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In this work, we extend the Constructive Cryptography (CC) frameworkintroduced 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 aclient-server protocol in CC, namely the Server Memory Resource. We then applyour ISMR construction to two types of interactive cryptographic protocols forremote 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, wegive the relevant security notion to consider according to a given leakagecontext. Letting USMR denote our ISMR adapted to the UE application, we provethat $\mathsf{IND}\text{-}\mathsf{UE}\text{-}\mathsf{CPA}$ security issufficient for a secure construction of a confidential USMR that hides the ageof ciphertexts; and$\mathsf{IND}\text{-}(\mathsf{ENC}+\mathsf{UPD})\text{-}\mathsf{CPA}$ securityis sufficient for a secure construction of a confidential USMR in case ofunrestricted leakage. As a consequence, contrary to what was claimed before,the $\mathsf{IND}\text{-}\mathsf{UE}$ security notion is not always strongerthan the $\mathsf{IND}\text{-}(\mathsf{ENC+UPD})$ one.  Concerning PIR, we also give a composable version of PIR protocols, yieldinga unique model that unifies different notions of PIR : IT-PIR, C-PIR, one- ormulti- server PIR. Using the flexibility of CC, we are also able to model PIRvariants, such as SPIR.

Interactivity in Constructive Cryptography : Modeling and Applications to Updatable Encryption and Private Information Retrieval