Improved Classical and Quantum Algorithms for the Shortest Vector Problem via Bounded Distance Decoding

Abstract

The most important computational problem on lattices is the Shortest Vector Problem (SVP). In this paper, we present new algorithms that improve the state-of-the-art for provable classical/quantum algorithms for SVP. We present the following results. $\bullet$ A new algorithm for SVP that provides a smooth tradeoff between time complexity and memory requirement. For any positive integer $4\le q\le \sqrt{n}$, our algorithm takes $q^{13n+o(n)}$ time and requires $poly(n)\cdot q^{16n/q^2}$ memory. This tradeoff which ranges from enumeration ($q=\sqrt{n}$) to sieving ($q$ constant), is a consequence of a new time-memory tradeoff for Discrete Gaussian sampling above the smoothing parameter. $\bullet$ A quantum algorithm for SVP that runs in time $2^{0.950n+o(n)}$ and requires $2^{0.5n+o(n)}$ classical memory and poly(n) qubits. In Quantum Random Access Memory (QRAM) model this algorithm takes only $2^{0.835n+o(n)}$ time and requires a QRAM of size $2^{0.293n+o(n)}$, poly(n) qubits and $2^{0.5n}$ classical space. This improves over the previously fastest classical (which is also the fastest quantum) algorithm due to [ADRS15] that has a time and space complexity $2^{n+o(n)}$. $\bullet$ A classical algorithm for SVP that runs in time $2^{1.669n+o(n)}$ time and $2^{0.5n+o(n)}$ space. This improves over an algorithm of [CCL18] that has the same space complexity. The time complexity of our classical and quantum algorithms are obtained using a known upper bound on a quantity related to the lattice kissing number which is $2^{0.402n}$. We conjecture that for most lattices this quantity is a $2^{o(n)}$. Assuming that this is the case, our classical algorithm runs in time $2^{1.292n+o(n)}$, our quantum algorithm runs in time $2^{0.750n+o(n)}$ and our quantum algorithm in QRAM model runs in time $2^{0.667n+o(n)}$.

SICOMP journal version and application to Lattice Isomorphism Problem over Z^n, 43 pages