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Introduction to Cryptography Lecture 2
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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
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
Permutations
Example: x 1 2 3 4 5
p(x) 3 1 5 4 2
x 1 2 3 4 5
q(x) 2 5 1 4 3
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
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.
