In this paper, we first show that many independent pseudorandom permutations over {0, 1}n can be obtained from a single public random permutation and secret n bits. We next prove that a slightly modified IAPM is secure even if the underlying block cipher F is publicly accessible (as a blackbox). We derive a similar result for OCB mode, too. The security proofs are based on our first result and are extremely simple. We finally show that our security bound is tight within a constant factor.