# Lattice problems in NP ∩ coNP

@article{Aharonov2005LatticePI, title={Lattice problems in NP ∩ coNP}, author={Dorit Aharonov and Oded Regev}, journal={J. ACM}, year={2005}, volume={52}, pages={749-765} }

We show that the problems of approximating the shortest and closest vector in a lattice to within a factor of &nradic; lie in NP intersect coNP. The result (almost) subsumes the three mutually-incomparable previous results regarding these lattice problems: Banaszczyk [1993], Goldreich and Goldwasser [2000], and Aharonov and Regev [2003]. Our technique is based on a simple fact regarding succinct approximation of functions using their Fourier series over the lattice. This technique might be…

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## References

SHOWING 1-10 OF 37 REFERENCES

### A note on the non-NP-hardness of approximate lattice problems under general Cook reductions

- Mathematics, Computer ScienceInf. Process. Lett.
- 2000

### The shortest vector problem in L2 is NP-hard for randomized reductions (extended abstract)

- Mathematics, Computer ScienceSTOC '98
- 1998

There is a prob-abilistic Turing-machine which in polynomial time reduces any problem in NP to instances of the shortest vector problem, provided that it can use an oracle which returns the solution of the longest vector problem if an instance of it is presented (by giving a basis of the corresponding lattice).

### A lattice problem in quantum NP

- Mathematics44th Annual IEEE Symposium on Foundations of Computer Science, 2003. Proceedings.
- 2003

This work gives a new characterization of QMA, called QMA+ formulation, and makes the important observation that autocorrelation functions are positive definite functions and using properties of such functions the authors severely restrict the prover's possibility to cheat.

### The shortest vector in a lattice is hard to approximate to within some constant

- Computer ScienceProceedings 39th Annual Symposium on Foundations of Computer Science (Cat. No.98CB36280)
- 1998

It is shown the shortest vector problem in the l/sub 2/ norm is NP-hard (for randomized reductions) to approximate within any constant factor less than /spl radic/2 and an alternative construction satisfying Ajtai's probabilistic variant of Sauer's lemma is given.

### Generating hard instances of lattice problems (extended abstract)

- Mathematics, Computer ScienceSTOC '96
- 1996

We give a random class of lattices in Zn whose elements can be generated together with a short vector in them so that, if there is a probabilistic polynomial time algorithm which finds a short vector…

### The inapproximability of lattice and coding problems with preprocessing

- Computer ScienceProceedings 17th IEEE Annual Conference on Computational Complexity
- 2002

It is shown that there exists a reduction from an NP-hard problem to the approximate closest vector problem such that the lattice depends only on the size of the original problem, and the specific instance is encoded solely, in the target vector.

### Hardness of approximating the shortest vector problem in lattices

- Computer Science45th Annual IEEE Symposium on Foundations of Computer Science
- 2004

A new (randomized) reduction from closest vector problem (CVP) to SVP that achieves some constant factor hardness is given, based on BCH codes, that enables the hardness factor to 2/sup log n1/2-/spl epsi//.

### A sieve algorithm for the shortest lattice vector problem

- Computer Science, MathematicsSTOC '01
- 2001

Several consequences of this algorithm for related problems on lattices and codes are obtained, including an improvement for polynomial time approximations to the shortest vector problem.

### A Hierarchy of Polynomial Time Lattice Basis Reduction Algorithms

- Computer ScienceTheor. Comput. Sci.
- 1987

### Korkin-Zolotarev bases and successive minima of a lattice and its reciprocal lattice

- MathematicsComb.
- 1990

It is proved that and, where andγj denotes Hermite's constant, are lower bounds for polynomial time computable quantities λ1(L) andΜ(x,L), where Μ( x,L) is the Euclidean distance fromx to the closest vector inL.