# What is the security implication of non-full-rank systematic matrix in McEliece cryptosystem?

The Classic McEliece cryptosystem has the following key generation procedure:

1. Choose a field $$\mathbb{F}_{2^m}$$, an irreducible polynomial $$g(x)$$ of degree $$t$$, and $$n$$ field elements $$\alpha_1, \cdots, \alpha_n$$.
2. Build the $$t \times n$$ matrix $$\tilde{H} = (h_{ij}), h_{ij} = \frac{\alpha_j^{i - 1}}{g(\alpha_j)}$$
3. Replace each component in $$\tilde{H}$$ with a binary vector of length $$m$$, to get a matrix $$\hat{H}$$ of size $$mt \times n$$.
4. Reduce $$\hat{H}$$ to the row-echelon form $$H$$ (called "systematic form")

The NIST proposal says that if $$H$$ does not have full-rank (meaning, there are empty rows in the row-echelon form), then the key should be abandoned. This paper (p.8) claims without proof that the non-full-rank situation is extremely rare, with probability less than $$2^{-256}$$.

What would be the security implication if one uses a non-full-rank public key? Since they are very rare, would it be much easier to find the private key if one knows the public key has $$r$$ empty rows?