The RSA Cryptosystem ppt

The RSA Cryptosystem
Symmetric key cryptosystem
Public key cryptosystem (PKC)
One-way function
More number Theory
MAPLE command:
igcd(a,b)
example
Exercise
MAPLE
command:
igcdex(a, b, ’s’, ’t’)
example
Exercise
• Proof
Exercise
example
MAPLE command:
a^(-1) mod b
MAPLE command:
chrem([a
1
, …, a
r
],[m
1
, …, m
r
])

Exercise
The order of group elements
• Definition: The order of an element g in G is the
smallest positive integer m such that g
m
= 1.
• Example: Find the order of 3 and 2 in Z
7
*
.
– 3
1
= 3; 3
2
= 2; 3
3
= 6; 3
4
= 4; 3
5
= 5; 3
6
= 1 (mod 7).
–
2
1
= 2; 2
2
= 4;
2

3
= 1
(mod 7).
–
2
1
= 2; 2
2
= 4;
2
3
= 1
(mod 7).
– The order of an element g in Z
7
*
must divides 6 {1, 2, 3, 6}.
– The order of an element g in Z
11
*
must divides 10 {1, 2, 5,
10}.
Facts
Definition: An element having order p – 1 modulo p is
call a primitive element modulo p.
Exercise
The RSA cryptosystem
Short break

Exercise: show that (x
b
)
a
= x (mod n) if x in Z
n
\ Z
n
*.