Near-optimal extractors against quantum storage

TL;DR

This paper proves the security of Trevisan's extractor against quantum storage adversaries, achieving near-optimal output lengths and introducing quantum functional access codes to improve cryptographic extraction methods.

Contribution

It demonstrates the first secure extractors against quantum storage with near-optimal output length and introduces quantum functional access codes for improved security analysis.

Findings

Achieved extractor output length (K-b) with poly-log seed length.

Constructed extractors with logarithmic key length and near-optimal output.

Developed quantum functional access codes to analyze security against quantum adversaries.

Abstract

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 $\Theta(K-b)$, 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 $\Theta((K-b)/K^\gamma)$ for any $\gamma>0$. 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.