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

SIAM Journal on Computing

Volumn 30, Issue 6, 2000, Pages 2008-2035

  Miccianciq, Daniele ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Geometry of numbers; NP hardness; Point lattices; Shortest vector problem; Sphere packing

Indexed keywords

ALGORITHMS; APPROXIMATION THEORY; COMPUTATIONAL COMPLEXITY; COMPUTATIONAL GEOMETRY; NUMBER THEORY; POLYNOMIALS; PROBABILITY; PROBLEM SOLVING; THEOREM PROVING;

POINT LATTICES; RANDOM POLYNOMIAL TIME; REVERSE UNFAITHFUL RANDOM REDUCTIONS; SHORTEST VECTOR PROBLEM; SPHERE PACKING; SQUARE-FREE SMOOTH NUMBERS;

VECTORS;

EID: 0035707359     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/s0097539700373039     Document Type: Article

References (35)

1. L. M. ADLEMAN, Factoring and Lattice Reduction, manuscript, 1995.
2. M. AJTAI, The shortest vector problem in L_2 is NP-hard for randomized reductions (extended abstract), in Proceedings of the 30th Annual ACM Symposium on Theory of Computing, Dallas, TX, 1998, pp. 10-19.
3. S. ARORA, L. BABAI, J. STERN, AND E. Z. SWEEDYK, The hardness of approximate optima in lattices, codes, and systems of linear equations, J. Comput. System Sei., 54 (1997), pp. 317-331. Preliminary version in FOCS'93.
4. L. BABAI, On Lovasz' lattice reduction and the nearest lattice point problem, Combinatorica, G (1986), pp. 1-13.
5. W. BANASZCZYK, New bounds in some transference theorems in the geometry of numbers, Math. Ann., 296 (1993), pp. 625-635
6. M. BELLARE, S. GOLDWASSER, AND D. MICCIANCIO, "Pseudo-random" generators within cryptographic applications: the DSS case, in Advances in Cryptology-CRYPTO '97, B. S. Kaliski Jr., ed., Lecture Notes in Comput. Sei. 1294, Springer-Verlag, New York, 1997, pp. 277-291
7. D. COPPERSMITH, Small solutions to polynomial equations, and low-exponent RSA vulnerabilities, J. Cryptology, 10 (1997), pp. 233-260.
8. M. J. COSTER, A. Joux, B. A. LAMACCHIA, A. M. ODLYZKO, C.-P. SCHNORR, AND J. STERN, Improved low-density subset sum algorithms, Comput. Complexity, 2 (1992), pp. 111-128.
9. I. DINUR, Approximating SVPoo to within almost-polynomial factors is NP-hard, in Algorithms and Complexity, 4th Italian Conference, CIAC 2000, Proceedings, G. Bongiovanni, G. Gambosi, and R. Petreschi, eds., Lecture Notes in Comput. Sei. 1767, Rome, Italy, 2000, Springer-Verlag, New York, pp. 263-276.
10. I. DINUR, G. KINDLER, AND S. SAFRA, Approximating CVP to within almost-polynomial factors is NP-hard, in 39th Annual IEEE Symposium on Foundations of Computer Science, Palo Alto, CA, 1998.
11. I. DUMER, D. MlCCIANCIO, AND M. SUDAN, Hardness of approximating the minimum distance of a linear code, in 40th Annual IEEE Symposium on Foundations of Computer Science, New York, 1999, pp. 475-484
12. A. M. FRIEZE, On the Lagarias-Odlyzko algorithm for the subset sum problem, SIAM J. Cornput., 15 (1986), pp. 536-539.
13. A. M. FRIEZE, J. HâSTAD, R. KANNAN, J. C. LAGARIAS, AND A. SHAMIR, Reconstructing truncated integer variables satisfying linear congruences, SIAM J. on Comput., 17 (1988), pp. 262-280
14. O. GOLDREICH, private communication, 1999.
15. O. GOLDREICH AND S. GOLDWASSER, On the limits of nonapproximability of lattice problems, J. Comput. System Sei., 60 (2000), pp. 540-563. Preliminary version in STOC'98.
