Polynomial Lattices for the BIKE Cryptosystem

In this paper we introduce a rank $2$ lattice over a polynomial ring arising from the public key of the BIKE cryptosystem \cite{aragon2022bike}. The secret key is a sparse vector in this lattice. We study properties of this lattice and generalize the recovery of weak keys from \cite{BardetDLO16}. 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.