## On the Limits of Non-Approximability of Lattice Problems (1998)

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@MISC{Goldreich98onthe,

author = {Oded Goldreich and Shafi Goldwasser},

title = { On the Limits of Non-Approximability of Lattice Problems},

year = {1998}

}

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### Abstract

We show simple constant-round interactive proof systems for problems capturing the approximability, to within a factor of p n, of optimization problems in integer lattices; specifically, the closest vector problem (CVP), and the shortest vector problem (SVP). These interactive proofs are for the "coNP direction"; that is, we give an interactive protocol showing that a vector is "far" from the lattice (for CVP), and an interactive protocol showing that the shortest-latticevector is "long" (for SVP). Furthermore, these interactive proof systems are Honest-Verifier Perfect Zero-Knowledge. We conclude that approximating CVP (resp., SVP) within a factor of p n is in NP " coAM. Thus, it seems unlikely that approximating these problems to within a p n factor is NPhard. Previously, for the CVP (resp., SVP) problem, Lagarias et. al., Hastad and Banaszczyk showed that the gap problem corresponding to approximating CVP (resp., SVP) within n is in NP " coNP . On the other hand, Ar...