The nearest-colattice algorithm

In this work, we exhibit a hierarchy of polynomial time algorithms solving approximate variants of the Closest Vector Problem (CVP). Our first contribution is a heuristic algorithm achieving the same distance tradeoff as HSVP algorithms, namely $\approx \beta^{\frac{n}{2\beta}}$\text{covol}$(\Lambda)^{\frac{1}{n}}$ for a random lattice $\mathrm{\Lambda }$ of rank $n$. Compared to the so-called Kannan’s embedding technique, our algorithm allows using precomputations and can be used for efficient batch CVP instances. This implies that some attacks on lattice-based signatures lead to very cheap forgeries, after a precomputation. Our second contribution is a proven reduction from approximating the closest vector with a factor $\approx n^{\frac{3}{2}}{\beta }^{\frac{3n}{2\beta }}$ to the Shortest Vector Problem (SVP) in dimension $\beta$.