Improvements on reductions among different variants of SVP and CVP

Abstract

It is well known that Search SVP is equivalent to Optimization SVP. However, the classical reduction from Search SVP to Optimization SVP by Kannan needs polynomial times of calls to the oracle that solves Optimization SVP. In this paper, a new rank-preserving reduction is presented with only one call to the Optimization SVP oracle. The idea also leads to a similar direct reduction from Search CVP to Optimization CVP with only one call to the corresponding oracle. Both of the reductions above can be generalized for l p norm with p Z +. On the other hand, whether the search and optimization variants of approximate SVP are computationally equivalent is an outstanding open problem. Recently, Cheng gave a reduction from Search SVP γ to Optimization SVP γ, where γ = γ 1 n (n - 1) log 2 γ n is much smaller than γ. We slightly improve the reduction by making γ = γ O (log 2 n) n (n - 1) log 2 γ n. In addition, a reduction from Search CVP γ to Optimization CVP γ with γ = γ 1 n n / 2 + log 2 γ · dist (t, L (B)) is also presented. © 2014 Springer International Publishing Switzerland.