Bit-flipping Decoder Failure Rate Estimation for (v,w)-regular Codes

Providing closed form estimates of the Decoding Failure Rates (DFR) of iterative decoder for low- and moderate-density parity check codes has attracted significant interest in the research community over the years. This interest has raised due to the use of iterative decoders in post-quantum cryptosystems, where the desired DFRs are impossible to estimate via Monte Carlo simulations. In this work, we propose a new technique to provide accurate estimates of the DFR of a two-iterations (parallel) bit-flipping decoder, which is also employable for cryptographic purposes. In doing so, we successfully tackle the estimation of the bit-flipping probabilities at the first and second decoder iteration, and provide a fitting estimate for the syndrome weight distribution. We numerically validate our results, providing comparisons of the modeled and simulated weight of the syndrome, incorrectly-guessed error bit distribution at the end of the first iteration, and two-iteration DFR, both in the floor and waterfall regime. Finally, we apply our method to estimate the DFR of LEDAcrypt, a post-quantum cryptosystem, improving by factors larger than 2^{70}, with respect to the previous estimation techniques.