  In this work, we extend the Constructive Cryptography (CC) framework
introduced by Maurer in 2011 so as to handle interactive protocols.
  We design and construct a so-called {\em Interactive Server Memory Resource}
(ISMR), that is an augmented version of the basic instantiation of a
client-server protocol in CC, namely the Server Memory Resource. We then apply
our ISMR construction to two types of interactive cryptographic protocols for
remote storage : Updatable Encryption (UE) and Private Information Retrieval
(PIR).
  Concerning UE, our results are a composable version of those protocols,
clarifying the security guarantees achieved by {\em any} UE scheme. Namely, we
give the relevant security notion to consider according to a given leakage
context. Letting USMR denote our ISMR adapted to the UE application, we prove
that $\mathsf{IND}\text{-}\mathsf{UE}\text{-}\mathsf{CPA}$ security is
sufficient for a secure construction of a confidential USMR that hides the age
of ciphertexts; and
$\mathsf{IND}\text{-}(\mathsf{ENC}+\mathsf{UPD})\text{-}\mathsf{CPA}$ security
is sufficient for a secure construction of a confidential USMR in case of
unrestricted leakage. As a consequence, contrary to what was claimed before,
the $\mathsf{IND}\text{-}\mathsf{UE}$ security notion is not always stronger
than the $\mathsf{IND}\text{-}(\mathsf{ENC+UPD})$ one.
  Concerning PIR, we also give a composable version of PIR protocols, yielding
a unique model that unifies different notions of PIR : IT-PIR, C-PIR, one- or
multi- server PIR. Using the flexibility of CC, we are also able to model PIR
variants, such as SPIR.
