Pseudorandomness and average-case complexity via uniform reductions

Computational Complexity

Volumn 16, Issue 4, 2007, Pages 331-364

  Trevisan, Luca ^a   Vadhan, Salil ^b  

^a UNIVERSITY OF CALIFORNIA   (United States)

^b HARVARD UNIVERSITY   (United States)

Author keywords

Average case complexity; Derandomization; Instance checkers; Pseudorandomness

Indexed keywords

EID: 38749129723     PISSN: 10163328     EISSN: 14208954     Source Type: Journal    
DOI: 10.1007/s00037-007-0233-x     Document Type: Article

References (63)

1. M. ABADI, J. FEIGENBAUM, and J. KILIAN. On hiding information from an oracle. Journal of Computer and System Sciences, 39:21-50, 1989.
2. E. ALLENDER, H. BUHRMAN, M. KOUCK'Y, D. VAN MELKEBEEK, and D. RONNEBURGER. Power from random strings. In Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, pages 669-678, 2002.
3. A. E. ANDREEV, A. E. F. CLEMENTI, and J. D. P. ROLIM. Worst-case hardness suffices for derandomization: A new method for hardness-randomness trade-offs. In P. Degano, R. Gorrieri, and A. Marchetti-Spaccamela, editors, Automata, Languages and Programming, 24th International Colloquium, volume 1256 of Lecture Notes in Computer Science, pages 177-187, Bologna, Italy, 7-11 July 1997. Springer-Verlag.
4. A. E. ANDREEV, A. E. F. CLEMENTI, J. D. P. ROLIM, and L. TREVISAN. Weak random sources, hitting sets, and BPP simulations. SIAM Journal on Computing, 28(6):2103-2116 (electronic), 1999.
5. L. BABAI, L. FORTNOW, and C. LUND. Nondeterministic exponential time has two-prover interactive protocols. Computational Complexity, 1(1):3-40, 1991.
6. L. BABAI, L. FORTNOW, N. NISAN, and A. WIGDERSON. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Computational Complexity, 3(4):307-318, 1993.
7. L. BABAI, L. FORTNOW, L. LEVIN, and M. SZEGEDY. Checking computations in poly logarithmic time. In Proceedings of the Twenty Third Annual ACM Symposium on Theory of Computing, pages 21-31, New Orleans, Louisiana, 6-8 May 1991.
8. B. BARAK. A probabilistic-time hierarchy theorem for "slightly nonuniform" algorithms. In Randomization and approximation techniques in computer science, volume 2483 of Lecture Notes in Comput. Sci, pages 194-208. Springer, Berlin, 2002.
9. D. BEAVER and J. FEIGENBAUM. Hiding instances in multioracle queries. In 7th Annual Symposium on Theoretical Aspects of Computer Science, volume 415 of Lecture Notes in Computer Science, pages 37-48, Rouen, France, 22-24 February 1990. Springer.
10. V. M. BLINOVSKY. Bounds for codes in the case of list decoding of finite volume. Problems of Information Transmission, 22(1):7-19, 1986.
11. M. BLUM, M. LUBY, and R. RUBINFELD. Self-testing/correcting with applications to numerical problems. Journal of Computer and System Sciences, 47(3):549-595, 1993.
12. M. BLUM and S. KANNAN. Designing programs that check their work. Journal of the ACM, 42(1):269-291, 1995.
13. M. BLUM and S. MICALI. HOW to generate cryptographically strong sequences of pseudo-random bits. SIAM Journal on Computing, 13(4):850-864, November 1984.
14. H. BUHRMAN, L. FORTNOW, D. VAN MELKEBEEK, and L. TORENVLIET. Using autoreducibility to separate complexity classes. SIAM Journal on Computing, 29:1497-1520, 2000.
15. J.-Y. CAI, A. NERURKAR, and D. SIVAKUMAR. Hardness and hierarchy theorems for probabilistic quasi-polynomial time. In Annual ACM Symposium on Theory of Computing (Atlanta, GA, 1999), pages 726-735 (electronic). ACM, New York, 1999.
16. J.-Y. CAI, A. PAVAN, and D. SIVAKUMAR. On the hardness of the permanent. In 16th International Symposium on Theoretical Aspects of Computer Science, Lecture Notes in Computer Science, Trier, Germany, March 4-6 1999. Springer-Verlag.
17. U. FEIGE, S. GOLDWASSER, L. LOVÁSZ, S. SAFRA, and M. SZEGEDY. Interactive proofs and the hardness of approximating cliques. Journal of the ACM, 43(2):268-292, 1996.
