introduction to cryptography
DESCRIPTION
Introduction to Cryptography. Lecture 2. x1. f. f(x1). x2. f(x3). f(x2). x3. Domain. Range. Functions. f. x1. f. x1. f(x1). x2. f(x1). x2. f(x2). f(x2). x3. Range. Range. Domain. Domain. Functions. - PowerPoint PPT PresentationTRANSCRIPT
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
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
Functions
f(x1)
x3
x2
x1
f(x2)
f
Domain Range
f(x1)
x3
x2
x1
f(x2)
f
Domain Range
Not one to one One to one
Onto Not onto
f(x1)
y
Functions
f
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 .
1f xyf )(1
yxf )(
xyf log)(1 xexf )(
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
1f1f
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.
Functions and Cryptography
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
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: 23176647 2
53290 2
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.
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
Greatest Common Divisors - GCD
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.
Division Principle
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
Euclidean Algorithm
Example: Let x = 4200 and y = 1485
i xi yi qi ri
0 4200 1485 2 1230
1 1485 1230 1 255
2 1230 255 4 210
3 255 210 1 45
4 210 45 4 30
5 45 30 1 15
6 30 15 2 0
7 15 0
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
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
0 4200 1485 2 1230 0 1
1 1485 1230 1 255 1 -2
2 1230 255 4 210 -1 3
3 255 210 1 45 5 -14
4 210 45 4 30 -6 17
5 45 30 1 15 29 -82
6 30 15 2 0 -35 99
7 15 0