1. Pick two prime numbers:
p = 7 q = 13
2. Multiply them together:
n = p * q n = 7 * 13 n = 91
3. Compute Euler's "totient" function of n:
φ(n) = (p - 1) * (q - 1) φ(91) = (7 - 1) * (13 - 1) φ(91) = 6 * 12 φ(91) = 72
4. Pick a random number e such that:
1 < e < φ(n) 1 < e < φ(91) 1 < e < 72
and e is "coprime" with φ(n), meaning it has no common factors.
e = 23 (because I said so)
5. Compute d, the modular multiplicative inverse of e (mod φ(n)):
e^-1 = d (mod φ(n)) 23^-1 = d (mod φ(91)) 23^-1 = d (mod 72) 23 * d = 1 (mod 72) 23 * 47 = 1 (mod 72) d = 47
Now you have all the magic numbers you need:
public key = (n = 91, e = 23) private key = (n = 91, d = 47)
Secret messages to you
1. Have someone else encrypt a message m using your public key (n, e):
m = 60 c(m) = m^e mod n c(60) = 60^23 mod 91 c(60) = 44 c = 44 (they send this to you)
2. Decrypt the message c using your private key (n, d):
c = 44 m(c) = c^d mod n m(44) = 44^47 mod 91 m(44) = 60 m = 60 (now you have the secret)
Prove you wrote something
1. Hash the message to broadcast (this is Knuth's insecure hash function):
broadcast = 102 hash(broadcast) = broadcast * K mod 2^32 hash(102) = 102 * 2654435761 mod 2^32 hash(102) = 169507974
2. Compute the signature with your private key (n, d):
signature = hash(broadcast) ^ d mod n signature = hash(102) ^ 47 mod 91 signature = 169507974 ^ 47 mod 91 signature = 90
3. People who receive the broadcast message (102) and the signature (90) can confirm with your public key (n, e):
broadcast = 102 hash(broadcast) = broadcast * K mod 2^32 hash(102) = 102 * 2654435761 mod 2^32 hash(102) = 169507974 confirmation = hash(broadcast)^e mod n confirmation = 169507974^23 mod 91 confirmation = 90 (matches your signature)
Why is this safe?
In this simple example it's totally not. You can trivially take n = 91 and guess p and q with simple factorization:
>>> set((91 / i) for i in xrange(1, 91) if 91 % i == 0) set([91, 13, 7])
Redo all the math above and you have the private key. Doh~
Now imagine if n was such a large number that it took a very long time to guess p and q:
>>> set((12031294782491844L / i) for i in xrange(1, 12031294782491844L) if 12031294782491844L % i == 0)
This takes a really long time. Even fancy solutions on the fastest computer on Earth would take until the end of the universe. That's why transmitting secrets with public key cryptography is safe. That's also why great leaps in prime number theory are frightening / exciting.