Why public key criptography requires longer keys than single key cryptography?

Answer 1

The answer is actually in the question.

First, you need to understand what happens with public key cryptography (asynchronous). In essence you use public keys to initiate a secure channel, by which to pass a "single" or more properly stated, synchronous key. The operation to encrypt with the large asynchronous key is very expensive (from a computation time perspective), so you only use that to encrypt the synchronous key, then you encrypt the actual data with this synchronous key.

The reason that synchronous keys are shorter is because (if you are being secure) the synchronous keys are transfered in a secure manner. For example, either by passing it to someone in person, or over a channel that is secure.

An asynchronous key is not transfered via a secure method, it is transfered in the clear. To really understand how this all works, you need to look into a Diffie Hellman key exchange. A public key is really just a really big Prime number put through some sort of algorithm. If we use multiplication as an example we can say, you take one huge prime number and multiply it by another huge prime number. This operation is easy to do, but very hard to undo without one of the numbers. The magic of how to get to secret numbers by transferring (in the clear) non secret numbers is what Diffie Hellman key exchange is all about.

So I guess the short and sweet answer to your question is:

Asynchronous keys are longer then synchronous keys because synchronous keys are only meant to be transfered via a secure channel. Asynchronous keys are transfered via insecure channels and must be longer to get an appropriately sized number to prevent factorization.