Error-control codes, which are being extensively used in communication systems, have found themselves very useful in data storage as well during the past decade. This thesis deals with two types of codes for data storage, one pertaining to the issue of privacy and the other to reliability. In many scenarios, user accessing some critical data from a server would not want the server to learn the identity of data retrieved. This problem, called Private Information Retrieval (PIR) was rst formally introduced by Chor et al and they gave protocols for PIR in the case where multiple copies of the same data is stored in non-communicating servers. The PIR protocols that came up later also followed this replication model. The problem with data replication is the high storage overhead involved, which will lead to large storage costs. Later, Fazeli, Vardy and Yaakobi, came up with the notion of PIR code that enables information-theoretic PIR with low storage overhead. In the rst part of this thesis, construction of PIR codes for certain parameter values is presented. These constructions are based on a variant of conventional Reed-Muller (RM) codes called binary Projective Reed-Muller (PRM) codes. A lower bound on block length of systematic PIR codes is derived and the PRM based PIR codes are shown to be optimal with respect to this bound in some special cases. The codes constructed here have smaller block lengths than the short block length PIR codes known in the literature. The generalized Hamming weights of binary PRM codes are also studied. Another work described here is the implementation and evaluation of an erasure code called Coupled Layer (CL) code in Ceph distributed storage system. Erasure codes are used in distributed storage to ensure reliability. An additional desirable feature required for codes used in this setting is the ability to handle node repair efficiently. The Minimum Storage Regenerating (MSR) version of CL code downloads optimal amount of data from other nodes during repair of a failed node and even disk reads during this process is optimum, for that storage overhead. The CL-Near-MSR code, which is a variant of CL-MSR, can efficiently handle a restricted set of multiple node failures also. Four example CL codes were evaluated using a 26 node Amazon cluster and performance metrics like network bandwidth, disk read and repair time were measured. Repair time reduction of the order of 3 was observed for one of those codes, in comparison with Reed Solomon code having same parameters. To the best of our knowledge, such large gains in repair performance have never been demonstrated before.

Le chiffrement totalement homomorphe permet d’effectuer des calculs sur des données chiffrées sans fuite d’information sur celles-ci. Pour résumer, un utilisateur peut chiffrer des données, tandis qu’un serveur, qui n’a pas accès à la clé de déchiffrement, peut appliquer à l’aveugle un algorithme sur ces entrées. Le résultat final est lui aussi chiffré, et il ne peut être lu que par l’utilisateur qui possède la clé secrète. Dans cette thèse, nous présentons des nouvelles techniques et constructions pour le chiffrement totalement homomorphe qui sont motivées par des applications en apprentissage automatique, en portant une attention particulière au problème de l’inférence homomorphe, c’est-à-dire l’évaluation de modèles cognitifs déjà entrainé sur des données chiffrées. Premièrement, nous proposons un nouveau schéma de chiffrement totalement homomorphe adapté à l’évaluation de réseaux de neurones artificiels sur des données chiffrées. Notre schéma atteint une complexité qui est essentiellement indépendante du nombre de couches dans le réseau, alors que l’efficacité des schéma proposés précédemment dépend fortement de la topologie du réseau. Ensuite, nous présentons une nouvelle technique pour préserver la confidentialité du circuit pour le chiffrement totalement homomorphe. Ceci permet de cacher l’algorithme qui a été exécuté sur les données chiffrées, comme nécessaire pour protéger les modèles propriétaires d’apprentissage automatique. Notre mécanisme rajoute un coût supplémentaire très faible pour un niveau de sécurité égal. Ensemble, ces résultats renforcent les fondations du chiffrement totalement homomorphe efficace pour l’apprentissage automatique, et représentent un pas en avant vers l’apprentissage profond pratique préservant la confidentialité. Enfin, nous présentons et implémentons un protocole basé sur le chiffrement totalement homomorphe pour le problème de recherche d’information confidentielle, c’est-à-dire un scénario où un utilisateur envoie une requête à une base de donnée tenue par un serveur sans révéler cette requête. / Fully homomorphic encryption enables computation on encrypted data without leaking any information about the underlying data. In short, a party can encrypt some input data, while another party, that does not have access to the decryption key, can blindly perform some computation on this encrypted input. The final result is also encrypted, and it can be recovered only by the party that possesses the secret key. In this thesis, we present new techniques/designs for FHE that are motivated by applications to machine learning, with a particular attention to the problem of homomorphic inference, i.e., the evaluation of already trained cognitive models on encrypted data. First, we propose a novel FHE scheme that is tailored to evaluating neural networks on encrypted inputs. Our scheme achieves complexity that is essentially independent of the number of layers in the network, whereas the efficiency of previously proposed schemes strongly depends on the topology of the network. Second, we present a new technique for achieving circuit privacy for FHE. This allows us to hide the computation that is performed on the encrypted data, as is necessary to protect proprietary machine learning algorithms. Our mechanism incurs very small computational overhead while keeping the same security parameters. Together, these results strengthen the foundations of efficient FHE for machine learning, and pave the way towards practical privacy-preserving deep learning. Finally, we present and implement a protocol based on homomorphic encryption for the problem of private information retrieval, i.e., the scenario where a party wants to query a database held by another party without revealing the query itself.