A Statistical Explanation of the Timing Attack on QC-MDPC Code Crypto-system

The McEliece cryptosystem based on quasi-cyclic moderate-density parity-check (QC-MDPC) codes is first purposed in 2013\cite{QCMDPC} and is considered a promising contender in the post-quantum era. Understanding its security is hence essential. Till now, the most effective attacks are the reaction attack\cite{Reaction} and the timing attack\cite{Timing}. Both of these attacks rely on the decoding performance to recover the private key. The reaction attack relies on the decoding failure rate and the timing attack relies on the iterations during decoding. However, the mechanics behind these attacks remain elusive. In this paper, a mathematical model is proposed to explain both attacks by connecting the spectrum of private key and first-layer performance of the decoder