On worst-case to average-case reductions for NP problems

SIAM Journal on Computing

Volumn 36, Issue 4, 2006, Pages 1119-1159

  Bogdanov, Andrej ^a   Trevisan, Luca ^b  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

^b UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Average case complexity; One way function; Worst case to average case reduction

Indexed keywords

ALGORITHMS; HIERARCHICAL SYSTEMS; PROBLEM SOLVING; QUERY PROCESSING; RANDOM PROCESSES;

AVERAGE-CASE COMPLEXITY; ONE-WAY FUNCTION; RANDOM SELF-REDUCTION; WORST-CASE TO AVERAGE-CASE REDUCTION;

POLYNOMIAL APPROXIMATION;

EID: 34547851140     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0097539705446974     Document Type: Article

References (42)

1. M. ABADI, J. FEIGENBAUM, AND J. KILIAN, On hiding information from an oracle, J. Comput. System Sci., 39 (1989), pp. 21-50.
2. W. AIELLO AND J. HÅSTAD, Statistical zero-knowledge languages can be recognized in two rounds, J. Comput. System Sci., 42 (1991), pp. 327-345.
3. M. AJTAI, Generating hard instances of lattice problems, in Proceedings of the 28th Annual ACM Symposium on Theory of Computing, ACM, New York, 1996, pp. 99-108.
4. M. AJTAI AND C. DWORK, A public-key cryptosystem with worst-case/average-case equivalence, in Proceedings of the 29th Annual ACM Symposium on Theory of Computing, ACM, New York, 1997, pp. 284-293.
5. D. BEAVER AND J. FEIGENBAUM, Hiding instances in multioracle queries, in Proceedings of the 7th Annual Symposium on Theoretical Aspects of Computer Science, Springer, Berlin, 1990, pp. 37-48.
6. D. BEAVER, J. FEIGENBAUM, J. KILIAN, AND P. ROGAWAY, Locally random reductions: Improvements and applications, J. Crypt., 10 (1997), pp. 17-36.
7. S. BEN-DAVID, B. CHOR, O. GOLDREICH, AND M. LUBY, On the theory of average-case complexity, in Proceedings of the 21st Annual ACM Symposium on Theory of Computing, ACM, New York, 1989, pp. 204-216.
8. M. BLUM, Designing Programs to Check Their Work, Tech. Report 88-09, International Computer Science Institute, Berkeley, CA, 1988. Available online at http://www.icsi.berkeley.edu/cgi-bin/pubs/publication.pl?ID- 000495
9. M. BLUM AND S. KANNAN, Designing programs that check their work, J. ACM, 41 (1995), pp. 269-291.
10. M. BLUM, M. LUBY, AND R. RUBINFELD, Self-testing/correcting with applications to numerical problems, J. Comput. System Sci., 47 (1993), pp. 549-595.
11. A. BOGDANOV AND L. TREVISAN, Average-Case Complexity: A Survey, in preparation, 2006.
12. G. BRASSARD, Relativized cryptography, in Proceedings of the 20th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 1979, pp. 383-391.
13. B. CHOR, O. GOLDREICH, E. KUSHILEVITZ, AND M. SUDAN, Private information retrieval, J. ACM, 45 (1998), pp. 965-982.
14. W. DIFFIE AND M. E. HELLMAN, New directions in cryptography, IEEE Trans. Inform. Theory, 122 (1976), pp. 644-654.
15. S. EVEN, A. L. SELMAN, AND Y. YACOBI, The complexity of promise problems with applications to public-key cryptography, Inform. Control, 61 (1984), pp. 159-173.
16. S. EVEN AND Y. YACOBI, Cryptography and NP-completeness, in Proceedings of the 7th Annual ICALP, Lecture Notes in Comput. Sci. 85, Springer-Verlag, Berlin, 1980, pp. 195-207.
17. J. FEIGENBAUM AND L. FORTNOW, Random-self- reducibility of complete sets, SIAM J. Comput., 22 (1993), pp. 994-1005.
18. J. FEIGENBAUM, S. KANNAN, AND N. NISAN, Lower bounds on random-self-reducibility, in Proceedings of the 5th Annual IEEE Conference on Structure in Complexity Theory, IEEE, Los Alamitos, CA, 1990, pp. 100-109.
19. L. FORTNOW, The complexity of perfect zero-knowledge, in Proceedings of the 19th Annual ACM Symposium on Theory of Computing, ACM, New York, 1987, pp. 204-209.
