CS338 Computer Security
Monday, 25 September 2023

+ Modular arithmetic
- Integers
- Multiplication
- Division

+ Diffie Hellman steps
- Alice comes up with some integers
    g, p     (often chosen from a standard list)
    x        (random, x < p)
    a = g^x mod p
- Alice sends (g, p, a) to Bob
- Bob comes up with              <-- if Bob doesn't like g, p, he can cancel here
    y        (random, y < p)
    b = g^y mod p
- Bob sends y to Alice
- Alice computes
    b^x mod p = (g^y)^x mod p = g^(xy) mod p    <-- K
- Bob computes
    a^y mod p = (g^x)^y mod p = g^(xy) mod p    <-- K

NOTE: Alice doesn't know y; Bob doesn't know x


+ RSA encryption
- Bob generates
    p, q            (random primes)
    n = p * q
    e               (with a bunch of properties--check the wikipedia page for details)
    d               (such that e * d = 1 mod (p-1)(q-1))
- Bob publicizes
    (n, e)          (Bob's "public key")
- Alice's message
    m < n
- Alice computes
    c = m^e mod n   (the ciphertext; i.e., the encrypted message)
- Alice sends c to Bob
- Bob computes
    c^d mod n = ...[math]... = m


+ Doing the RSA part
- I have n and e
- Brute-force, factor n into p * q (observe that p and q are prime)
- Brute-force, find d with the appropriate property with respect to e, p, and q
- For each message block M, compute M^d mod n
- Take the resulting decrypted blocks and try to make ASCII characters out of them