Analysis of the error correction capability of LDPC and MDPC codes under parallel bit-flipping decoding and application to cryptography
This work addresses the need for guaranteed error rates in code-based cryptography to achieve strong security levels, representing an incremental improvement over existing models that are computationally intractable or rely on assumptions.
The authors tackled the problem of guaranteeing low error rates for LDPC and MDPC codes under iterative decoding, which is critical for applications like code-based cryptography, by developing a theoretical analysis that provides tight bounds on error rates, especially for codes with small girth and single-iteration decoding without assumptions.
Iterative decoders used for decoding low-density parity-check (LDPC) and moderate-density parity-check (MDPC) codes are not characterized by a deterministic decoding radius and their error rate performance is usually assessed through intensive Monte Carlo simulations. However, several applications, like code-based cryptography, need guaranteed low values of the error rate, which are infeasible to assess through simulations, thus requiring the development of theoretical models for the error rate of these codes under iterative decoding. Some models of this type already exist, but become computationally intractable for parameters of practical interest. Other approaches approximate the code ensemble behaviour through some assumptions, which may not hold true for a specific code. We propose a theoretical analysis of the error correction capability of LDPC and MDPC codes that allows deriving tight bounds on the error rate at the output of parallel bit-flipping decoders. Special attention is devoted to the case of codes with small girth; moreover, single-iteration decoding is investigated through a rigorous approach, which does not require any assumption and hence results in a guaranteed error correction capability for any single code. We show an example of application of the new bound to the context of code-based cryptography, where guaranteed error rates are needed to achieve some strong security levels.