Transcript Lecture 9
Introduction to
Cryptography
Lecture 9
Public – Key Cryptosystems
Each participant has a public key and
a private key.
It should be infeasible to determine the
private key from knowledge of the
public key.
Public – Key Cryptosystems
message
Alice
Bob
Bob encrypts message using Alice’s
public key
Alice decrypt message using her
private key
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: 6647 17 2 23
90 2 3 5
2
The RSA Public – Key Cryptosystem
In 1978, Ronald Rives, Adi Shamir, and
Leonard Adelman wrote a paper called “A
Method for Obtaining Digital Signatures
and Public Key Cryptosystem”.
They described a cipher system in which
senders encrypt message using a method
and a key that are publicly distributed.
The RSA Public – Key Cryptosystem
Alice:
Selects two prime numbers p and q.
Calculates m = pq and n = (p - 1)(q - 1).
Selects number e relatively prime to n
Finds inverse of e modulo n
Publishes e and m
The RSA Public – Key Cryptosystem
To encrypt the message x:
e
y
x
mod m .
Bob computes:
Bob sends y to Alice.
To Decrypt the message y:
d
Alice computes: x y mod m .
The RSA Public – Key Cryptosystem
Example:
p =127, q = 223.
Then m = 28321 and n = 27972
Let e = 5623, check gcd(n,e) = 1.
Then using Extended Euclidean Algorithm
d = 22495.
Public Key: (5623, 28321).
The RSA Public – Key Cryptosystem
Example:
Let the message be x = 3620.
5623
y
3620
mod 28321 27845.
Then
Alice gets one and decrypts it
22495
mod 28321 3620.
Then x 27845
The RSA Public – Key Cryptosystem
Why does this method work?
y d ( x e mod m) d ( x e ) d mod m x ed mod m x.
Last step is a little bit more complicate
How secure is RSA?
Can opponent deduce d and n from (m,e)?
The opponent can find n and d only if he can
factor m.
Factoring
Problem of factoring a number is very hard
Fermat’s factoring method sometimes can
be used to find any large factors of a
number fair quickly (pg.251)
Want to make sure Fermat’s factoring
method does not work for your key
p and q should be at least 155 decimal
digits each
Homework
Read pg.286-293.
Exercises: 2(a), 4(c), 5(a) on pg.294.
Those questions will be a part of your
collected homework.