With the development of sensor network, mobile computing, and web applications, data are now collected from many distributed sources to form big datasets. Such datasets can be hosted in the cloud to achieve economical processing. However, these data might be highly sensitive requiring secure storage and processing. We envision a cloud-based data storage and processing framework that enables users to economically and securely share and handle big datasets. Under this framework, we study the matrix-based data mining algorithms with a focus on the secure top-k eigenvector algorithm. Our approach uses an iterative processing model in which the authorized user interacts with the cloud to achieve the result. In this process, both the source matrix and the intermediate results keep confidential and the client-side incurs low costs. The security of this approach is guaranteed by using Paillier Encryption and a random perturbation technique. We carefully analyze its security under a cloud-specific threat model. Our experimental results show that the proposed method is scalable to big matrices while requiring low client-side costs.
A framework for securely outsourcing general computations is given in @cite_16 . However, this framework is based on Gentry's fully homomorphic encryption scheme @cite_9 rendering it impractical due to the high computational costs and ciphertext sizes. A simple test with the Scarab FHE library (hcrypt.com scarab-library) yielded ciphertext sizes more than ten times those generated using Paillier @cite_14 . Very recent work by @cite_12 propose a secure outsourcing solution for problems which only require the encryption scheme to be somewhat'' homomorphic (SHE). They use the SHE scheme of @cite_10 which provides reasonably efficient computational performance but still suffers from large ciphertexts.
@cite_14 @cite_9 @cite_16 @cite_10 @cite_12
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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 describe a working implementation of a variant of Gentry's fully homomorphic encryption scheme (STOC 2009), similar to the variant used in an earlier implementation effort by Smart and Vercauteren (PKC 2010). Smart and Vercauteren implemented the underlying "somewhat homomorphic" scheme, but were not able to implement the bootstrapping functionality that is needed to get the complete scheme to work. We show a number of optimizations that allow us to implement all aspects of the scheme, including the bootstrapping functionality. Our main optimization is a key-generation method for the underlying somewhat homomorphic encryption, that does not require full polynomial inversion. This reduces the asymptotic complexity from O(n2.5) to O(n1.5) when working with dimension-n lattices (and practically reducing the time from many hours days to a few seconds minutes). Other optimizations include a batching technique for encryption, a careful analysis of the degree of the decryption polynomial, and some space time trade-offs for the fully-homomorphic scheme. We tested our implementation with lattices of several dimensions, corresponding to several security levels. From a "toy" setting in dimension 512, to "small," "medium," and "large" settings in dimensions 2048, 8192, and 32768, respectively. The public-key size ranges in size from 70 Megabytes for the "small" setting to 2.3 Gigabytes for the "large" setting. The time to run one bootstrapping operation (on a 1-CPU 64- bit machine with large memory) ranges from 30 seconds for the "small" setting to 30 minutes for the "large" setting. In 2009, Gentry proposed the first Fully Homomorphic Encryption (FHE) scheme, an extremely powerful cryptographic primitive that enables to perform computations, i.e., to evaluate circuits, on encrypted data without decrypting them first. This has many applications, in particular in cloud computing. In all currently known FHE schemes, encryptions are associated to some (non-negative integer) noise level, and at each evaluation of an AND gate, the noise level increases. This is problematic because decryption can only work if the noise level stays below some maximum level @math at every gate of the circuit. To ensure that property, it is possible to perform an operation called to reduce the noise level. However, bootstrapping is time-consuming and has been identified as a critical operation. This motivates a new problem in discrete optimization, that of choosing where in the circuit to perform bootstrapping operations so as to control the noise level; the goal is to minimize the number of bootstrappings in circuits. In this paper, we formally define the , we design a polynomial-time @math -approximation algorithm using a novel method of rounding of a linear program, and we show a matching hardness result: @math -inapproximability for any @math . We describe two improvements to Gentry’s fully homomorphic scheme based on ideal lattices and its analysis: we provide a more aggressive analysis of one of the hardness assumptions (the one related to the Sparse Subset Sum Problem) and