The University of Babylon
Department of Software
LECTURE NOTES ON INFORMATION SYSTEM SECURITY
(Basic Mathematical Concepts)
By
Dr. Samaher Hussein Ali
College of Information Technology, University of Babylon, Iraq
Samaher_hussein@yahoo.com
10/29/2012
The Modulo Operation
Definition
Let a, r, n be integers and let m > 0
We write a  r mod n if n divides a – r (or r – a) and 0  r < n
n is called the modulus
r is called the remainder
Note that r is positive or zero
Note that a = n.q + r where q is another integer (quotient)
Example: 42  6 mod 9
9 divides 42 ‐ 6 = 36
9 also divides 6 ‐ 42 = ‐36
Note that 42= 9x4 + 6 (q = 4)
10/29/2012
Dr. Samaher Hussein Ali
Notes of Lecture 7
Prime Numbers
 An integer p is said to be a prime number if its only positive divisors are 1 and itself
 Examples 1, 3, 7, 11, ..
 Any integer can be expressed as a unique product of prime numbers raised to positive integral powers
n=p1e1 p2e2 …pkek // n: ingterger, pi:prime, e,: positive integer
 Examples
7569 = 3 x 3 x 29 x 29 = 32 x 292
5886 = 2 x 27 x 109 = 2 x 33 x 109
4900 = 72 x 52 x 22
100 = ? , 250 = ?
 This process is called Prime Factorization
10/29/2012
Dr. Samaher Hussein Ali
Notes of Lecture 7
Greatest Common Divisor (GCD)
 Definition: Greatest Common Divisor
This is the largest divisor of both a and b
 Given two integers a and b, the positive integer c is called their GCD or greatest common divisor if and only if
 c | a and c | b
 Any divisor of both a and b also divides c
 Notation: gcd(a, b) = c
 Notation: gcd(0, 2) = 2
 Notation: gcd(0, 0) = ????
10/29/2012
Dr. Samaher Hussein Ali
Notes of Lecture 7
Greatest Common Divisor (GCD)
 Example: gcd(15,100) = ?
a= q*b+r
100=6*15+10
15=1*10+5
10=2*5+0
Gcd(15,100)=5
 Example: gcd(49,63) = ?
a= q*b+r
63=1*49+14
49=3*14+7
14=2*7+0
Gcd(49,63)=7
10/29/2012
Dr. Samaher Hussein Ali
Notes of Lecture 7
Relatively Prime Numbers
 Two numbers are said to be relatively prime if their gcd is 1
 Example: 63 and 22 are relatively prime
Gcd(63,22)
63= 2(22)+19
22=1(19)+3
19=6(3)+1
3= 3(1)+0
‐ Gcd(63,22)=1
 How do you determine if two numbers are relatively prime?
 Find their gcd or
 Find their prime factors
 If they do not have a common prime factor other than 1, they are relatively prime
 Example: 63 = 9 x 7 = 32 x 7 and
22 = 11 x 2
10/29/2012
Dr. Samaher Hussein Ali
Notes of Lecture 7
Euler phi (or totient) function For n ≥ 1, (n) : is the number of integers in [1,n] which are relatively prime to n // (n) is the Euler phi or totient function
 If p is prime, then (p)=p‐1
 If R=p1e1 p2e2 …pkek
then
( R ) =IIi=1n(pi-1)ei Piei-1
 If gcd(m,n)=1, then (mn)= (m).(n)
Examples:
(7)=7‐1=6
(96)=?
96=2^5*3
(96)=(2)5‐1*(2‐1)5*(3‐1)1*(3)0
=2^4*2
=32
10/29/2012
Dr. Samaher Hussein Ali
Notes of Lecture 7