arXiv subjects
Publications and source records attributed to Anindya De.
This is a paper about private data analysis, in which a trusted curator holding a confidential database responds to real vector-valued queries. A common approach to ensuring privacy for the database elements is to add appropriately generated random noise to the answers, releasing only these {\em noisy} responses. In this paper, we investigate various lower bounds on the noise required to maintain different kind of privacy guarantees.
We show that Trevisan's extractor and its variants \cite{T99,RRV99} are secure against bounded quantum storage adversaries. One instantiation gives the first such extractor to achieve an output length $Θ(K-b)$, where $K$ is the source's entropy and $b$ the adversary's storage, together with a poly-logarithmic seed length. Another instantiation achieves a logarithmic key length, with a slightly smaller output length $Θ((K-b)/K^γ)$ for any $γ>0$. In contrast, the previous best construction \cite{TS09} could only extract $(K/b)^{1/15}$ bits. Some of our constructions have the additional advantage that every bit of the output is a function of only a polylogarithmic number of bits from the source, which is crucial for some cryptographic applications. Our argument is based on bounds for a generalization of quantum random access codes, which we call \emph{quantum functional access codes}. This is crucial as it lets us avoid the local list-decoding algorithm central to the approach in \cite{TS09}, which was the source of the multiplicative overhead.
We give an $O(N\cdot \log N\cdot 2^{O(\log^*N)})$ algorithm for multiplying two $N$-bit integers that improves the $O(N\cdot \log N\cdot \log\log N)$ algorithm by Schönhage-Strassen. Both these algorithms use modular arithmetic. Recently, Fürer gave an $O(N\cdot \log N\cdot 2^{O(\log^*N)})$ algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from Fürer's algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a $p$-adic version of Fürer's algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar.