In this paper we introduce a rank $2$ lattice over a polynomial ring arising from the public key of the BIKE cryptosystem. The secret key is a sparse vector in this lattice. We study properties of this lattice and generalize the recovery of weak keys from "Weak keys for the quasi-cyclic MDPC public key encryption scheme". In particular, we show that they implicitly solved a shortest vector problem in the lattice we constructed. Rather than finding only a shortest vector, we obtain a reduced basis of the lattice which makes it possible to check for more weak keys.

翻译：本文从BIKE密码系统的公钥出发，引入了一个多项式环上的秩为$2$的格。该系统中的私钥为此格中的一个稀疏向量。我们研究了此格的性质，并推广了"准循环MDPC公钥加密方案的弱密钥"中的弱密钥恢复方法。特别地，我们证明了该工作本质上是在我们所构造的格中求解了一个最短向量问题。相较于仅寻找最短向量，我们通过获得该格的约化基，使得检测更多弱密钥成为可能。