18. J. FEIGENBAUM, S. KANNAN, and N. NISAN. Lower bounds on random-self-reducibility. In Proceedings of the 5th IEEE Conference on Structure in Complexity Theory, pages 100-109, 1990.
19. J. FEIGENBAUM and L. FORTNOW. Random-self-reducibility of complete sets. SIAM Journal on Computing, 22(5):994-1005, October 1993.
20. L. FORTNOW. Comparing notions of full derandomization. In Proceedings of the Sixteenth Annual Conference on Computational Complexity, pages 28-34. IEEE, June 18-21 2001.
21. L. FORTNOW and R. SANTHANAM. Hierarchy theorems for probabilistic polynomial time. In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, pages 316-324, 2004.
22. L. FORTNOW, R. SANTHANAM, and L. TREVISAN. Hierarchies for semantic classes. In Proceedings of the 37th ACM Symposium on Theory of Computing, pages 348-355, 2005.
23. O. GOLDREICH. On promise problems (in memory of Shimon Even, 1935-2004). Technical Report TR05-018, Electronic Colloquium on Computational Complexity, 2005. See June 2005 revision, available from author's homepage.
24. O. GOLDREICH, D. RON, and M. SUDAN. Chinese remaindering with errors. In Proceedings of the 31st ACM Symposium on Theory of Computing, pages 225-234, 1999.
25. O. GOLDREICH, M. SUDAN, and L. TREVISAN. From logarithmic advice to single-bit advice. Technical Report TR04-093, ECCC, 2004.
26. O. GOLDREICH and D. ZUCKERMAN. Another proof that BPP C PH (and more). Electronic Colloquium on Computational Complexity Technical Report TR97-045, September 1997. http://www.eccc.uni-trier.de/eccc.
27. V. GURUSWAMI and S. VADHAN. A lower bound on list size for list decoding. In C. Chekuri, K. Jansen, J. D. P. Rolim, and L. Trevisan, editors, Proceedings of the 8th International Workshop on Randomization and Computation (RANDOM '05), number 3624 in Lecture Notes in Computer Science, pages 318-329, Berkeley, CA, August 2005. Springer.
28. D. GUTFREUND, R. SHALTIEL, and A. TA- SHMA. Uniform hardness versus randomness tradeoffs for Arthur-Merlin games. Computational Complexity, 12(3-4):85-130, 2003.
29. J. HÅSTAD, R. IMPAGLIAZZO, L. A. LEVIN, and M. LUBY. A pseudorandom generator from any one-way function. SIAM Journal on Computing, 28(4): 1364-1396 (electronic), 1999.
30. R. IMPAGLIAZZO. Hard-core distributions for somewhat hard problems. In 36th Annual Symposium on Foundations of Computer Science, pages 538-545, Milwaukee, Wisconsin, 23-25 October 1995. IEEE.
31. R. IMPAGLIAZZO. A personal view of average-case complexity. In Proceedings of the 10th IEEE Conference on Structure in Complexity Theory, pages 134-147, 1995.
32. R. IMPAGLIAZZO, V. KABANETS, and A. WIGDERSON. In search of an easy witness: exponential time vs. probabilistic polynomial time. Journal of Computer and System Sciences, 65(4):672-694, 2002.
33. R. IMPAGLIAZZO, R. SHALTIEL, and A. WIGDERSON. Extractors and pseudo-random generators with optimal seed length. In Proceedings of the Thirty-second Annual ACM Symposium on Theory of Computing, pages 1-10, Portland, Oregon, May 2000. See also ECCC TR00-009.
34. R. IMPAGLIAZZO and A. WIGDERSON. P = BPP if E requires exponential circuits: Derandomizing the XOR lemma. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 220-229, El Paso, Texas, 4-6 May 1997.
35. R. IMPAGLIAZZO and A. WIGDERSON. Randomness vs. time: Derandomization under a uniform assumption. In 36th Annual Symposium on Foundations of Computer Science, Palo Alto, CA, November 8-111998. IEEE.
36. V. KABANETS. Easiness assumptions and hardness tests: trading time for zero error. Journal of Computer and System Sciences, 63(2):236-252, 2001.
37. R. M. KARP and R. J. LIPTON. Turing machines that take advice. L'Enseignement Mathématique. Revue Internationale. Ile Série, 28(3-4):191-209, 1982.
38. R. LIPTON. New directions in testing. In Proceedings of DIMACS Workshop on Distributed Computing and Cryptography, 1989.
