Ok, away from politics, and back to the good stuff. When I left off talking about
encryption, we were looking at
RSA, as an example of an asymmetric encryption system.
Since it's been a while, I'll start off with a quick review of RSA.
Asymmetric encryption systems like RSA are based on computations that are easy to perform if you know the keys, and incredibly hard to perform if you don't. In the specific case
of RSA, everything is based on a pair of very large prime numbers. If you know those two
primes, and you know the public key, it's really easy to compute the private key. But
if you don't know the two prime numbers, then even given the public key,
it's incredibly difficult to compute the private key.
To be a bit more specific, in RSA, you get a pair of large prime numbers, P and Q. You
compute from them a totient of their product, which is the number
N=(P-1)×(Q-1). Then you arbitrarily pick a public exponent, E, which is
smaller than N, and which is prime relative to N. You can then compute
the private key exponent, D. If you know what P and Q are, it's pretty easy to compute
Once you've got D and E, your public key is the pair is (N,E), and the private key is the pair is (N,D).
If you've got a plaintext message M, then encrypting it with the public key
is done by computing ME mod N. If you've got a ciphertext C encrypted
with the public key, then decrypting it is done by computing CD mod N.
Encrypting a plaintext with the public key is exactly the same process
as decrypting a ciphertext produced with the private key. And vice versa.