I explained this at Why does key generation take an input $1^k$, and how do I represent it in practice?. Take a look at that one. Basically, it's an artificial little detail that's only there to keep the theoretical details squared away. You can safely ignore it on first reading.
In concrete security, this $1^k$ is no longer needed; it only shows up in asymptotic complexity theory. Concrete security is more relevant in practice anyway.
In asymptotic crypto-theory, we allow the adversary to run in time polynomial in the length of the input. The longer the input, the longer the adversary is allowed to take. Thus, it's not more convenient for the adversary if the input is shorter. Notice that this is a purely artificial element of asymptotic theory, where "efficient" is defined as "runs in asymptotically polynomial time in the length of the input" -- that definition is artificial and doesn't really correspond to reality (but it's sometimes convenient for theoretical analysis).
