Improved Reduction from the Bounded Distance Decoding Problem to the Unique Shortest Vector Problem in Lattices

We present a probabilistic polynomial-time reduction from the lattice Bounded Distance Decoding (BDD) problem with parameter 1/( √ 2 · γ) to the unique Shortest Vector Problem (uSVP) with parameter γ for any γ > 1 that is polynomial in the lattice dimension n. It improves the BDD to uSVP reductions of [Lyubashevsky and Micciancio, CRYPTO, 2009] and [Liu, Wang, Xu and Zheng, Inf. Process. Lett., 2014], which rely on Kannan’s embedding technique. The main ingredient to the improvement is the use of Khot’s lattice sparsification [Khot, FOCS, 2003] before resorting to Kannan’s embedding, in order to boost the uSVP parameter.