20. O. GOLDREICH, Notes on Levin's Theory of Average-Case Complexity, Tech. Report TR97-058, Electronic Colloquium on Computational Complexity (ECCC), University of Trier, Trier, Germany, 1997. Available online at http://eccc.hpi-web.de/eccc-reports/1997/TR97-058/index.html
21. O. GOLDREICH, Modern Cryptography, Probabilistic Proofs and Pseudorandomness, Algorithms Combin. 17, Springer-Verlag, Berlin, 1999.
22. O. GOLDREICH, On Promise Problems, Tech. Report TR05-018, Electronic Colloquium on Computational Complexity (ECCC), University of Trier, Trier, Germany, 2005. Available online at http://eccc.hpi-web.de/eccc-reports/ 2005/TR05-018/index.html
23. O. GOLDREICH AND S. GOLDWASSER, On the Possibility of Basing Cryptography on the Assumption that P ≠ NP, unpublished manuscript, 1998.
24. O. GOLDREICH, H. KARLOFF, L. SCHULMAN, AND L. TREVISAN, Lower bounds for linear locally decodable codes and private information retrieval, in Proceedings of the 17th Annual IEEE Conference on Computational Complexity, IEEE, Los Alamitos, CA, 2002, pp. 175-183.
25. O. GOLDREICH, S. VADHAN, AND A. WIGDERSON, On interactive proofs with a laconic prover, in Proceedings of the 28th Annual ICALP, Lecture Notes in Comput. Sci. 2076, Springer-Verlag, Berlin, 2001, pp. 334-345.
26. S. GOLDWASSER AND M. SIPSER. Private coins versus public coins in interactive proof systems, in Proceedings of the 18th Annual ACM Symposium on Theory of Computing, ACM, New York, 1986, pp. 59-68.
27. A. HEALY, S. VADHAN, AND E. VIOLA, Using nondeterminism to amplify hardness, in Proceedings of the 36th Annual ACM Symposium on Theory of Computing, ACM, New York, 2004, pp. 192-201.
28. R. IMPAGLIAZZO, A personal view of average-case complexity, in Proceedings of the 10th Annual IEEE Conference on Structure in Complexity Theory, IEEE, Los Alamitos, CA, 1995, pp. 134-147.
29. R. IMPAGLIAZZO AND L. LEVIN, No better ways to generate hard NP instances than picking uniformly at random, in Proceedings of the 31st Annual IEEE Symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 1990, pp. 812-821.
30. J. KATZ AND L. TREVISAN, On the efficiency of local decoding procedures for error correcting codes, in Proceedings of the 32nd Annual ACM Symposium on Theory of Computing, ACM, New York, 2000, pp. 80-86.
31. A. LEMPEL, Cryptology in transition, ACM Computing Surveys, 11 (1979), pp. 285-303.
32. L. A. LEVIN, Average case complete problems, SIAM J. Comput., 15 (1986), pp. 285-286.
33. R. LIPTON, New directions in testing, in Proceedings of DIMACS Workshop on Distributed Computing and Cryptography, AMS, Providence, RI, 1989, pp. 191-202.
34. D. MICCIANCIO, Almost perfect lattices, the covering radius problem, and applications to Ajtai's connection factor, SIAM J. Comput., 34 (2004), pp. 118-169.
35. D. MICCIANCIO AND O. REGEV, Worst-case to average-case reductions based on Gaussian measure, in Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, IEEE, Los Alamitos, CA, 2004, pp. 372-381.
36. R. MOTWANI AND P. RAGHAVAN, Randomized Algorithms, Cambridge University Press, Cambridge, UK, 1995.
37. R. O'DONNELL, Hardness amplification within NP, in Proceedings of the 34th Annual ACM Symposium on Theory of Computing, ACM, New York, 2002, pp. 751-760.
38. O. REGEV, New lattice based cryptographic constructions, in Proceedings of the 35th Annual ACM Symposium on Theory of Computing, ACM, New York, 2003, pp. 407-416.
39. M. SUDAN, L. TREVISAN, AND S. VADHAN, Pseudorandom generators without the XOR lemma, J. Comput. System Sci., 62 (2001), pp. 236-266.
40. L. TREVISAN, On uniform amplification of hardness in NP, in Proceedings of the 37th Annual ACM Symposium on Theory of Computing, ACM, New York, 2005, pp. 31-38.
41. L. VALIANT AND V. VAZIRANI, NP is as easy as detecting unique solutions, Theoret. Comput. Sci., 47 (1986), pp. 85-93.
42. C. K. YAP, Some consequences of nonuniform conditions on uniform classes, Theoret. Comput. Sci., 26 (1983), pp. 287-300.