 
Summary: Lattice Problems in NP # coNP
Dorit Aharonov # Oded Regev +
July 17, 2004
Abstract
We show that the problems of approximating the shortest and closest vector in a lattice to within a
factor of # n lie in NP intersect coNP. The result (almost) subsumes the three mutuallyincomparable
previous results regarding these lattice problems: Banaszczyk [7], Goldreich and Goldwasser [13], and
Aharonov and Regev [2]. Our technique is based on a simple fact regarding succinct approximation
of functions using their Fourier transform over the lattice. This technique might be useful elsewhere 
we demonstrate this by giving a simple and e#cient algorithm for one other lattice problem (CVPP)
improving on a previous result of Regev [25]. An interesting fact is that our result emerged from a
``dequantization'' of our previous quantum result in [2]. This route to proving purely classical results
might be beneficial elsewhere.
# School of Computer Science and Engineering, The Hebrew University, Jerusalem, Israel. doria@cs.huji.ac.il. Research
supported by ISF grant 0329738.
+ Department of Computer Science, TelAviv University, TelAviv 69978, Israel. Work supported by an Alon Fellowship and
the Army Research O#ce grant DAAD190310082.
1 Introduction
A lattice is the set of all integer combinations of n linearly independent vectors v 1 , . . . , v n in R n . These
