## The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations (1993)

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@MISC{Arora93thehardness,

author = {Sanjeev Arora and László Babai and Jacques Stern and Z. Sweedyk},

title = {The Hardness of Approximate Optima in Lattices, Codes, and Systems of Linear Equations},

year = {1993}

}

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### Abstract

We prove the following about the Nearest Lattice Vector Problem (in any `p norm), the Nearest Codeword Problem for binary codes, the problem of learning a halfspace in the presence of errors, and some other problems. 1. Approximating the optimum within any constant factor is NP-hard. 2. If for some ffl ? 0 there exists a polynomial-time algorithm that approximates the optimum within a factor of 2 log 0:5\Gammaffl n , then every NP language can be decided in quasi-polynomial deterministic time, i.e., NP ` DTIME(n poly(log n) ). Moreover, we show that result 2 also holds for the Shortest Lattice Vector Problem in the `1 norm. Also, for some of these problems we can prove the same result as above, but for a larger factor such as 2 log 1\Gammaffl n or n ffl . Improving the factor 2 log 0:5\Gammaffl n to p dimension for either of the lattice problems would imply the hardness of the Shortest Vector Problem in `2 norm; an old open problem. Our proofs use reductions from few-pr...