# Space-efficient classical and quantum algorithms for the shortest vector problem. (arXiv:1709.00378v2 [cs.DS] UPDATED)

A lattice is the integer span of some linearly independent vectors. Lattice

problems have many significant applications in coding theory and cryptographic

systems for their conjectured hardness. The Shortest Vector Problem (SVP),

which is to find the shortest non-zero vector in a lattice, is one of the

well-known problems that are believed to be hard to solve, even with a quantum

computer. In this paper we propose space-efficient classical and quantum

algorithms for solving SVP. Currently the best time-efficient algorithm for

solving SVP takes $2^{n+o(n)}$ time and $2^{n+o(n)}$ space. Our classical

algorithm takes $2^{2.05n+o(n)}$ time to solve SVP with only $2^{0.5n+o(n)}$

space. We then modify our classical algorithm to a quantum version, which can

solve SVP in time $2^{1.2553n+o(n)}$ with $2^{0.5n+o(n)}$ classical space and

only poly(n) qubits.