Efficient reductions among lattice problems

Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms

Volumn , Issue , 2008, Pages 84-93

  Micciancio, Daniele ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

POLYNOMIALS;

APPROXIMATION FACTORS; APPROXIMATION PROBLEMS; CLOSEST VECTOR PROBLEM (CVP); LATTICE PROBLEMS; OPEN PROBLEMS; POLYNOMIAL-TIME;

POLYNOMIAL APPROXIMATION;

EID: 51949098877     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (30)

1. 2 is NP-hard for randomized reductions (extended abstract). In Proceedings of STOC '98, pages 10-19. ACM, May 1998.
2. M. Ajtai. Generating hard instances of lattice problems. Complexity of Computations and Proofs, Quaderni di Maiemaiica, 13:1-32, 2004. Preliminary version in STOC 1996.
3. M. Ajtai and C. Dwork. A public-key cryptosystem with worst-case/average-case equivalence. In Proceedings of STOC '97, pages 284-293. ACM, May 1997.
4. M. Ajtai, R. Kumar, and D. Sivakumar. A sieve algorithm for the shortest lattice vector problem. In Proceedings of STOC '01, pages 266-275. ACM, July 2001.
5. M. Ajtai, R. Kumar, and D. Sivakumar. Sampling short lattice vectors and the closest lattice vector problem. In Proceedings of CCC '02, pages 53-57. IEEE, May 2002.
6. L. Babai. On Lovasz' lattice reduction and the nearest lattice point problem. Combinatorica, 6(1):1-13, 1986.
7. W. Banaszezyk. New bounds in some transference theorems in the geometry of numbers. Mathematische Annalen, 296:625-635, 1993.
8. J. Blömer. Closest vectors, successive minima and dual HKZ-bases of lattices. In Proceedings of ICALP '00, volume 1853 of LNCS, pages 248-259. Springer, July 2000.
9. J. Blömer and S. Naewe. Sampling methods for shortest vectors, closest vectors and successive minima. In Proceedings of ICALP '07, volume 4596 of LNCS, pages 65-77. Springer, July 2007.
10. J. Blömer and J.-P. Seifert. On the complexity of computing short linearly independent vectors and short bases in a lattice. In Proceedings of STOC '99, pages 711-720. ACM, May 1999.
11. E. F. Brickell and A. M. Odlyzko. Cryptanalysis: A survey of recent results. In G. J. Simmons, editor, Contemporary Cryptology, chapter 10, pages 501-540. IEEE Press, 1991.
12. J.-Y. Cai. A new transference theorem in the geometry of numbers and new bounds for Ajtai's connection factor. Discrete Applied Mathematics, 126(1):9-31, Mar. 2003. Preliminary versions in CCC 1999 and COCOON 1999.
13. H. Cohen. A course in computational algebraic number theory. Springer-Verlag, 1993.
14. I. Dinur, G. Kindler, R. Raz, and S. Safra. Approximating CVP to within almost-polynomial factors is NP-hard. Combinatorica, 23(2):205-243, 2003. Preliminary version in FOCS 1998.
15. O. Goldreich, S. Goldwasser, and S. Halevi. Publickey cryptosystems from lattice reduction problems. In Proceedings of CRYPTO '97, volume 1294 of LNCS, pages 112-131. Springer, Aug. 1997.
16. O. Goldreich, D. Micciancio, S. Safra, and J.-P. Seifert. Approximating shortest lattice vectors is not harder than approximating closest lattice vectors. Information Processing Letters, 71(2):55-61, 1999.
17. V. Guruswami, D. Micciancio, and O. Regev. The complexity of the covering radius problem. Computational Complexity, 14(2):90-120, 2005. Preliminary version in CCC 2004.
18. I. Haviv and O. Regev. Tensor-based hardness of the shortest vector problem to within almost polynomial factors. In Proceedings of STOC '07, pages 469-477. ACM, June 2007.
19. A. Joux and J. Stern. Lattice reduction: A toolbox for the cryptanalyst. Journal of Cryptology, 11(3):161-185, 1998.
20. R. Kannan. Minkowski's convex body theorem and integer programming. Mathematics of operation research, 12(3):415-440, Aug. 1987.
21. S. Khot. Hardness of approximating the shortest vector problem in lattices. Journal of the ACM, 52(5):789-808, Sept. 2005. Preliminary version in FOCS 2004.
22. A. K. Lenstra, H. W. Lenstra, Jr., and L. Lovász. Factoring polynomials with rational coefficients. Mathematische Annalen, 261(4):513-534, Dec. 1982.
23. D. Micciancio. The shortest vector problem is NP-hard to approximate to within some constant. SIAM Journal on Computing, 30(6):2008-2035, Mar. 2001. Preliminary version in FOCS 1998.
24. D. Micciancio. Almost perfect lattices, the covering radius problem, and applications to Ajtai's connection factor. SIAM Journal on Computing, 34(1):118-169, 2004. Preliminary version in STOC 2002.
25. D. Micciancio. Generalized compact knapsaks, cyclic lattices, and efficient one-way functions. Computational Complexity, 2007. To appear in special issue on average-case complexity. Preliminary version in FOCS 2002.
26. D. Micciancio and S. Goldwasser. Complexity of Lattice Problems: a cryptographic perspective, volume 671 of The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers, Boston, Massachusetts, Mar. 2002.
27. D. Micciancio and O. Regev. Worst-case to averagecase reductions based on Gaussian measure. SIAM Journal on Computing, 37(1):267-302, 2007. Preliminary version in FOCS 2004.
28. P. Nguyen and J. Stern. The two faces of lattices in cryptology. In Proceedings of CaLC '01, volume 2146 of LNCS, pages 146-180. Springer, Mar. 2001.
29. O. Regev. New lattice based cryptographic constructions. Journal of the ACM, 51(6):899-942, 2004. Preliminary version in STOC 2003.
30. C.-P. Schnorr. A hierarchy of polynomial time lattice basis reduction algorithms. Theoretical Computer Science, 53(2-3):201-224, 1987.