[Resource Topic] 2023/1224: Theoretical analysis of decoding failure rate of non-binary QC-MDPC codes

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Theoretical analysis of decoding failure rate of non-binary QC-MDPC codes

Authors: Kirill Vedenev, Yury Kosolapov


In this paper, we study the decoding failure rate (DFR) of non-binary QC-MDPC codes using theoretical tools, extending the results of previous binary QC-MDPC code studies. The theoretical estimates of the DFR are particularly significant for cryptographic applications of QC-MDPC codes. Specifically, in the binary case, it is established that exploiting decoding failures makes it possible to recover the secret key of a QC-MDPC cryptosystem. This implies that to attain the desired security level against adversaries in the CCA2 model, the decoding failure rate must be strictly upper-bounded to be negligibly small. In this paper, we observe that this attack can also be extended to the non–binary case as well, which underscores the importance of DFR estimation. Consequently, we study the guaranteed error-correction capability of non-binary QC-MDPC codes under one-step majority logic (OSML) decoder and provide a theoretical analysis of the 1-iteration parallel symbol flipping decoder and its combination with OSML decoder. Utilizing these results, we estimate the potential public-key sizes for QC-MDPC cryptosystems over \mathbb{F}_4 for various security levels. We find that there is no advantage in reducing key sizes when compared to the binary case.

ePrint: https://eprint.iacr.org/2023/1224

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