Polynomial time bounded distance decoding near Minkowski’s bound in discrete logarithm lattices

Abstract

We propose a concrete family of dense lattices of arbitrary dimension n in which the lattice bounded distance decoding (BDD) problem can be solved in deterministic polynomial time. This construction is directly adapted from the Chor–Rivest cryptosystem (IEEE-TIT 1988). The lattice construction needs discrete logarithm computations that can be made in deterministic polynomial time for well-chosen parameters. Each lattice comes with a deterministic polynomial time decoding algorithm able to decode up to large radius. Namely, we reach decoding radius within $$O(\log n)$$ Minkowski’s bound, for both $$\ell _1$$ and $$\ell _2$$ norms.