Abhishek Bhowmick

Noiseless database privacy

Differential Privacy (DP) has emerged as a formal, flexible framework for privacy protection, with a guarantee that is agnostic to auxiliary information and that admits simple rules for composition. Benefits notwithstanding, a major drawback of DP is that it provides noisy responses to queries, making it unsuitable for many applications. We propose a new notion called Noiseless Privacy that provides exact answers to queries, without adding any noise whatsoever. While the form of our guarantee is similar to DP, where the privacy comes from is very different, based on statistical assumptions on the data and on restrictions to the auxiliary information available to the adversary. We present a first set of results for Noiseless Privacy of arbitrary Boolean-function queries and of linear Real-function queries, when data are drawn independently, from nearly-uniform and Gaussian distributions respectively. We also derive simple rules for composition under models of dynamically changing data.

Update efficient codes for distributed storage

This paper determines mechanisms for distributed storage that are simultaneously repair and update efficient. Repair efficiency demands that minimum information be downloaded from surviving nodes to reconstruct failed storage nodes. Update efficiency desires that changes in the original data require minimal updates at the storage nodes. These two requirements can be seen as counteracting one another, as the latter imposes a sparsity constraint on the encoding process that is not desirable for the former. In this paper we establish the existence of the codes that meet both requirements: require only logarithmic updates when data changes, while simultaneously minimizing repair bandwidth for exact reconstruction. To show this, we use a combination of KG codes for update efficiency with interference-alignment strategies for distributed storage.

New bounds for matching vector families

A Matching Vector (MV) family modulo m is a pair of ordered lists U=(u₁,...,u_t) and V=(v₁,...,v_t) where u_i,v_j ∈ Z_mⁿ with the following inner product pattern: for any i, {u_i,v_i}=0, and for any i ≠ j, {u_i,v_j} ≠ 0. A MV family is called q-restricted if inner products {u_i,v_j} take at most q different values. Our interest in MV families stems from their recent application in the construction of sub-exponential locally decodable codes (LDCs). There, q-restricted MV families are used to construct LDCs with q queries, and there is special interest in the regime where q is constant. When m is a prime it is known that such constructions yield codes with exponential block length. However, for composite m the behaviour is dramatically different. A recent work by Efremenko [8] (based on an approach initiated by Yekhanin [24]) gives the first sub-exponential LDC with constant queries. It is based on a construction of a MV family of super-polynomial size by Grolmusz [10] modulo composite m. In this work, we prove two lower bounds on the block length of LDCs which are based on black box construction using MV families. When q is constant (or sufficiently small), we prove that such LDCs must have a quadratic block length. When the modulus m is constant (as it is in the construction of Efremenko [8]) we prove a super-polynomial lower bound on the block-length of the LDCs, assuming a well-known conjecture in additive combinatorics, the polynomial Freiman-Ruzsa conjecture over Z_m.

Nonclassical polynomials as a barrier to polynomial lower bounds

The problem of constructing explicit functions which cannot be approximated by low degree polynomials has been extensively studied in computational complexity, motivated by applications in circuit lower bounds, pseudo-randomness, constructions of Ramsey graphs and locally decodable codes. Still, most of the known lower bounds become trivial for polynomials of super-logarithmic degree. Here, we suggest a new barrier explaining this phenomenon. We show that many of the existing lower bound proof techniques extend to nonclassical polynomials, an extension of classical polynomials which arose in higher order Fourier analysis. Moreover, these techniques are tight for nonclassical polynomials of logarithmic degree.

On primitive elements in finite fields of low characteristic

We discuss the problem of constructing a small subset of a finite field containing primitive elements of the field. Given a finite field,

New Bounds for Matching Vector Families

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A note on character sums in finite fields

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Deterministic Extractors for Additive Sources: Extended Abstract

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The List Decoding Radius of Reed-Muller Codes over Small Fields

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Bias vs structure of polynomials in large fields, and applications in effective algebraic geometry and coding theory

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