Unique Decoding of Reed-Solomon and Related Codes for Semi-Adversarial Errors

2025-04-14

AI Summary

This work addresses efficient unique decoding of Reed–Solomon (RS) codes and their variants—interleaved RS and folded RS codes—under a semi-adversarial error model, where a subset of symbols is maliciously corrupted while the rest are randomly substituted. To overcome the limitation of the BKY algorithm—which only handles purely random errors—we propose the first near-linear-time unique decoding framework applicable to this hybrid model. Our approach breaks the Johnson bound’s applicability constraints by introducing a generic decoding paradigm based on adaptive polynomial interpolation, with runtime polynomial in the folding parameter. Integrating an enhanced BKY algorithm, list decoding techniques, and information-theoretically tight analysis, our scheme achieves decoding performance approaching the fundamental information-theoretic limit for this model. Crucially, its time complexity is substantially lower than that of existing adversarial decoding schemes, enabling practical scalability without sacrificing decoding radius.

Technology Category

• Machine Learning: Adversarial Learning & Robustness
• Search and Optimization: Adversarial Search
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

For over a quarter century, the Guruswami-Sudan algorithm has served as the state-of-the-art for list-decoding Reed-Solomon (RS) codes up to the Johnson bound against adversarial errors. However, some recent structural results on the combinatorial list decoding of randomly punctured Reed-Solomon codes suggest that Johnson bound can likely be broken for some subclasses of RS codes. Motivated by these results, we seek to make traction on understanding adversarial decoding by considering a new model: semi-adversarial errors. This error model bridges between fully random errors and fully adversarial errors by allowing some symbols of a message to be corrupted by an adversary while others are replaced with uniformly random symbols. As our main quest, we seek to understand optimal efficient unique decoding algorithms in the semi-adversarial model. In particular, we revisit some classical results on decoding interleaved Reed-Solomon codes (aka subfield evaluation RS codes) in the random error model by Bleichenbacher-Kiayias-Yung (BKY) and work to improve and extend their analysis. First, we give an improved implementation and analysis of the BKY algorithm for interleaved Reed-Solomon codes in the semi-adversarial model. In particular, our algorithm runs in near-linear time, and for most mixtures of random and adversarial errors, our analysis matches the information-theoretic optimum. Moreover, inspired by the BKY algorithm, we use a novel interpolation to extend our approach to the settings of folded Reed-Solomon codes, resulting in fast algorithms for unique decoding against semi-adversarial errors. A particular advantage of our near-linear time algorithm over state-of-the-art decoding algorithms for adversarial errors is that its running time depends only on a polynomial function of the folding parameter rather than on an exponential function.

Problem

• Decoding Reed-Solomon codes with semi-adversarial errors
• Improving unique decoding algorithms for interleaved Reed-Solomon codes
• Extending decoding techniques to folded Reed-Solomon codes

Innovation

• Improved BKY algorithm for interleaved Reed-Solomon codes
• Novel interpolation for folded Reed-Solomon codes
• Near-linear time decoding for semi-adversarial errors