Amortized Locally Decodable Codes

2025-02-14

AI Summary

This work addresses the high query overhead of locally decodable codes (LDCs) when batch-recovering multiple message symbols. We introduce *amortized LDCs*, a new paradigm that amortizes query complexity over the number of decoded symbols. Our construction builds upon the Hadamard code, leveraging shared randomness and cryptographic security modeling to enable efficient batch decoding over resource-constrained channels. Key contributions are: (1) the first proof that the Hadamard code achieves an amortized query complexity strictly less than 2—breaking the long-standing barrier of ≥2 queries per symbol in standard LDCs; (2) achieving simultaneous optimality in rate, error tolerance, and amortized query complexity—yielding the first construction satisfying all three criteria concurrently, with direct applicability to secret sharing and related primitives; and (3) establishing a formal theoretical framework for amortized locality, thereby extending the applicability of LDCs to low-latency, high-throughput settings.

Technology Category

• Search and Optimization: Learning to Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Machine Learning: Quantum Machine Learning

Application Category

• Security and Privacy: Applications of cryptography
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs

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 $ ilde {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 $ ilde{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.

Problem

• Improve trade-offs in Locally Decodable Codes
• Enable efficient multi-symbol recovery using amortization
• Achieve constant rate, error tolerance, and locality

Innovation

• Amortized Locally Decodable Codes
• Overcomes prior impossibility results
• Achieves constant rate and locality