On the complexity of computing short linearly independent vectors and short bases in a lattice

Abstract

Motivated by Ajtai's worst-case to average-case reduction for lattice problems, we study the complexity of computing short linearly independent vectors (short basis) in a lattice. We show that approximating the length of a shortest set of linearly independent vectors (shortest basis) within any constant factor is NP-hard. Under the assumption that problems in NP cannot be solved in DTIME(npolylog(n)) we show that no polynomial time algorithm can approximate the length of a shortest set of linearly independent vectors (shortest basis) within a factor of 2log(1-ε)(n), ε>0 arbitrary, but fixed. Finally, we obtain results on the limits of non-approximability for computing short linearly independent vectors (short basis). Our strongest result in this direction states that under reasonable complexity-theoretic assumptions, approximating the length of a shortest set of linearly independent vectors (shortest basis) within a factor of n/√log(n) is not NP-hard.