Can someone give me a short explanation as to why it is so hard to crack the private key? Won't the motivations to crack private keys be so high right now that someone is going to figure out how to do it? Won't we have enough computing power to eventually be able to do it?
Also, how is it possible that from the private key you can generate the public key and that fact in no way makes it easier to do the opposite???
There is no easy way and I'm no specialist on the suibject.
First wait for a spend transaction on the account you want to hack, only then will the public key be known.
The bitcoin address is a sha-256 hash of the public key. Since a hash doesn't have all the data you are unable to find a public key easily. So you must wait for an outgoing transaction.
(This is the reason offline wallets are mega safe ;-) )
Once you have the public key you can use Eliptic curve calculations with some bitcoin specifics to work back the public key towards the factorization problem.
Once you have that big ass number (c) you simply need to break it in to two numbers (a and b) like the formula c=a*b so grab your quantum-computer and do the math.
Using a and b you can then calculate a private key that fits the lock. Then turn it into the private key form for BTC.
http://en.wikipedia.org/wiki/Elliptic_Curve_DSAAlso there is some dude on the internet who hacked weak private keys. He explains the process and in the weak-key case a random number generator failed (same number every time) making the private key easily hackable.
If you understand his math then you can go hack private keys if you can solve the factorization problem (if you can you are the most powerful person on the planet).
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people that expect 100% profit per month 
