CIS 5371 Cryptography
6. An Introduction to Number Theory
1
C
Congruence
and
d Residue
R id classes
l

Arithmetic modulo n, Zn

Solving
g linear equations
q

The Chinese Remainder Theorem

Euler’s p
phi function

The theorems of Fermat and Euler

Quadratic residues

Legendre & Jacobi symbols
2
A ith ti modulo
Arithmetic
d l n
Examples

Zn ={0,1,2,
{ , , , … , n-1},
},

Zp* = {1,2, … , p-1}, for prime p,

Zn* = {all integers k,
k 0 < k < n,
n with gcd(k,n)
gcd(k n) = 1}
1}.
3
S l i li
Solving
linear equations
ti
Theorem
For any integer n > 1,
ax  b (mod
( d n))
is solvable, if and only if, gcd (a,n) | b .
Examples
6x  18 (mod 36) has 6 solutions: 3, 9, 15, 21, 27, 33.
2x  5 (mod 10), has no solutions.
4
Th Chi
The
Chinese R
Remainder
i d Th
Theorem
Let m1 ,  , mn be positive integers with gcd(mi , m j )  1,
and let M  m1m 2  mr .
Then the congruence
x  c1 (mod m1 )
x  c2 (mod m2 )
 

x  cr (mod mr )
has a unique solution z  Z M .
5
Th Chi
The
Chinese R
Remainder
i d Th
Theorem
Let ( M / mi ). yi  1 (mod mi ). Then for
zi  ( M / mi ). yi
it is easy to see that
z  i 1 zi ci (mod
( d M)
r
is a solution.
6
E
Example
l
S l the
Solve
th modular
d l conguence :
x  2 (mod 3)
x  3 ((mod 5)
x  4 (mod 7)
W have
We
h
: M  105,
105
y1  35 1 (mod 3)  2 (mod 3)
y 2  211 (mod
( d 5)  1 (mod
( d 5)
y 3  15 1 (mod 7)  1 (mod 7)
Then z  2  35  2  3  21  1  4  15  1  263 mod 105  53 mod 105
7
E
Example
l
Solve
S
l the
th modular
d l conguence :
x  1 (mod 2)
x  2 (mod 5)
x  3 (mod 7)
8
E l ’ phi
Euler’s
hi f
function
ti
 (n) is the number of positive integers k for which
gcd(k,n)  1. We have
 (1)  1
 ( p )  p  1, p a prime,
 ( pq)  ( p-1)(q-1), for p,q primes,
1
p
 ( p e )  p e  p e 1  p e (1  ), p a prime, e  1.
It follows that if n  p1 1  p1 2    p1 1 , is the p
prime
e
e
e
factorization of n, then
1
1
1
 (n)  n (1  )  (1  )    (1  )).
p1
p2
pk
9
Th theorems
The
h
of
f Fermat
F
and
d Euler
E l
Fermat’ss little theorem
Fermat
If p is prime then,
ap-11  1 (mod
( d p),
)
for all integers a: 0 < a < p.
Euler’s theorem
If gcd (a,n)=1 then a(n)  1 (mod n),
10
L
Legendre
d &J
Jacobi
bi symbols
b l
L QR p be
Let
b the
h set off quadratic
d i residues
id modd p andd let
l
QNR p  Z p \ QR p be the set of quadratic nonresidues.
For x  Z p , p prime, the Legendre symbol of x mod p is :
pe
 1 if x  QR p
x
   
 p
 1 if x  QNR p
Let n  p1 p2  pk be the prime factorization of n.
For x  Z n , the Jakobi symbol of n mod p is :
 x  x   x 
 x
         
n
 p1   p2   pk 
11
L
Legendre
d &J
Jacobi
bi symbols
b l
It is easy to see that
 x
   x ( p 1) / 2 (mod p ), for p prime.
 p
12
L
Legendre
d &J
Jacobi
bi symbols
b l
We have :
1
 
n
 x

 mn
 xy   x   y 
 1, and 
    
n
 n n

  x   x
    
 m n
 x  y
If x  y ((mod n ) then     
n n
For m , n odd :
  1
(n -1)/2

  (-1)
 n 
2
(n 2 -1)/8
   (-1)
n
m n 
If gcd( m,n )  1 and m , n  2 then      (-1) (m -1)(n -1)/4
 n m
13
E
Example
l
Compute
 7 
 4 
 and 


 15 
 15 
14