On Locally Decodable Codes in Resource Bounded Channels

TL;DR

This paper introduces a method to construct explicit locally decodable codes with constant rate and polylogarithmic locality in resource-constrained channels, overcoming previous limitations of low rate or high locality.

Contribution

It demonstrates how to bootstrap private key LDC constructions to weaker resource-bounded channels using cryptographic assumptions and random oracles.

Findings

Constructs explicit constant rate LDCs with polylogarithmic locality.

Shows how to transmit secret keys over resource-bounded channels.

Answers an open question on applying bootstrapping techniques to weaker channel models.

Abstract

Constructions of locally decodable codes (LDCs) have one of two undesirable properties: low rate or high locality (polynomial in the length of the message). In settings where the encoder/decoder have already exchanged cryptographic keys and the channel is a probabilistic polynomial time (PPT) algorithm, it is possible to circumvent these barriers and design LDCs with constant rate and small locality. However, the assumption that the encoder/decoder have exchanged cryptographic keys is often prohibitive. We thus consider the problem of designing explicit and efficient LDCs in settings where the channel is slightly more constrained than the encoder/decoder with respect to some resource e.g., space or (sequential) time. Given an explicit function $f$ that the channel cannot compute, we show how the encoder can transmit a random secret key to the local decoder using $f(\cdot)$ and a random oracle $H(\cdot)$. This allows bootstrap from the private key LDC construction of Ostrovsky, Pandey and Sahai (ICALP, 2007), thereby answering an open question posed by Guruswami and Smith (FOCS 2010) of whether such bootstrapping techniques may apply to LDCs in weaker channel models than just PPT algorithms. Specifically, in the random oracle model we show how to construct explicit constant rate LDCs with locality of polylog in the security parameter against various resource constrained channels.