1. Introduction to Cryptography
Lecture 2

2. Functions
f(x1) x3 x2 x1
f(x3)
f(x2) f
Domain
Range

3. Functions
Definition: A function is a relation, such that each element of a set (the domain) is associated with a unique element of another (possibly the same) set (the range)
f(x1) x3 x2 x1
f(x2) f
Domain
Range
f(x1) x2 x1
f(x2) f
Domain
Range
Function
Not a function

4. Functions
Definition: A function is called one to one if each element of domain is associated with precisely one element of the range.
Definition: A function is called onto if each element of range is associated with at least one element of the domain.

5. Functions f f f(x1) x3 x2 x1 f(x2) Domain Range x1 f(x1) x2 f(x1)y
Domain
Range
One to one
Not one to one
Onto
Not onto

6. Functions
A one to one and onto function always has an inverse function
Definition: Given a function an inverse function is computed by rule: if
Example: If , then

7. Functions and Cryptography
Cipher can be represented as a function
Example 1:
f(Secret message)= YpbzobqjbZqqyec
Example 2:
f(son) = girl (girl) = son
f(girl) = son (son) = girl

8. Functions and Cryptography
For each key, an encryption method defines a one-to-one and onto function; and the corresponding decryption method is the inverse of this function.

9. Permutations
Definition: A permutation of n ordered objects is a way of reordering them.
It is a mathematical function
It is one-to-one and onto
An inverse of permutation is a permutation

10. Permutations
Example: x 1 2 3 4 5
p(x) x 1 2 3 4 5
q(x)

11. Prime Numbers
Definition: A prime number is an integer number that has only two divisors: one and itself.
Example: 1, 2,17, 31.
Prime numbers distributed irregularly among the integers
There are infinitely many prime numbers

12. Factoring
The Fundamental Theorem of Arithmetic tells us that every positive integer can be written as a product of powers of primes in essentially one way.
Example:

13. Factoring Problem of factoring a number is very hard
The decision if n is a prime or composite number is much easier
Fermat’s factoring method sometimes can be used to find any large factors of a number fair quickly (pg.251)

14. Greatest Common Divisors - GCD
Definition: Let x and y be two integers. The greatest common divisor of x and y is number d such that d divides x and d divides y.
Definition: x and y are relatively prime if gcd(x,y)=1.

15. Greatest Common Divisors - GCD
Example: gcd(3,16) = 1
gcd(-28,8) = 4
One way to find gcd is by finding factorization of both numbers
Euclidean Algorithm is usually used in order to find gcd

16. 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.

17. Euclidean Algorithm Input x and y Output gcd(x,y) = xi x0 = x, y0 = y
For I >= 0 do
xi+1 = yi, yi+1 = xi mod yi
If yi =0, stop
Output gcd(x,y) = xi

18. Euclidean Algorithm Example: Let x = 4200 and y = 1485 i xi yi qi ri
4200
1485 2
1230 1
255 4
210 3 45 30 5 15 6 7

19. Extended Euclidean Algorithm
For every x and y there are integers s and t such that sx + ty = gcd(x,y)
We can find s and t using Euclidean Algorithm

20. Extended Euclidean Algorithm
Input x and y
x0 = x, y0 = y, s0 = t-1 = 0, t0 = s-1 = 1
For I >= 0 do
xi+1 = yi, yi+1 = xi mod yi,
si+1 = si-1 – qisi, ti+1 = ti-1 - qiti
If yi =0, stop
Output gcd(x,y) = xi, si-1,ti-1

21. Extended Euclidean Algorithm
Example: Let x = 4200 and y = 1485 i xi yi qi ri si ti
4200
1485 2
1230 1
255 -2 4
210 -1 3 45 5
-14 30 -6 17 15 29
-82 6
-35 99 7

22. Homework Read Section 1.2. Exercises: 4, 5 on pg.46-47.
Exercises: 6(a,c), 11(b,d), on pg
Those questions will be a part of your collected homework.