 82.
Those questions will be a part of your collected homework.