Numerical linear algebra plays an important role in computer science. In this paper, we initiate the study of performing linear algebraic tasks while preserving privacy when the data is streamed online. Our main focus is the space requirement of the privacy-preserving data-structures. We give the first sketch-based algorithm for differential privacy. We give optimal, up to logarithmic factor, space datastructures that can compute low rank approximation, linear regression, and matrix multiplication, while preserving differential privacy with better additive error bounds compared to the known results. Notably, we match the best known space bound in the non-private setting by Kane and Nelson (J. ACM, 61(1):4).Our mechanism for differentially private low-rank approximation reuses the random Gaussian matrix in a specific way to provide a single-pass mechanism. We prove that the resulting distribution also preserve differential privacy. This can be of independent interest. We do not make any assumptions, like singular value separation or normalized row assumption, as made in the earlier works. The mechanisms for matrix multiplication and linear regression can be seen as the private analogues of the known nonprivate algorithms. All our mechanisms, in the form presented, can also be computed in the distributed setting.
In this paper, we analyze the complexity of the construction of the 2 k -diamond structure proposed by Kelsey and Kohno [9]. We point out a flaw in their analysis and show that their construction may not produce the desired diamond structure. We then give a more rigorous and detailed complexity analysis of the construction of a diamond structure. For this, we appeal to random graph theory, which allows us to determine sharp necessary and sufficient conditions for the message complexity (i.e., the number of hash computations required to build the required structure). We also analyze the computational complexity for constructing a diamond structure, which has not been previously studied in the literature. Finally, we study the impact of our analysis on herding and other attacks that use the diamond structure as a subroutine. Precisely, our results shows the following:1. The message complexity for the construction of a diamond structure is √ k times more than what was previously stated in literature. 2. The time complexity is n times the message complexity, where n is the size of hash value.Due to above two results, the complexity of the herding attack [9] and the second preimage attack [3] on iterated hash functions have increased complexity. We also show that the message complexity of herding and second preimage attacks on "hash twice" is n times the complexity claimed by [2], by giving a more detailed analysis of the attack.
There has been considerable recent interest in "cloud storage" wherein a user asks a server to store a large file. One issue is whether the user can verify that the server is actually storing the file, and typically a challenge-response protocol is employed to convince the user that the file is indeed being stored correctly. The security of these schemes is phrased in terms of an extractor which will recover or retrieve the file given any "proving algorithm" that has a sufficiently high success probability.This paper treats proof-of-retrievability schemes in the model of unconditional security, where an adversary has unlimited computational power. In this case retrievability of the file can be modelled as error-correction in a certain code. We provide a general analytical framework for such schemes that yields exact (non-asymptotic) reductions that precisely quantify conditions for extraction to succeed as a function of the success probability of a proving algorithm, and we apply this analysis to several archetypal schemes. In addition, we provide a new methodology for the analysis of keyed POR schemes in an unconditionally secure setting, and use it to prove the security of a modified version of a scheme due to Shacham and Waters under a slightly restricted attack model, thus providing the first example of a keyed POR scheme with unconditional security. We also show how classical statistical techniques can be used to evaluate whether the responses of the prover are accurate enough to permit successful extraction. Finally, we prove a new lower bound on storage and communication complexity of POR schemes.
This paper initiates the study of preserving differential privacy (DP) when the data-set is sparse. We study the problem of constructing efficient sanitizer that preserves DP and guarantees high utility for answering cut-queries on graphs. The main motivation for studying sparse graphs arises from the empirical evidences that social networking sites are sparse graphs. We also motivate and advocate the necessity to include the efficiency of sanitizers, in addition to the utility guarantee, if one wishes to have a practical deployment of privacy preserving sanitizers. We show that the technique of Blocki et al. [3] (BBDS) can be adapted to preserve DP for answering cut-queries on sparse graphs, with an asymptotically efficient sanitizer than BBDS. We use this as the base technique to construct an efficient sanitizer for arbitrary graphs. In particular, we use a preconditioning step that preserves the spectral properties (and therefore, size of any cut is preserved), and then apply our basic sanitizer. We first prove that our sanitizer preserves DP for graphs with high conductance. We then carefully compose our basic technique with the modified sanitizer to prove the result for arbitrary graphs. In certain sense, our approach is complementary to the Randomized sanitization for answering cut queries [17]: we use graph sparsification, while Randomized sanitization uses graph densification. Our sanitizers almost achieves the best of both the worlds with the same privacy guarantee, i.e., it is almost as efficient as the most efficient sanitizer and it has utility guarantee almost as strong as the utility guarantee of the best sanitization algorithm. We also make some progress in answering few open problems by BBDS. We make a combinatorial observation that allows us to argue that the sanitized graph can also answer (S, T)-cut queries with same asymptotic efficiency, utility, and DP guarantee as our sanitization algorithm for S,S-cuts. Moreover, we achieve a better utility guarantee than Gupta, Roth, and Ullman [17]. We give further optimization by showing that fast Johnson-Lindenstrauss transform of Ailon and Chazelle [2] also preserves DP.
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