Performance Bounds for QC-MDPC Codes Decoders

Abstract

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.

Fast computation of cyclic convolutions and their applications in code-based asymmetric encryption schemes

“…There are a number of approaches to assessing DFR. In [11], a lower theoretical DFR estimate was obtained for an ML decoder (Maximum Likelihood decoder), which differs from the iterative decoder from [6]. However, this estimate may be redundant: the real DFR may be lower when using modern fast decoders.…”

“…The study, including experimental, of the inflection point for values of 𝜆 exceeding 20, seems to be an urgent task, since this can allow more accurate extrapolation to the values of 𝜆 corresponding to the values recommended in practice (𝜆 = 128, 192, 256). Note that the efficient implementation of the BIKE cryptosystem is beneficial for obtaining estimates of the probability of decryption errors [11][12][13]. The parallel computations mentioned in [12,13] alone may not provide significant acceleration.…”