# Outline for October 20, 2003

## Discussion Problem

"To fight and conquer in all your battles is not supreme excellence; supreme excellence consists in breaking the enemy's resistance without fighting. In the practical art of war, the best thing of all is to take the enemy's country whole and intact; to shatter and destroy it is not so good. So, too, it is better to capture an army entire than to destroy it, to capture a regiment, a detachment, or a company entire than to destroy it."1
What does this paragraph say to a system administrator or security officer seeking insight to defend her systems?

## Outline for the Day

1. Public-Key Cryptography
1. Basic idea: 2 keys, one private, one public
2. Cryptosystem must satisfy:
1. given public key, CI to get private key;
2. cipher withstands chosen plaintext attack;
3. encryption, decryption computationally feasible [note: commutativity not required]
3. Benefits: can give confidentiality or authentication or both
2. RSA
1. Provides both authenticity and confidentiality
2. Go through algorithm:
Idea: C = Me mod n, M = Cd mod n, with ed mod φ(n) = 1.
Proof: Mφ(n) mod n = 1 [by Fermat's theorem as generalized by Euler]; follows immediately from ed mod φ(n) = 1.
Public key is (e, n); private key is d. Choose n = pq; then φ(n) = (p-1)(q-1).
3. Example:
p = 5, q = 7; n = 35, φ(n) = (5-1)(7-1) = 24. Pick e = 11. Then de mod φ(n) = 1, so choose d = 11. To encipher 2, C = Me mod n = 211 mod 35 = 2048 mod 35 = 18, and M = Cd mod n = 1811 mod 35 = 2.
4. Example: p = 53, q = 61, n = 3233, φ(n) = (53-1)(61-1) = 3120. Take e = 71; then d = 791. Encipher M = RENAISSANCE: A = 00, B = 01, ..., Z = 25, blank = 26. Then:
M = RE NA IS SA NC Eblank = 1704 1300 0818 1800 1302 0426
C= (1704)71 mod 3233 = 3106; etc. = 3106 0100 0931 2691 1984 2927
3. Cryptographic Checksums
1. Function y = h(x): easy to compute y given x; computationally infeasible to compute x given y
2. Variant: given x and y, computationally infeasible to find a second x' such that y = h(x').
3. Keyed vs. keyless

1. Sun Tzu, The Art of War, James Clavell, ed., Dell Publishing, New York, NY ©1983, p. 15