1. Introduction to Cryptography
Lecture 3

2. Division Principle
Let m be a positive integer and let b be any integer. Then there is exactly one pair of integers q (quotient) and r (remainder) such that b = qm + r, r < m.
If x and y are relatively prime, than exists s and t such that sx + ty = gcd(x,y) = 1.
If exist s and t such that sx + ty = 1, than x and y are relatively prime.

3. Modular Arithmetic
Definition: Let m be a positive integer. We say that two integers x and y are congruent modulo m if x - y is evenly divisible by m
Notation:
Example:

4. Modular Arithmetic
Rules:
Example:

5. Modular Arithmetic
Question: What are the possible reminders when the number n is divided by m?
{0,1,….,m-1}
Example: If we have mod 2 operation, only reminders we will have are {0,1}.
0 - for evens
1 - for odds

6. Modular Arithmetic Example: Make a table of y-values for
We only need to check possible reminder of division by 12. x 1 2 3 4 5 6 7 8 9 10 11
x-11
-11
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 y

7. Modular Arithmetic
Theorem: Let m be a positive integer and let 0 < x < m. There is 0 < s < m, such that xs = 1 if and only if gcd(x,m) = 1.
Definition: Such s is called the inverse of x mod m.
s is unique

8. Modular Arithmetic Example: Let m = 17 and x = 11.
Then gcd(17,11) = 1 and using extended Euclidean algorithm we can find:
and
14 is the inverse of 11 mod 17.

9. Fermat’s Little Theorem
If p is prime, then for every x,
Example:

10. Methods of Attacks Ciphertext-Only Attack Known-Plaintext Attack
Chosen-Plaintext Attack
Chosen-Ciphertext Attack

11. Ciphertext-Only Attack
Know
A portion of ciphertext
Method of encryption and decryption
Find
Plaintext
Key

12. Known-Plaintext Attack
A portion of ciphertext and corresponding plaintext
Method of encryption and decryption
Find
Key

13. Chosen-Plaintext Attack
Know
Method of encryption and decryption
Can choose
Plaintext
Find
Key

14. Chosen-Ciphertext Attack
Know
Method of encryption and decryption
Can choose
Ciphertext
Find
Key

15. Homework
Read pg.51-54,
Exercises: 4(b,c), 5(b,d), 6 on pg
Exercises: 2(a,c) on pg.80.
Those questions will be a part of your collected homework.