39. C.-J. LU. Derandomizing Arthur-Merlin games under uniform assumptions. In Algorithms and computation (Taipei, 2000), pages 302-312. Springer, Berlin, 2000.
40. C.-J. LU, S.-C TSAI, and H.-L. WU. On the complexity of hardness amplification. In Proceedings of the 20th Annual IEEE Conference on Computational Complexity (CCC '05), pages 170-182, San Jose, 11-15 June 2005. IEEE.
41. C. LUND, L. FORTNOW, H. KARLOFF, and N. NISAN. Algebraic methods for interactive proof systems. Journal of the ACM, 39(4):859-868, October 1992.
42. D. VAN MELKEBEEK and K. PERVYSHEV. A generic time hierarchy for semantic models with one bit of advice. In Proceedings of the 21st Annual IEEE Conference on Computational Complexity, pages 129-142, 2006.
43. P. B. MILTERSEN and N. V. VINODCHANDRAN. Derandomizing Arthur-Merlin games using hitting sets. In 40th Annual Symposium on Foundations of Computer Science (New York, 1999), pages 71-80. IEEE Computer Soc., Los Alamitos, CA, 1999.
44. P. B. MILTERSEN, N. V. VINODCHANDRAN, and O. WATANABE. Superpolynomial versus half-exponential circuit size in the exponential hierarchy. In Computing and combinatorics (Tokyo, 1999), pages 210-220. Springer, Berlin, 1999.
45. N. NISAN and A. WIGDERSON. Hardness vs randomness. Journal of Computer and System Sciences, 49(2): 149-167, October 1994.
46. N. NISAN and D. ZUCKERMAN. Randomness is linear in space. Journal of Computer and System Sciences, 52(1):43-52, February 1996.
47. J. RADHAKRISHNAN and A. TA-SHMA. Bounds for dispersers, extractors, and depth-two superconcentrators. SIAM Journal on Discrete Mathematics, 13(1):2-24 (electronic), 2000.
48. O. REINGOLD, L. TREVISAN, and S. VADHAN. Notions of reducibility between cryptographic primitives. In M. Naor, editor, Proceedings of the First Theory of Cryptography Conference (TCC '04), volume 2951 of Lecture Notes in Computer Science, pages 1-20. Springer-Verlag, 19-21 February 2004.
49. R. SHALTIEL. Recent developments in extractors. In G. Paun, G. Rozenberg, and A. Salomaa, editors, Current Trends in Theoretical Computer Science, volume 1: Algorithms and Complexity. World Scientific, 2004.
50. R. SHALTIEL and C. UMANS. Simple extractors for all min-entropies and a new pseudo-random generator. Journal of the ACM, 52(2):172-216, 2005.
51. A. SHAMIR. IP = PSPACE. Journal of the ACM, 39(4):869-877, October 1992.
52. V. SHOUP. New algorithms for finding irreducible polynomials over finite fields. Mathematics of Computation, 54(189):435-447, 1990.
53. M. SIPSER. Introduction to the Theory of Computation. Course Technology, 2nd edition, 2005.
54. M. SUDAN, L. TREVISAN, and S. VADHAN. Pseudorandom generators without the XOR lemma. Journal of Computer and System Sciences, 62:236-266, 2001.
55. S. TODA. PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20(5):865-877, 1991.
56. L. TREVISAN. Extractors and pseudorandom generators. Journal of the ACM, 48(4):860-879 (electronic), 2001.
57. L. TREVISAN. List decoding using the XOR lemma. In Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, pages 126-135, Cambridge, MA, October 2003.
58. L. TREVISAN and S. VADHAN. Pseudorandomness and average-case complexity via uniform reductions. In Proceedings of the 17th Annual IEEE Conference on Computational Complexity (CCC '02), pages 129-138, Montréal, CA, May 2002. IEEE.
59. C. UMANS. Pseudo-random generators for all hardnesses. Journal of Computer and System Sciences, 67(2):419-440, 2003.
60. L. G. VALIANT. The complexity of computing the permanent. Theoretical Computer Science, 8(2): 189-201, 1979.
61. exp ⊈ (NP ∩ coNP)/poly. Information Processing Letters, 89(1):43-47, 2004.
62. E. VIOLA. The complexity of constructing pseudorandom generators from hard functions. Computational Complexity, 13(3-4): 147-188, 2004.
63. A. C. YAO. Theory and applications of trapdoor functions (extended abstract). In 23rd Annual Symposium on Foundations of Computer Science, pages 80-91, Chicago, Illinois, 3-5 November 1982. IEEE.