we introduce a probabilistic decryption algorithm that can be implemented with an algebraic circuit of low multiplicative degree. Combined together, these improvements lead to a faster fully homomorphic scheme, with a O(λ 3.5) bit complexity per elementary binary add mult gate, where λ is the security parameter. These improvements also apply to the fully homomorphic schemes of Smart and Vercauteren [PKC’2010] and van [Eurocrypt’2010]. We present complexity results for the verification of security protocols. Since the perfect cryptography assumption is unrealistic for cryptographic primitives with visible algebraic properties, we extend the classical Dolev-Yao model by permitting the intruder to exploit these properties. More precisely, we are interested in theories such as Exclusive or and Abelian groups in combination with the homomorphism axiom. We show that the intruder deduction problem is in PTIME in both cases, improving the EXPTIME complexity results of Lafourcade, Lugiez and Treinen. iez, R. Treinen, Intruder deduction for AC-like equational theories with homomorphisms, in: Proc. 16th Internat. Conf. Rewriting Techniques and Applications (RTA'05), Nara, Japan, Lecture Notes in Comput. Sci., vol. 3467, Springer, Berlin, 2005, pp. 308-322]. We propose a method to compute the exact minimal number of bootstrappings required to homomorphically evaluate any circuit. Given a circuit (typically over ( F _2 ) although our method readily extends to circuits over any ring), the maximal noise level supported by the considered fully homomorphic encryption (FHE) scheme and the desired noise level of circuit inputs and outputs, our algorithms return a minimal subset of circuit variables such that boostrapping these variables is enough to perform an evaluation of the whole circuit. We introduce a specific algorithm for 2-level encryption (first generation of FHE schemes) and an extended algorithm for l max -level encryption with arbitrary l max ≥ 2 to cope with more recent FHE schemes. We successfully applied our method to a range of real-world circuits that perform various operations over plaintext bits. Practical results show that some of these circuits benefit from significant improvements over the naive evaluation method where all multiplication outputs are bootstrapped. In particular, we report that a circuit for the AES S-box put forward by Boyar and Peralta admits a solution in 17 bootstrappings instead of 32, thereby leading to a 88 faster homomorphic evaluation of AES for any 2-level FHE scheme. 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 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. 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). Several public key cryptosystems with additional homomorphic properties have been proposed so far. They allow to perform computation with encrypted data without the knowledge of any secret information. In many applications, the ability to perform decryption, i.e. the knowledge of the secret key, gives a huge power. A classical way to reduce the trust in such a secret owner, and consequently to increase the security, is to share the secret between many entities in such a way that cooperation between them is necessary to decrypt. In this paper, we propose a distributed version of the Paillier cryptosystem presented at Eurocrypt '99. This shared scheme can for example be used in an electronic voting scheme or in a lottery where a random number related to the winning ticket has to be jointly chosen by all participants. The rapid expansion and increased popularity of cloud computing comes with no shortage of privacy concerns about outsourcing computation to semi-trusted parties. Leveraging the power of encryption, in this paper, we introduce Cryptoleq: an abstract machine based on the concept of one instruction set computer, capable of performing general-purpose computation on encrypted programs. The program operands are protected using the Paillier partially homomorphic cryptosystem, which supports addition on the encrypted domain. Full homomorphism over addition and multiplication, which is necessary for enabling general-purpose computation, is achieved by inventing a heuristically obfuscated software re-encryption module written using Cryptoleq instructions and blended into the executing program. Cryptoleq is heterogeneous, allowing mixing encrypted and unencrypted instruction operands in the same program memory space. Programming with Cryptoleq is facilitated using an enhanced assembly language that allows the development of any advanced algorithm on encrypted data sets. In our evaluation, we compare Cryptoleq's performance against a popular fully homomorphic encryption library, and demonstrate correctness using a typical private information retrieval problem. We construct a simple fully homomorphic encryption scheme, using only elementary modular arithmetic. We use Gentry’s technique to construct a fully homomorphic scheme from a “bootstrappable” somewhat homomorphic scheme. However, instead of using ideal lattices over a polynomial ring, our