Here is the response I sent to Peter Simons:
Well, your example that you sent isn't even correct! Your example
said:
p = 5 q = 7
xy = 4*6+1 = 25 x=5
y=5
Well, you just chose some bad primes. Here is a better example:
p = 5 q = 11 N = pq = 55
m = (p-1)(q-1) = 4*10 = 40
Now, we need to choose our public and private decryptors, E and d,
such that Ed = 1 mod (m):
E = 3 d = 27
So, the Public Key (N, E) is (55, 3) and the Private (Secret)
Key (p, q, m, d) is (5, 11, 40, 27). Now, to encrypt a message,
S, you take C = S^E mod N, and to decrypt you get S = C^d mod N.
So, say the message you want to send is, oh, "8" (for lack of a
better example off the top of my head). So, you try to encrypt this
message, and you get:
C = 8^3 mod 55 = 512 mod 55 = 17
You then send this message to the recipient, who then calculates
the message back:
S = 17^27 mod 55 = 1667711322168688287513535727415473 mod 55 = 8
And you get the original message back.
-derek atkins