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Internet Engineering
Czesław Smutnicki
Discrete Mathematics – Numbers Theory
CONTENTS
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Basic notions
Greatest common divisor
Modular arithmetics
Euclidean algorithm
Modular equations
Chinese theorem
Modular powers
Prime numbers
RSA algorithm
Decomposition into factors
BASIC NOTIONS
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Natural/integer numbers
Divisor d|a, a = kd for some integer k
d|a if and only if -d|a
Divisor: 24: 1,2,3,4,6,8,12,24
Trivial divisors 1 and a
Prime number 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59
Composite number, 27 (3|27)
For any integer a and any positive integer n there exist unique integers q and r,
0<=r<n so that a = qn + r
Residue r = a mod n
Division q = [a/n]
Congruence: a  b (mod n) if (a mod n) = (b mod n)
Equivalence class (mod n): [a]n = {a + kn : k  Z}
GREATEST COMMON DIVISOR
• Common divisor: if d|a and d|b
• d|(ax + by)
• Relatively prime numbers a and b : gcd(a,b)=1
gcd(a, b)  gcd(b, a )
gcd(a, b)  gcd( a, b)
gcd(a, b)  gcd( a , b )
gcd(a,0)  a
gcd(a, ka)  a
gcd(a, b)  gcd(b, a modb)
MODULAR ARITHMETIC
[a]n  n [b]n  [a  b]n
[a]n n [b]n  [a  b]n
[a]n  n [b]n  [a  b]n
[a]n n [a 1 ]n  [1]n
1
[a]n / n [b]n  [a]n n [b ]n
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EUCLIDEAN ALGORITHM
mx  ny  k , k  gcd(m, n)
a0  q1a1  a2
a1  q2 a2  a3
.......... .......... ......
as 1  qs as  as 1
a0  n, a1  m, as 1  0, as  gcd(m, n)
x0 , x1, x2 ,..., xs , y0 , y1, y2 ,..., ys
ai  mxi  nyi , i  0,1,2,..., s
x0  0, y0  1, x1  1, y1  0,
xi 1  qi xi  xi 1 , yi 1  qi yi  yi 1 ,
EUCLIDEAN ALGORITHM.
INSTANCE
333 x  1234 y  1
1234  3*333  235  1234  3*333  235
333  1*235  98  333  1*235  98
235  2*98  39  235  2*98  39
98  2*39  20  98  2*39  20
39  1*20  19  39  1*20  19
20  1*19  1  20  1*19  1
19  19*1  0  19  19*1  0
EUCLIDEAN ALGORITHM.
INSTANCE cont.
20  1*  39  1* 20  1  1* 39  2* 20  1
1* 39  2*  98  2* 39  1  5* 39  2*98  1
5*  235  2*98  2*98  1  5* 235  12*98  1
5* 235  12*  333  1* 235  1  17* 235  12* 333  1
17* 1234  3* 333  12* 333  1  63* 333  17*1234  1
333*63  1234*17  1
x  63
y  17
MODULAR EQUATIONS
ax  b (modn)
d  gcd(a, n)
EQUATION EITHER HAS d VARIOUS SOLUTIONS mod n
OR DOES NOT HAVE ANY SOLUTION
CASE b = 1: MULTIPLICATIVE INVERSE
(IF gcd(a,n)=1 THEN IT EXISTS AND IS UNIQUE)
LEAST COMMON MULTIPLIER
lcm(a, b) 
ab
gcd(a, b)
CHINESE THEOREM
n  n1n2 ...nk , gcd(ni , n j )  1, i  j
a  (a1 , a2 ,..., ak )
ai  a mod ni
b  (b1 , b2 ,..., bk )
(a  b) mod n  (( a1  b1 ) mod n1 ,..., (ak  bk ) mod nk )
(a  b) mod n  (( a1  b1 ) mod n1 ,..., (ak  bk ) mod nk )
(a  b) mod n  (( a1  b1 ) mod n1 ,..., (ak  bk ) mod nk )
MODULAR POWERS
a 2c mod n  (a c ) 2 mod n
a 2c1 mod n  a  (a c ) 2 mod n
RSA ALGORITHM
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Find two big prime numbers p and q
Calculate n=p*q and z=(p-1)*(q-1)
Find any number d relatively prime with z
Find number e so that (e*d) mod z=1
Public key (e,n)
Private key (d,n)
Encryption message P
Decryption C
C  P e mod n
P  C d mod n
Thank you for your attention
DISCRETE MATHEMATICS
Czesław Smutnicki