bootstrappable encryption scheme merely uses addition and multiplication over the integers. The main appeal of our scheme is the conceptual simplicity. We reduce the security of our scheme to finding an approximate integer gcd – i.e., given a list of integers that are near-multiples of a hidden integer, output that hidden integer. We investigate the hardness of this task, building on earlier work of Howgrave-Graham. Fully homomorphic encryption is an encryption method with the property that any computation on the plaintext can be performed by a party having access to the ciphertext only. Here, we formally define and give schemes for quantum homomorphic encryption, which is the encryption of quantum information such that quantum computations can be performed given the ciphertext only. Our schemes allow for arbitrary Clifford group gates, but become inefficient for circuits with large complexity, measured in terms of the non-Clifford portion of the circuit (we use the “ ( 8 )” non-Clifford group gate, also known as the ( T )-gate). We demonstrate that, by using a recently proposed leveled homomorphic encryption scheme, it is possible to delegate the execution of a machine learning algorithm to a computing service while retaining confidentiality of the training and test data. Since the computational complexity of the homomorphic encryption scheme depends primarily on the number of levels of multiplications to be carried out on the encrypted data, we define a new class of machine learning algorithms in which the algorithm's predictions, viewed as functions of the input data, can be expressed as polynomials of bounded degree. We propose confidential algorithms for binary classification based on polynomial approximations to least-squares solutions obtained by a small number of gradient descent steps. We present experimental validation of the confidential machine learning pipeline and discuss the trade-offs regarding computational complexity, prediction accuracy and cryptographic security. Let ψ be a 2-DNF formula on boolean variables x1,...,xn ∈ 0,1 . We present a homomorphic public key encryption scheme that allows the public evaluation of ψ given an encryption of the variables x1,...,xn. In other words, given the encryption of the bits x1,...,xn, anyone can create the encryption of ψ(x1,...,xn). More generally, we can evaluate quadratic multi-variate polynomials on ciphertexts provided the resulting value falls within a small set. We present a number of applications of the system: In a database of size n, the total communication in the basic step of the Kushilevitz-Ostrovsky PIR protocol is reduced from @math to @math . An efficient election system based on homomorphic encryption where voters do not need to include non-interactive zero knowledge proofs that their ballots are valid. The election system is proved secure without random oracles but still efficient. A protocol for universally verifiable computation. We propose a new cryptographic primitive called Delegatable Homomorphic Encryption (DHE). This allows a Trusted Authority to control delegate the evaluation of circuits over encrypted data to untrusted workers evaluators by issuing tokens. This primitive can be both seen as a public-key counterpart to Verifiable Computation, where input generation and output verification are performed by different entities, or as a generalisation of Fully Homomorphic Encryption enabling control over computations on encrypted data. Our primitive comes with a series of extra features: 1) there is a one-time setup procedure for all circuits; 2) senders do not need to be aware of the functions which will be evaluated on the encrypted data, nor do they need to register keys; 3) tokens are independent of senders and receiver; and 4) receivers are able to verify the correctness of computation given short auxiliary information on the input data and the function, independently of the complexity of the computed circuit. We give a modular construction of such a DHE scheme from three components: Fully Homomorphic Encryption (FHE), Functional Encryption (FE), and a (customised) MAC. As a stepping stone, we first define Verifiable Functional Encryption (VFE), and then show how one can build a secure DHE scheme from a VFE and an FHE scheme. We also show how to build the required VFE from a standard FE together with a MAC scheme. All our results hold in the standard model. Finally, we show how one can build a verifiable computation (VC) scheme generically from a DHE. As a corollary, we get the first VC scheme which remains verifiable even if the attacker can observe verification results. We propose a general multiparty computation protocol secure against an active adversary corrupting up to @math of the n players. The protocol may be used to compute securely arithmetic circuits over any finite field @math . Our protocol consists of a preprocessing phase that is both independent of the function to be computed and of the inputs, and a much more efficient online phase where the actual computation takes place. The online phase is unconditionally