Performance Bounds for QC-MDPC Codes Decoders

• Published in: Code-Based Cryptography Vol. 13150; pp. 95 - 122
• Main Authors: Baldi, Marco, Barenghi, Alessandro, Chiaraluce, Franco, Pelosi, Gerardo, Santini, Paolo
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2022; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Bit-Flipping decoder; Code-based cryptography; Decoding failure rate; Error floor; Maximum likelihood decoder; Post-quantum cryptography; QC-MDPC codes
• ISBN: 9783030983642; 3030983641
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-98365-9_6

Quasi-Cyclic Moderate-Density Parity-Check (QC-MDPC) codes are receiving increasing attention for their advantages in the context of post-quantum asymmetric cryptography based on codes. However, a fundamentally open question concerns modeling the performance of their decoders in the region of a low decoding failure rate (DFR). We provide two approaches for bounding the performance of these decoders, and study their asymptotic behavior. We first consider the well-known Maximum Likelihood (ML) decoder, which achieves optimal performance and thus provides a lower bound on the performance of any sub-optimal decoder. We provide lower and upper bounds on the performance of ML decoding of QC-MDPC codes and show that the DFR of the ML decoder decays polynomially in the QC-MDPC code length when all other parameters are fixed. Secondly, we analyze some hard to decode error patterns for Bit-Flipping (BF) decoding algorithms, from which we derive some lower bounds on the DFR of BF decoders applied to QC-MDPC codes.