1. 1. RSA basics 2. Key generation 3. What it would take to break RSA
RSA encryption
1. RSA basics
2. Key generation
3. What it would take to break RSA

2. 1. RSA basics Two large prime numbers (p, q): n = pq
Small number e relatively prime to (p-1)(q-1)
(e,n) is the public key
Number d (multiplicative inverse of d, modulo (p-1)(q-1))
d e = 1 ( modulo (p-1)(q-1) )
(d,n) is the secret key
It works because (Me)d = Med = M , [1]
[1] Proof of RSA:

3. 1. RSA basics C  decrypt: M = SecretKey(C) = Cd (mod n)
M  encrypt: C = PublicKey(M) = Me (mod n) M
Public Key
(e,n) C
Secret key
(d,n) M
C  decrypt: M = SecretKey(C) = Cd (mod n)

4. 2. Key generation
Each pair public/private key requires two large primes (around 512 bits)
RSA widely used  needs lots of large primes
Primality tests:
- try all possible factors (good for small numbers)
- probable tests (may be enough)
- recently shown (2002): primality can be proven in just polynomial time in the number of digits [2]
[2] Agrawal, M., N. Kayal, and N. Saxena. Preprint. Primes is in P. Available at

5. 3. What it would take to break RSA
What does “breaking RSA” mean? A:
Factor n  find d
C = Me (mod n)  computing e-th roots (mod n) B:
Guessing the message C:
Attacking a particular implementation

6. Experiment: Breaking DES using brute force
Project statistics:
Start of contest: January 29, 1997
Announcement of DESCHALL project: February 18, 1997
End of contest: June 17, 1997
Size of keyspace: 72,057,594,037,927,936
Keys searched: 17,731,502,968,143,872
Peak keys/day: 601,296,394,518,528
Peak keys/second: 7,000,000,000 (approx)
Peak clients/day: 14,000 (approx, based on IP address)
Total clients, since start: 78,000 (approx, based on IP address)
The computer that found the key: CPU: Pentium 90
RAM: 16 megabytes
Operating System: FreeBSD 2.2.1
Speed (keys/second): 250,000 (approx)
Client: FreeBSD v0.214, built March 12, 1997
Owner: iNetZ Corporation, Salt Lake City,
Utah Operator: Michael K. Sanders