secure and has total computational and communication complexity linear in n, the number of players, where earlier work was quadratic in n. Moreover, the work done by each player is only a small constant factor larger than what one would need to compute the circuit in the clear. We show this is optimal for computation in large fields. In practice, for 3 players, a secure 64-bit multiplication can be done in 0.05 ms. Our preprocessing is based on a somewhat homomorphic cryptosystem. We extend a scheme by , so that we can perform distributed decryption and handle many values in parallel in one ciphertext. The computational complexity of our preprocessing phase is dominated by the public-key operations, we need @math operations per secure multiplication where s is a parameter that increases with the security parameter of the cryptosystem. Earlier work in this model needed @math operations. In practice, the preprocessing prepares a secure 64-bit multiplication for 3 players in about 13 ms. This paper proposes a novel public-key cryptosystem, which is practical, provably secure and has some other interesting properties as follows: 1. Its trapdoor technique is essentially different from any other previous schemes including RSA-Rabin and Diffie-Hellman. 2. It is a probabilistic encryption scheme. 3. It can be proven to be as secure as the intractability of factoring n = p2q (in the sense of the security of the whole plaintext) against passive adversaries. 4. It is semantically secure under the p-subgroup assumption, which is comparable to the quadratic residue and higher degree residue assumptions. 5. Under the most practical environment, the encryption and decryption speeds of our scheme are comparable to (around twice slower than) those of elliptic curve cryptosystems. 6. It has a homomorphic property: E(m0, r0)E(m1, r1) mod n = E(@#@ m0 + m1, r2), where E(m, r) means a ciphertext of plaintext m as randomized by r and m0+ m1 < p. 7. Anyone can change a ciphertext, C = E(m, r), into another ciphertext, C′ = Chr' mod n, while preserving plaintext of C (i.e., C′ = E(m,r″)), and the relationship between C and C′ can be concealed. We give improved protocols for the secure and private outsourcing of linear algebra computations, that enable a client to securely outsource expensive algebraic computations (like the multiplication of large matrices) to a remote server, such that the server learns nothing about the customer's private input or the result of the computation, and any attempted corruption of the answer by the server is detected with high probability. The computational work performed at the client is linear in the size of its input and does not require the client to locally carry out any expensive encryptions of such input. The computational burden on the server is proportional to the time complexity of the current practically used algorithms for solving the algebraic problem (e.g., proportional to n3 for multiplying two n x n matrices). The improvements we give are: (i) whereas the previous work required more than one remote server and assumed they do not collude, our solution works with a single server (but readily accommodates many, for improved performance); (ii) whereas the previous work required a server to carry out expensive cryptographic computations (e.g., homomorphic encryptions), our solution does not make use of any such expensive cryptographic primitives; and (iii) whereas in previous work collusion by the servers against the client revealed to them the client's inputs, our scheme is resistant to such collusion. As in previous work, we maintain the property that the scheme enables the client to detect any attempt by the server(s) at corruption of the answer, even when the attempt is collusive and coordinated among the servers. We propose a privacy-preserving framework for learning visual classifiers by leveraging distributed private image data. This framework is designed to aggregate multiple classifiers updated locally using private data and to ensure that no private information about the data is exposed during and after its learning procedure. We utilize a homomorphic cryptosystem that can aggregate the local classifiers while they are encrypted and thus kept secret. To overcome the high computational cost of homomorphic encryption of high-dimensional classifiers, we (1) impose sparsity constraints on local classifier updates and (2) propose a novel efficient encryption scheme named doubly-permuted homomorphic encryption (DPHE) which is tailored to sparse high-dimensional data. DPHE (i) decomposes sparse data into its constituent non-zero values and their corresponding support indices, (ii) applies homomorphic encryption only to the non-zero values, and (iii) employs double permutations on the support indices to make them secret. Our experimental evaluation on several public datasets shows that the proposed approach achieves comparable performance against state-of-the-art visual recognition methods while preserving privacy and significantly outperforms other privacy-preserving methods.