Streaming computation of combinatorial objects

Bar-Yossef, Ziv; Reingold, Omer; Shaltiel, Ronen; Trevisan, Luca

doi:10.1109/ccc.2002.1004352

Cited by 16 publications

(18 citation statements)

References 47 publications

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“…Thus, in addition to the usual goals of minimizing d and maximizing m, we also wish to minimize t. Bar-Yossef, Reingold, Shaltiel, and Trevisan [12] studied a related notion of on-line extractors, which are required to be computable in small space in one pass. They show that space approximately m is necessary and sufficient to evaluate extractors with output length m. Since the smallspace requirement is weaker than being locally computable, their lower bound also applies here.…”

Section: Definition 4 ([9]) Ext : {0 1}mentioning

confidence: 99%

On Constructing Locally Computable Extractors and Cryptosystems in the Bounded Storage Model

Vadhan

2003

Advances in Cryptology - CRYPTO 2003

133

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Abstract. We consider the problem of constructing randomness extractors that are locally computable; that is, read only a small number of bits from their input. As recently shown by Lu (CRYPTO '02 ), locally computable extractors directly yield secure private-key cryptosystems in Maurer's bounded storage model (J. Cryptology, 1992). We suggest a general "sample-then-extract" approach to constructing locally computable extractors. Plugging in known sampler and extractor constructions, we obtain locally computable extractors, and hence cryptosystems in the bounded storage model, whose parameters improve upon previous constructions and come quite close to the lower bounds. The correctness of this approach follows from a fundamental lemma of Nisan and Zuckerman (J. Computer and System Sciences, 1996), which states that sampling bits from a weak random source roughly preserves the min-entropy rate. We also present a refinement of this lemma, showing that the min-entropy rate is preserved up to an arbitrarily small additive loss, whereas the original lemma loses a logarithmic factor.

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Section: Definition 4 ([9]) Ext : {0 1}mentioning

confidence: 99%

On Constructing Locally Computable Extractors and Cryptosystems in the Bounded Storage Model

Vadhan

2003

Advances in Cryptology - CRYPTO 2003

133

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“…Our SXE (Sample-extract-encrypt) MLE scheme is inspired by locallycomputable extractors [9,36,46] and the sample-then-extract paradigm [39,46]. The idea is to put a random subset of the message bits through an extractor to get a key used to encrypt the rest of the bits, and the only assumption made is a standard, ROR-secure symmetric encryption scheme.…”

Section: Theoretical Contributionsmentioning

confidence: 99%

“…In the first category we analyze existing schemes and new variants, breaking some and justifying others with proofs in the random-oracle-model (ROM) [16]. In the second category we address the challenging question of finding a standard-model MLE scheme, making connections with deterministic public-key encryption [11], correlated-input-secure hash functions [29] and locally-computable extractors [9,36,46] to provide schemes exhibiting different trade-offs between assumptions made and the message distributions for which security is proven. From our treatment MLE emerges as a primitive that combines practical impact with theoretical depth and challenges, making it well worthy of further study and a place in the cryptographic pantheon.…”

Section: Introductionmentioning

confidence: 99%

Message-Locked Encryption and Secure Deduplication

Bellare

Keelveedhi

Ristenpart

2013

Lecture Notes in Computer Science

419

310

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We formalize a new cryptographic primitive, Message-Locked Encryption (MLE), where the key under which encryption and decryption are performed is itself derived from the message. MLE provides a way to achieve secure deduplication (space-efficient secure outsourced storage), a goal currently targeted by numerous cloud-storage providers. We provide definitions both for privacy and for a form of integrity that we call tag consistency. Based on this foundation, we make both practical and theoretical contributions. On the practical side, we provide ROM security analyses of a natural family of MLE schemes that includes deployed schemes. On the theoretical side the challenge is standard model solutions, and we make connections with deterministic encryption, hash functions secure on correlated inputs and the sample-then-extract paradigm to deliver schemes under different assumptions and for different classes of message sources. Our work shows that MLE is a primitive of both practical and theoretical interest.

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“…These are generally worst-case bounds, such as the work by [1]. Related to our applications is the work of [4] (see also [3]) on lower bounds for combinatorial objects such as hash functions and linear codes. We note that our model is different in that we give tight bounds for each single matrix A.…”

Section: Definition 12 (Streaming Rank Withmentioning

confidence: 99%

Adaptive Matrix Vector Product

Vempala¹,

Woodruff²

2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the following streaming problem: given a hardwired m × n matrix A together with a poly(mn)-bit hardwired string of advice that may depend on A, for any x with coordinates x1, . . . xn presented in order, output the coordinates of A · x in order. Our focus is on using as little memory as possible while computing A · x; we do not count the size of the output tape on which the coordinates of A · x are written; for some matrices A such as the identity matrix, a constant number of words of space is achievable. Such an algorithm has to adapt its memory contents to the changing structure of A and exploit it on the fly. We give a nearly tight characterization, for any number of passes over the coordinates of x, of the space complexity of such a streaming algorithm. Our characterization is constructive, in that we provide an efficient algorithm matching our lower bound on the space complexity. The essential parameters, streaming rank and multi-pass streaming rank of A, might be of independent interest, and we show they can be computed efficiently. We give several applications of our results to computing Johnson-Lindenstrauss transforms. Finally, we note that we can characterize the optimal space complexity when the coordinates of A · x can be written in any order.

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Streaming computation of combinatorial objects

Cited by 16 publications

References 47 publications

On Constructing Locally Computable Extractors and Cryptosystems in the Bounded Storage Model

On Constructing Locally Computable Extractors and Cryptosystems in the Bounded Storage Model

Message-Locked Encryption and Secure Deduplication

Adaptive Matrix Vector Product

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