Biblio

We consider the secure search problem of retrieving from an unsorted data cost=(x\_1,...,xm) an item (i,xi) matching a given lookup value l (for a generic matching criterion either hardcoded or given as part of the query), where both input and output are encrypted by a Fully Homomorphic Encryption (FHE). The secure search problem is central in applications of secure outsourcing to an untrusted party ("the cloud"). Prior secure search algorithms on FHE encrypted data are realized by polynomials of degree Ømega(m), evaluated in Ømega(log m) sequential homomorphic multiplication steps (ie., multiplicative depth) even using an unbounded number of parallel processors. This is too slow with current FHE implementations, especially as the size of the array grows. We present the first secure search algorithm that is realized by a polynomial of logarithmic degree, log3 m, evaluated in O(log log m) sequential homomorphic multiplication steps (ie., multiplicative depth) using m parallel processors. We implemented our algorithm in an open source library based on HElib and ran experiments on Amazon's EC2 cloud with up to 100 processors. Our experiments show that we can securely search in m= millions of entries in less than an hour on a standard EC2 64-cores machine. We achieve our result by: (1) Employing modern data summarization techniques known as sketching for returning as output (the encryption of) a short sketch C from which the matching item (i,xi) can be decoded in time polynomial in log m. (2) Designing for this purpose a novel sketch that returns the first strictly-positive entry in a (not necessarily sparse) array of non-negative integers; this sketch may be of independent interest. (3) Suggesting a multi-ring evaluation of FHE for degree reduction from linear to logarithmic.

We introduce a new sub-linear space sketch—the Weight-Median Sketch—for learning compressed linear classifiers over data streams while supporting the efficient recovery of large-magnitude weights in the model. This enables memory-limited execution of several statistical analyses over streams, including online feature selection, streaming data explanation, relative deltoid detection, and streaming estimation of pointwise mutual information. Unlike related sketches that capture the most frequently-occurring features (or items) in a data stream, the Weight-Median Sketch captures the features that are most discriminative of one stream (or class) compared to another. The Weight-Median Sketch adopts the core data structure used in the Count-Sketch, but, instead of sketching counts, it captures sketched gradient updates to the model parameters. We provide a theoretical analysis that establishes recovery guarantees for batch and online learning, and demonstrate empirical improvements in memory-accuracy trade-offs over alternative memory-budgeted methods, including count-based sketches and feature hashing.

Obtaining frequency information of data streams, in limited space, is a well-recognized problem in literature. A number of recent practical applications (such as those in computational advertising) require temporally-aware solutions: obtaining historical count statistics for both time-points as well as time-ranges. In these scenarios, accuracy of estimates is typically more important for recent instances than for older ones; we call this desirable property Time Adaptiveness. With this observation, [20] introduced the Hokusai technique based on count-min sketches for estimating the frequency of any given item at any given time. The proposed approach is problematic in practice, as its memory requirements grow linearly with time, and it produces discontinuities in the estimation accuracy. In this work, we describe a new method, Time-adaptive Sketches, (Ada-sketch), that overcomes these limitations, while extending and providing a strict generalization of several popular sketching algorithms. The core idea of our method is inspired by the well-known digital Dolby noise reduction procedure that dates back to the 1960s. The theoretical analysis presented could be of independent interest in itself, as it provides clear results for the time-adaptive nature of the errors. An experimental evaluation on real streaming datasets demonstrates the superiority of the described method over Hokusai in estimating point and range queries over time. The method is simple to implement and offers a variety of design choices for future extensions. The simplicity of the procedure and the method's generalization of classic sketching techniques give hope for wide applicability of Ada-sketches in practice.