Solving Closest Vector Instances Using an Approximate Shortest Independent Vectors Oracle

Abstract

Given a lattice L ⊂ R n and some target vector, this paper studies the algorithms for approximate closest vector problem (CVP γ ) by using an approximate shortest independent vectors problem oracle (SIVP γ ). More precisely, if the distance between the target vector and the lattice is no larger than c γn λ 1 (L) for any constant c > 0, we give randomized and deterministic polynomial time algorithms to find a closest vector, which improves the known result by a factor of 2c. Moreover, if the distance between the target vector and the lattice is larger than some quantity with respect to λ n (L), using SIVP γ oracle and Babai's nearest plane algorithm, we can solve CVP γ √ n in deterministic polynomial time. Specially, if the approximate factor γ ∈ (1, 2) in the SIVP γ oracle, we obtain a better reduction factor for CVP.