Introduction to Cryptography Lecture 2

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

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

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.

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

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 .

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

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.

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

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

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

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:

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)

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.

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

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.

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

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  

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

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

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  

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