On the Complexity of Computing Short Linearly Independent Vectors and Short Bases in a Lattice

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(n p”‘y’og(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 2’“g’-‘(“), E > 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/a is not NP-hard. Key Words. Algori...

• Type: Conference
• Year: 1999
• Where: STOC
• Authors: Johannes Blömer, Jean-Pierre Seifert
• Keywords: Algorithms, Independent Vectors, Linearly Independent, Shortest Set, STOC 1999