Introduction to Cryptography

1. Introduction to Cryptography Lecture 2

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

3. f x1 f x1 f(x1) x2 f(x1) x2 f(x2) f(x2) x3 Range Range Domain Domain 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) Not a function Function

4. Functions Definition: A function is called one to oneif 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. x1 f x2 f(x1) f(x2) x3 Range Domain Functions f x1 f(x1) x2 f(x1) f(x2) x3 y Range Domain 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 functionan 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:

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 greatestcommondivisor of x and y is number d such that d divides x and d divides y. Definition: x and y are relativelyprime 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 • 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

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

22. Homework • Read Section 1.2. • Exercises: 4, 5 on pg.46-47. • Read Section 4.1. • Exercises: 6(a,c), 11(b,d), on pg.260-262 • Those questions will be a part of your collected homework.