Cloud computing is an important part of today's world because offloading computations is a method to reduce costs. In this paper, we investigate computing the Speeded Up Robust Features (SURF) using Fully Homomorphic Encryption (FHE). Performing SURF in FHE enables a method to offload the computations while maintaining security and privacy of the original data. In support of this research, we developed a framework to compute SURF via a rational number based compatible with FHE. Although floating point (R) to rational numbers (Q) conversion introduces error, our research provides tight bounds on the magnitude of error in terms of parameters of FHE. We empirically verified the proposed method against a set of images at different sizes and showed that our framework accurately computes most of the SURF keypoints in FHE.
This paper investigates the problem of performing signal processing via remote execution methods while maintaining the privacy of the data. Primary focus on this problem is a situation where there are two parties; a client with data or a signal that needs to be processed and a server with computational resources. Revealing the signal unencrypted causes a violation of privacy for the client. One solution to this problem is to process the data or signal while encrypted. Problems of this type have been attracting attention recently; particularly with the growing capabilities of cloud computing. We contribute to solving this type of problem by processing the signals in an encrypted form, using fully homomorphic encryption FHE. Three additional contributions of this manuscript includes 1 extending FHE to real numbers, 2 bounding the error related to the FHE process against the unencrypted variation of the process, and 3 increasing the practicality of FHE as a tool by using graphical processing units GPU. We demonstrate our contributions by applying these ideas to two classical problems: natural logarithm calculation and signal prbrightness contrast filter. We present a new tensoring technique for LWE-based fully homomorphic encryption. While in all previous works, the ciphertext noise grows quadratically @math with every multiplication before "refreshing", our noise only grows linearly @math . We use this technique to construct a scale-invariant fully homomorphic encryption scheme, whose properties only depend on the ratio between the modulus q and the initial noise level B, and not on their absolute values. Our scheme has a number of advantages over previous candidates: It uses the same modulus throughout the evaluation process no need for "modulus switching", and this modulus can take arbitrary form. In addition, security can be classically reduced from the worst-case hardness of the GapSVP problem with quasi-polynomial approximation factor, whereas previous constructions could only exhibit a quantum reduction from GapSVP.  We present a fully homomorphic encryption scheme that is based solely on the(standard) learning with errors (LWE) assumption. Applying known results on LWE, the security of our scheme is based on the worst-case hardness of short vector problems'' on arbitrary lattices. Our construction improves on previous works in two aspects: We show that somewhat homomorphic'' encryption can be based on LWE, using a new re-linearization technique. In contrast, all previous schemes relied on complexity assumptions related to ideals in various rings. We deviate from the "squashing paradigm'' used in all previous works. We introduce a new dimension-modulus reduction technique, which shortens the cipher texts and reduces the decryption complexity of our scheme, without introducing additional assumptions . Our scheme has very short cipher texts and we therefore use it to construct an asymptotically efficient LWE-based single-server private information retrieval (PIR) protocol. The communication complexity of our protocol (in the public-key model) is @math bits per single-bit query (here, @math is a security parameter). We present a novel approach to fully homomorphic encryption (FHE) that dramatically improves performance and bases security on weaker assumptions. A central conceptual contribution in our work is a new way of constructing leveled fully homomorphic encryption schemes (capable of evaluating arbitrary polynomial-size circuits), without Gentry's bootstrapping procedure. Specifically, we offer a choice of FHE schemes based on the learning with error (LWE) or ring-LWE (RLWE) problems that have 2λ security against known attacks. For RLWE, we have: • A leveled FHE scheme that can evaluate L-level arithmetic circuits with O(λ · L3) per-gate computation -- i.e., computation quasi-linear in the security parameter. Security is based on RLWE for an approximation factor exponential in L. This construction does not use the bootstrapping procedure. • A leveled FHE scheme that uses bootstrapping as an optimization, where the per-gate computation (which includes the bootstrapping procedure) is O(λ2), independent of L. Security is based on the hardness of RLWE for quasi-polynomial factors (as opposed to the sub-exponential factors needed in previous schemes). We obtain similar results to the above for LWE, but with worse performance. Based on the Ring LWE assumption, we introduce a number of further optimizations to our schemes. As an example, for circuits of large width -- e.g., where a constant fraction of levels have width at least λ -- we can reduce the per-gate computation of the bootstrapped version to O(λ), independent of L, by batching the bootstrapping operation. Previous FHE schemes all required Ω(λ3.5) computation per gate. At the core of our construction is a much more effective approach for managing the noise level of lattice-based ciphertexts as homomorphic operations are performed, using some new techniques recently introduced by Brakerski and Vaikuntanathan (FOCS 2011). We propose the first fully homomorphic encryption scheme, solving an old open problem. Such a scheme allows one to compute arbitrary functions over encrypted data without the decryption key—i.e., given encryptions E(m1), ..., E( mt) of m1, ..., m t, one can efficiently compute a compact ciphertext that encrypts f(m1, ..., m t) for any efficiently computable function f. Fully homomorphic encryption has numerous applications. For example, it enables encrypted search engine queries—i.e., a search engine can give you a succinct encrypted answer to your (boolean) query without even knowing what your query was. It also enables searching on encrypted data; you can store your encrypted data on a remote server, and later have the server retrieve only files that (when decrypted) satisfy some boolean constraint, even though the server cannot decrypt the files on its own. More broadly, it improves the efficiency of secure multiparty computation. In our solution, we begin by designing a somewhat homomorphic "boostrappable" encryption scheme that works when the function f is the scheme's own decryption function. We then show how, through recursive self-embedding, bootstrappable encryption gives fully homomorphic encryption.