This thesis deals with the security of the storage, the access and the maintenance of outsourced data. Indeed, outsourced storage raises new threats for users. We focus on the three following protocols. First, Proofs of Retrievability (PoR) allows a user who rarely accesses its data to be sure that it is stored on the server and that it did not suffer any alterations. Second, Updatable Encryption (UE) permits the user of an encrypted database to rotate its cryptographic keys with low bandwith usage. Third, Private Information Retrieval (PIR) allows a user to make the way it accesses outsourced data confidential. The goal of this thesis can be summarized in three steps. In step one, we develop modular security notions and models that closely match the security expectations of real-world solutions for the three above problems. Then, in step two, we check if existing security definitions are sufficient, and sometimes also necessary, to provide the security guarantees identified in step one. Finally, we determine if existing cryptographic schemes reach our security definitions and, if not, we improve them or propose new constructions that do. Our security statements are phrased in the Constructive Cryptography (CC) model of Maurer. The contributions made in this thesis are the following. We study the problem of building UE in the group action framework. First, we propose the first post-quantum UE scheme that supports an unbounded number of key updates. Second, our new scheme is the first post-quantum UE scheme whose security does not rely on lattice-based problems. We also show how to obtain a post-quantum UE scheme secure against chosen ciphertexts attacks using group actions. Unfortunately, we do not have any practical instantiation for this last scheme currently. As for PoRs, we show that the security of a PoR of Lavauzelle and Levy-dit-Vehel was overestimated. We propose a framework for the design of secure and efficient PoR schemes based on Locally Correctable Codes. We use our framework to give a secure generalization of the previously mentioned PoR. Furthermore, we use expander codes to design another PoR scheme with better parameters. We also extend CC so as to handle interactive protocols which delegate computations to an adversary. We use it to model UE and PIR. As for UE, we give the first composable modeling of UE by circumventing an impossibility result known as the commitment problem. We use this modeling to understand which security notion for UE is best suited for different real-world applications. As for PIR, we give a composable and unified treatment of many PIR variants.