16. O. GOLDREICH, D. MICCIANCIO, S. SAFRA, AND J.-P. SEIFERT, Approximating shortest lattice vectors is not harder than approximating closest lattice vectors, Inform. Process. Lett., 71 (1999), pp. 55-61.
17. J. HâSTAD, Solving simultaneous modular equations of low degree, SIAM J. Comput., 17 (1988), pp. 336-341. Preliminary version in CryptoSS.
18. D. S. JOHNSON, A catalog of complexity classes, in Handbook of Theoretical Computer Science, vol. A (Algorithms and Complexity), Elsevier Science, Amsterdam, 1990, chap. 2, pp. 67-161.
19. R. KANNAN, Algorithmic geometry of numbers, in Annual Reviews of Computer Science, Vol. 2, Annual Review Inc., Palo Alto, CA, 1987, pp. 231-267.
20. R. KANNAN, Minkowski's convex body theorem and integer programming, Math. Oper. Res., 12 (1987), pp. 415-440.
21. R. M. KARP AND R. J. LIPTON, Some connections between nonuniform and uniform complexity classes, in Proceedings of the Twelfth Annual ACM Symposium on Theory of Computing, Los Angeles, CA, 1980, pp. 28-30.
22. Appeared in journal form as R. M. Karp and R. J. Lipton, Turing machines that take advice, Enseign. Math., 28 (1982) pp. 191-209.
23. J. C. LAGARIAS, H. W. LENSTRA, JR., AND C.-P. SCHNORR, Korkine-Zolotarev bases and successive minima of a lattice and its reciprocal lattice, Combinatorica, 10 (1990), pp. 333-348.
24. J. C. LAGARIAS AND A. M. ODLYZKO, Solving low-density subset sum problems, J. ACM, 32 (1985), pp. 229-246.
25. A. K. LENSTRA, H. \V. LENSTRA, JR., AND L. LOVäSZ, Factoring polynomials with rational coefficients, Math. Ann., 261 (1982), pp. 513-534.
26. H. W. LENSTRA, Integer programming with a fixed number of variables, Math. Oper. Res., 8 (1983), pp. 538-548.
27. A. R. MEYER AND L. J. STOCKMEYER, The equivalence problem for regular expression with squaring requires exponential space, in Proceedings of the 13th IEEE Symposium on Switching and Automata Theory, 1972, pp. 25-29.
28. D. MICCIANCIO, On the Hardness of the Shortest Vector Problem, Ph.D. thesis, Massachusetts Institute of Technology, Cambridge, MA, 1998.
29. D. MlCCIANCIO, The hardness of the closest vector problem with preprocessing, IEEE Trans. Inform. Theory, 47 (2001).
30. C.-P. SCHNORR, A hierarchy of polynomial time lattice basis reduction algorithms, Theoret. Comput. Sei., 53 (1987), pp. 201-224
31. C.-P. SCHNORR, Factoring integers and computing discrete logarithms via Diophantine approximation, in Advances in Computational Complexity 13, DIMACS Ser. Discrete Math. Theoret. Comput. Sei., AMS, Providence, RI, 1993, pp. 171-182. Preliminary version in Eurocrypt'91, Springer-Verlag, LNCS 547
32. A. SCHöNHAGE, Factorization of univariate integer polynomials by Diophantine approximation and an improved basis reduction algorithm, in Automata, Languages and Programming, llth Colloquium, J. Paredaens, ed., Lecture Notes in Comput. Sei. 172, Antwerp, Belgium, 1984, Springer-Verlag, New York, pp. 436-447.
33. A. SHAMIR, A polynomial time algorithm for breaking the basic Merkle-Hellman cryptosystem, IEEE Trans. Inform. Theory, 30 (1984), pp. 699-704. Preliminary version in FOCS'82.
34. L. J. STOCKMEYER, The polynomial-time hierarchy, Theoret. Comput. Sei., 3 (1977), pp. 1-22.
35. P. VAN EMDE BOAS, Another NP-Complete Problem and the Complexity of Computing Short Vectors in a Lattice, Tech. Report 81-04, Mathematische Instituut, Universiry of Amsterdam, 1981; also available online from http://turing.wins.uva.nl/~peterJ.