The nearest-colattice algorithm

TL;DR

This paper introduces a hierarchy of polynomial-time algorithms for approximate Closest Vector Problem (CVP), including a heuristic that improves efficiency and a reduction linking CVP approximation to the Shortest Vector Problem (SVP).

Contribution

It presents a new heuristic algorithm for approximate CVP with improved distance tradeoff and a proven reduction from CVP approximation to SVP, enabling more efficient lattice-based computations.

Findings

Heuristic algorithm achieves HSVP-like distance tradeoff.

Allows precomputations for batch CVP instances, enabling faster attacks.

Reduces CVP approximation to SVP in dimension 3n.

Abstract

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 $\approx \beta^{\frac{n}{2\beta}}\textrm{covol}(\Lambda)^{\frac{1}{n}}$ for a random lattice $\Lambda$ 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 $\approx n^{\frac32}\beta^{\frac{3n}{2\beta}}$ to the Shortest Vector Problem (SVP) in dimension $\beta$.