In the last few years, the need to design new cryptographic hashfunctions has led to the intense study of when desired hash

multi-properties are preserved or assured under compositions and

domain extensions. In this area, it is important to identify the

exact notions and provide often complex proofs of the resulting

properties. Getting this analysis right (as part of provable security

studies) is, in fact, analogous to cryptanalysis. We note that it is

important and quite subtle to get indeed the ``right\'\' notions and

properties, and ``right\'\' proofs in this relatively young

area. Specifically, the security notion we deal with is ``adaptive

preimage resistance\'\' (apr) which was introduced by Lee and Park as an extension of ``preimage resistance\'\' (pr). In

Eurocrypt 2010, in turn, Lee and Steinberger already

used the apr security notion to prove ``preimage awareness\'\' and

``indifferentiable security\'\' of their new double-piped mode of

operation. They claimed that if $H^P$ is collision-resistant (cr) and apr,

then $F(M)=\\mathcal{R}(H^P(M))$ is indifferentiable from a variable

output length (VIL) random oracle $\\mathcal{F}$, where $H^P$ is a

function based on an ideal primitive $P$ and $\\mathcal{R}$ is a fixed

input length (FIL) random oracle. However, there are some limitations in their claim, because they considered only indifferentiability security notion in the information-theoretic adversarial model, not in the computation-theoretic adversarial model. As we show in the current

work, the above statement is \\textit{not} correct in the computation-theoretic adversarial model. First in our

studies, we give a counterexample to the above. Secondly, we describe

\\textit{a new requirement} on $H^P$ (called ``admissibility\'\') so that

the above statement is correct even in the computation-theoretic adversarial model. Thirdly, we show that apr is, in fact,

not a strengthened notion of preimage resistance. Fourthly, we

explain the relation between preimage awareness and cr+apr+(our new

requirement) in the computation-theoretic adversarial model. Finally, we show that a polynomial-based mode of

operation \\cite{LeSt10} satisfies our new requirement; namely, the

polynomial-based mode of operation with fixed-input-length random

oracles is indifferentiable from a variable-input-length random oracle in the computation-theoretic adversarial model.