Near-Codewords Aware Bit Flipping Decoding of QC-MDPC Codes

2026-04-20

AI Summary

This work addresses the vulnerability of bit-flipping decoders for QC-MDPC codes to near-codeword trapping sets, which can cause decoding failures even at extremely low error rates. For the first time, the authors endow generic bit-flipping decoders with awareness of near-codewords by leveraging their structural characteristics, thereby enhancing several decoding algorithms—including BF-Max—without introducing significant computational overhead. Large-scale simulations under NIST security level 1 parameters and simplified BIKE settings demonstrate that the improved BF-Max decoder achieves a substantially reduced decoding failure rate, outperforming both original BIKE decoders in reliability.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Stochastic Optimization
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Bit-Flipping (BF) decoders are a family of decoders widely employed in post-quantum cryptographic schemes based on Quasi-Cyclic Moderate-Density Parity-Check (QC-MDPC) codes, such as BIKE. BF decoders suffer from trapping sets, corresponding to low-weight error patterns that likely lead to decoding failures. For QC-MDPC codes, the most relevant family of trapping sets is that of near-codewords, which are error patterns associated to low-weight syndromes. Indeed, recent works show that error patterns having a large overlap with near-codewords are the main culprits for decoding failures at very low Decoding Failure Rate (DFR) values. In this paper, we show that any BF decoder can be tweaked and made somehow aware of near-codewords, which means being able to recognize, and recover from, bad configurations due to near-codewords. We show that this modification results in minimal computational overhead. Through intensive numerical simulations, we evaluate the effectiveness of this approach on several BF decoders, considering both toy code parameters and BIKE parameters for NIST security category 1. Our results show drastic reductions in the DFR. We also find that, with this modification, a recently proposed BF variant called BF-Max outperforms the two decoders used by BIKE within the NIST competition.

Problem

• QC-MDPC codes
• Bit-Flipping decoding
• near-codewords
• trapping sets
• Decoding Failure Rate

Innovation

• near-codewords
• bit-flipping decoding
• QC-MDPC codes
• decoding failure rate
• post-quantum cryptography