p = 7 q = 13

n = p * q n = 7 * 13 n = 91

φ(n) = (p - 1) * (q - 1) φ(91) = (7 - 1) * (13 - 1) φ(91) = 6 * 12 φ(91) = 72

1 < e < φ(n) 1 < e < φ(91) 1 < e < 72

e = 23 (because I said so)

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

public key = (n = 91, e = 23) private key = (n = 91, d = 47)

m = 60 c(m) = m^e mod n c(60) = 60^23 mod 91 c(60) = 44 c = 44 (they send this to you)

c = 44 m(c) = c^d mod n m(44) = 44^47 mod 91 m(44) = 60 m = 60 (now you have the secret)

broadcast = 102 hash(broadcast) = broadcast * K mod 2^32 hash(102) = 102 * 2654435761 mod 2^32 hash(102) = 169507974

signature = hash(broadcast) ^ d mod n signature = hash(102) ^ 47 mod 91 signature = 169507974 ^ 47 mod 91 signature = 90

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)

>>> set((91 / i) for i in xrange(1, 91) if 91 % i == 0) set([91, 13, 7])

>>> set((12031294782491844L / i) for i in xrange(1, 12031294782491844L) if 12031294782491844L % i == 0)

I was trying to explain public key cryptography today and totally failed at it. Here are notes to myself based on various Wikipedia pages. There's a lot more to it than this (like padding ) but this is the gist of it.1. Pick two prime numbers:2. Multiply them together:3. Compute Euler's " totient " function of n:4. Pick a random number e such that:and e is " coprime " with φ(n), meaning it has no common factors.5. Compute d, the modular multiplicative inverse of e (mod φ(n)):Now you have all the magic numbers you need:1. Have someone else encrypt a message m using your public key (n, e):2. Decrypt the message c using your private key (n, d):1. Hash the message to broadcast (this is Knuth's insecure hash function):2. Compute the signature with your private key (n, d):3. People who receive the broadcast message (102) and the signature (90) can confirm with your public key (n, e):In this simple example it's totally not. You can trivially take n = 91 and guess p and q with simple factorization 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: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.