RSA padding: risk of using constants
By narrowing the space of messages and ciphertexts you are willing to consider to a tiny fraction of fewer than $1/2^{1000}$ of them, you cannot prove that an algorithm for breaking this translates to an algorithm for computing cube roots modulo $n$ in general.

Consequently you can't rely on the decades of work that have been put into failing to find a way to compute cube roots modulo $n$, and you have to redo those decades of failure anew in order to get confidence in the system's security.