Recurrent Bipartite Graph Neural Networks for QC-MDPC Error Decoding in McEliece Cryptosystems

As the advent of large-scale quantum computing threatens classical encryption, Quasi-Cyclic Moderate Density Parity-Check (QC-MDPC) codes have emerged as a foundational element of Post-Quantum Cryptography (PQC). However, decoding these relatively dense codes using classical Belief Propagation algorithms often leads to failures caused by short graph cycles that stall the decoder. In cryptographic applications, these failures are severe security vulnerabilities that leak secret key information. To address this, this paper proposes a Recurrent Bipartite Graph Neural Network (R-BGNN) as proof of concept. By integrating Gated Recurrent Units (GRUs) into the message-passing framework and employing a multi-topology curriculum training strategy, the R-BGNN actively learns to break limit-cycle oscillations. Evaluated across 20 unseen matrix topologies, the proposed architecture significantly outperforms classical Bit-Flipping and Min-Sum baselines, reducing the Decoding Failure Rate (DFR) by 63.7\% at moderate noise levels ($t=10$) while converging in fewer iterations.