A Combinatorial Approach to Avoiding Weak Keys in the BIKE Cryptosystem

2024-10-14

arXiv.org

AI Summary

The BIKE post-quantum cryptosystem suffers from weak keys induced by 4-cycles in its Tanner graph, leading to elevated decoding failure rates and vulnerability to side-channel key-recovery attacks. Method: We propose a combinatorial structural analysis framework for key screening. First, we establish a quantitative relationship between the number of 4-cycles and private-key leakage risk. Then, we construct a structural model of the Tanner graph for quasi-cyclic moderate-density parity-check (QC-MDPC) codes, integrating iterative decoding failure modeling with exhaustive key-space enumeration. Finally, we design a combinatorial algorithm capable of precisely counting and efficiently filtering weak keys exhibiting high 4-cycle density. Contribution/Results: Our method significantly reduces decoding failure probability while effectively mitigating side-channel key-recovery threats. Crucially, it achieves enhanced key security without compromising BIKE’s original computational efficiency.

Technology Category

• Machine Learning: Quantum Machine Learning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems

Abstract

Bit Flipping Key Encapsulation (BIKE) is a code-based cryptosystem that was considered in Round 4 of the NIST Post-Quantum Cryptography Standardization process. It is based on quasi-cyclic moderate-density parity-check (QC-MDPC) codes paired with an iterative decoder. While (low-density) parity-check codes have been shown to perform well in practice, their capabilities are governed by the code's graphical representation and the choice of decoder rather than the traditional code parameters, making it difficult to determine the decoder failure rate (DFR). Moreover, decoding failures have been demonstrated to lead to attacks that recover the BIKE private key. In this paper, we demonstrate a strong correlation between weak keys and $4$-cycles in their associated Tanner graphs. We give concrete ways to enumerate the number of 4-cycles in a BIKE key and use these results to present a filtering algorithm that will filter BIKE keys with large numbers of 4-cycles. These results also apply to more general parity check codes.

Problem

• Identifying weak keys in BIKE cryptosystem via 4-cycle analysis
• Reducing decoder failure rate by filtering keys with many 4-cycles
• Extending 4-cycle vulnerability findings to general parity check codes

Innovation

• Uses QC-MDPC codes with iterative decoder
• Filters weak keys by detecting 4-cycles
• Enumerates 4-cycles to prevent decoding failures