Most padding schemes for asymmetric encryption (OAEP, OAEP+) are only proven secure in the random oracle model. Although no attacks are known, it would be nice to find a padding scheme with provable security in the standard model. Are such padding schemes known?
To the best of my knowledge, there is no padded scheme for RSA (or general trapdoor permutation) that has been proven secure in the standard model. To be exact, let's call a padded scheme one where a padding transformation is carried out independently of the public key, and then the trapdoor permutation is applied once to the result.
Of course, as noted, we can do this to encrypt log(k) bits. However, for more than that, we don't know much. There is a very interesting result by Smith and Zhang that shows that under a "lossy" assumption about RSA, it's possible to prove security for about 1/4 of the bits of RSA. Concretely, you can encrypt 249 bits using a 2048-bit modulus. This is quite good.