Amortized Locally Decodable Codes

TL;DR

This paper introduces Amortized Locally Decodable Codes, enabling efficient decoding of multiple message symbols with improved locality by amortizing queries, surpassing previous limitations in error correction and cryptographic settings.

Contribution

The paper pioneers the study of amortized LDCs, demonstrating that amortization can achieve low locality and high efficiency, including for Hadamard codes and cryptographic scenarios.

Findings

Hadamard code achieves amortized locality below 2

Constant rate and error tolerance with constant amortized locality in cryptographic settings

Overcomes prior barriers and impossibility results in LDCs

Abstract

Locally Decodable Codes (LDCs) are error correcting codes that admit efficient decoding of individual message symbols without decoding the entire message. Unfortunately, known LDC constructions offer a sub-optimal trade-off between rate, error tolerance and locality, the number of queries that the decoder must make to the received codeword $\tilde {y}$ to recovered a particular symbol from the original message $x$, even in relaxed settings where the encoder/decoder share randomness or where the channel is resource bounded. We initiate the study of Amortized Locally Decodable Codes where the local decoder wants to recover multiple symbols of the original message $x$ and the total number of queries to the received codeword $y$ can be amortized by the total number of message symbols recovered. We demonstrate that amortization allows us to overcome prior barriers and impossibility results. We first demonstrate that the Hadamard code achieves amortized locality below $2$ -- a result that is known to be impossible without amortization. Second, we study amortized locally decodable codes in cryptographic settings where the sender and receiver share a secret key or where the channel is resource-bounded and where the decoder wants to recover a consecutive subset of message symbols $[L,R]$. In these settings we show that it is possible to achieve a trifecta: constant rate, error tolerance and constant amortized locality.