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In recent years, cloud computing has become the most popular and promising service platform. A cloud user can outsource its heavy computation overhead to a cloud service provider (CSP) and let the CSP make the computation instead. In order to guarantee the correctness of the outsourced processing (e.g., machine learning and data mining), a proof should be provided by the CSP in order to make sure that the processing is carried out properly. On the other hand, from the security and privacy points of view, users will always encrypt their sensitive data first before they are outsourced to the CSP rather than sending the raw data directly. However, processing and verifying of encrypted data computation has always been a challenging problem. Homomorphic Encryption (HE) has been proposed to tackle this task on computations over encrypted data and ensure the confidentiality of the data. However, original HE cannot provide an efficient approach to verify the correctness of computation over encrypted data that is processed by CSP. In this paper, we propose a verifiable outsourced computation scheme over encrypted data with the help of fully homomorphic encryption and polynomial factorization algorithm. Our scheme protects user data security in outsourced processing and allows public verification on the computation result processed by CSP with zero knowledge. We then prove the security of our scheme and analyze its performance by comparing it with some latest related works. Performances analysis shows that our scheme reduces the overload of both the cloud users and the verifier.

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Data deduplication is a technique to eliminate duplicate data in order to save storage space and enlarge upload bandwidth, which has been applied by cloud storage systems. However, a cloud storage provider (CSP) may tamper user data or cheat users to pay unused storage for duplicate data that are only stored once. Although previous solutions adopt message-locked encryption along with Proof of Retrievability (PoR) to check the integrity of deduplicated encrypted data, they ignore proving the correctness of duplication check during data upload and require the same file to be derived into same verification tags, which suffers from brute-force attacks and restricts users from flexibly creating their own individual verification tags. In this paper, we propose a verifiable deduplication scheme called VeriDedup to address the above problems. It can guarantee the correctness of duplication check and support flexible tag generation for integrity check over encrypted data deduplication in an integrative way. Concretely, we propose a novel Tag-flexible Deduplication-supported Integrity Check Protocol (TDICP) based on Private Information Retrieval (PIR) by introducing a novel verification tag called note set, which allows multiple users holding the same file to generate their individual verification tags and still supports tag deduplication at the CSP. Furthermore, we make the first attempt to guarantee the correctness of data duplication check by introducing a novel User Determined Duplication Check Protocol (UDDCP) based on Private Set Intersection (PSI), which can resist a CSP from providing a fake duplication check result to users. Security analysis shows the correctness and soundness of our scheme. Simulation studies based on real data show the efficacy and efficiency of our proposed scheme and its significant advantages over prior arts.