Hardness of approximating the shortest vector problem in lattices

Journal of the ACM

Volumn 52, Issue 5, 2005, Pages 789-808

  Khot, Subhash ^a,b  

^a GEORGIA INSTITUTE OF TECHNOLOGY   (United States)

^b Georgia Institute of Technology   (United States)

Author keywords

Approximation algorithms; Cryptography; Hardness of approximation; Lattices; Shortest vector problem

Indexed keywords

APPROXIMATION ALGORITHMS; HARDNESS OF APPROXIMATION; LATTICES; SHORTEST VECTOR PROBLEM;

ALGORITHMS; APPROXIMATION THEORY; COMPUTATIONAL COMPLEXITY; POLYNOMIALS; PROBLEM SOLVING; TENSORS; VECTORS;

CRYSTAL LATTICES;

EID: 27344453570     PISSN: 00045411     EISSN: 00045411     Source Type: Journal    
DOI: 10.1145/1089023.1089027     Document Type: Article

References (33)

1. AHARONOV, D., AND REGEV, O. 2004. Lattice problems in np ∩ conp. In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, CA.
2. AJTAI, M. 1996. Generating hard instances of lattice problems. In Proceedings of the 28th ACM Symposium on the Theory of Computing. ACM, New York, 99-108.
3. 2 is NP-hard for randomized reductions. In Proceedings of the 30th ACM Symposium on the Theory of Computing. ACM, New York, 10-19.
4. AJTAI, M., AND DWORK, C. 1997. A public-key cryptosystem with worst-case/average-case equivalence. In Proceedings of the 29th ACM Symposium on the Theory of Computing. ACM, New York, 284-293.
5. AJTAI, M., KUMAR, R., AND SIVAKUMAR, D. 2001. A sieve algorithm for the shortest lattice vector problem. In Proceedings of the 33rd ACM Symposium on the Theory of Computing. ACM, New York, 601-610.
6. ALON, N., SPENCER, J., AND ERDOS, P. 1991. The Probabilistic Method. Wiley-Interscience Series.
7. ARORA, S., BABAI, L., STERN, J., AND SWEEDYK, E. 1997. The hardness of approximate optima in lattices, codes and systems of linear equations. J. Comput. Syst. Sci. 54, 317-331.
8. BANASZCZYK, W. 1993. New bounds in some transference theorems in the geometry of numbers. Math. Ann. 296, 625-635.
9. CAI, J. 2003. Applications of a new transference theorem to Ajtai's connection factor. Discr. Appli. Math. 126, 1, 9-31.
10. CAI, J., AND NERURKAR, A. 1997. An improved worst-case to average-case connection for lattice problems. In Proceedings of the 38th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, CA.
11. ε) is NP-hard under randomized reductions. J. Comput. Syst. Sci. 59, 2, 221-239.
12. ∞ to within almost polynomial factors is NP-hard. Combinatorica 23, 2, 205-243.
13. DINUR, I., KINDLER, G., AND SAFRA, S. 1998. Approximating CVP to within almost-polynomial factors is NP-hard. In Proceedings of the 39th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, CA.
14. DUMER, I., MICCIANCIO, D., AND SUDAN, M. 1999. Hardness of approximating the minimum distance of a linear code. In Proceedings of the 40th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, CA.
15. GAUSS, C. 1801. Disquisitiones arithmetica (Leipzig, 1801: art. 171). Yale Univ. Press. (English translation by A. A. Clarke, 1966.)
16. GOLDREICH, O., AND GOLDWASSER, S. 2000. On the limits of non-approximability of lattice problems. J. Comput. Syst. Sci. 60, 3, 540-563.
17. GOLDREICH, O., MICCIANCIO, D., SAFRA, S., AND SEIFERT, J. 1999. Approximating shortest lattice vectors is not harder than approximating closest lattice vectors. Inf. Proc. Lett. 71, 2, 55-61.
18. HASTAD, J. 1988. Dual vectors and lower bounds for the nearest lattice point problem. Combinatorica 8, 75-81.
19. KANNAN, R. 1983. Improved algorithms for integer programming and related lattice problems. In Proceedings of the 15th ACM Symposium on Theory of Computing. ACM, New York, 193-206.
20. KANNAN, R. 1987. Minkowski's convex body theorem and integer programming. Math. Oper. Res. 12, 415-440.
21. p norms. In Proceedings of the 44th IEEE Symposium on Foundations of Computer Science. IEEE Computer Society Press, Los Alamitos, CA.
22. KUMAR, R., AND SIVAKUMAR, D. 2001. Complexity of SVP-A reader's digest. Complexity Theory Column, L. Hemaspaandra, Ed. SIGACT News 32, 3.
23. LAGARIAS, J., LENSTRA, H., AND SCHNORR, C. 1990. Korkine-Zolotarev bases and successive minima of a lattice and its reciprocal lattice. Combinatorica 10, 333-348.
24. LAGARIAS, J., AND ODLYZKO, A. 1985. Solving low-density subset sum problems. J. ACM 32, 1, 229-246.
25. LANDAU, S., AND MILLER, G. 1985. Solvability of radicals is in polynomial time. J. Comput. Syst. Sci. 30, 2, 179-208.
26. LENSTRA, A., LENSTRA, H., AND Lovász, L. 1982. Factoring polynomials with rational coefficients. Math. Ann. 261, 513-534.
27. LENSTRA, H. 1981. Integer programming with a fixed number of variables. Tech. Report 81-03. Univ. of Amsterdam, Amsterdam, The Netherland.
28. MICCIANCIO, D. 2000. The shortest vector problem is NP-hard to approximate to within some constant. SIAM J. Comput. 30, 6, 2008-2035.
29. MICCIANCIO, D., AND GOLDWASSER, S. 2002. Complexity of Lattice Problems, A Cryptographic Perspective. Kluwer Academic Publishers.
30. MINKOWSKI, H. 1910. Geometrie der zahlen. Tuebner.
31. REGEV, O. 2003. New lattice based cryptographic constructions. In Proceedings of the 35th ACM Symposium on the Theory of Computing. ACM, New York.
32. SCHNORR, C. 1987. A hierarchy of polynomial-time basis reduction algorithms. Theoret. Comput. Sci. 53, 2-3, 201-224.
33. VAN EMDE BOAS, P. 1981. Another NP-complete problem and the complexity of computing short vectors in a lattice. Tech. Report 81-04. Mathematische Instiut, Univ. of Amsterdam, Amsterdam, The Netherland.