Power from random strings

SIAM Journal on Computing

Volumn 35, Issue 6, 2006, Pages 1467-1493

  Allender, Eric ^a   Buhrman, Harry ^b   Koucký, Michal ^c   Van Melkebeek, Dieter ^d   Ronneburger, Detlef ^e  

^a RUTGERS UNIVERSITY   (United States)

^b UNIVERSITY OF AMSTERDAM   (Netherlands)

^c MCGILL UNIVERSITY   (Canada)

^d UNIVERSITY OF WISCONSIN MADISON   (United States)

^e CITY UNIVERSITY OF NEW YORK   (United States)

Author keywords

Completeness; Derandomization; Kolmogorov complexity; Randomness; Reductions

Indexed keywords

DERANDOMIZATION; KOLMOGOROV COMPLEXITY; RANDOMNESS; REDUCTIONS;

CRYSTAL LATTICES; PROBLEM SOLVING; SET THEORY;

COMPUTATIONAL COMPLEXITY;

EID: 33845570500     PISSN: 00975397     EISSN: None     Source Type: Journal    
DOI: 10.1137/050628994     Document Type: Article

References (63)

1. M. AGRAWAL, The first-order isomorphism theorem, in FST TCS 2001: Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Comput. Sci. 2245, Springer-Verlag, Berlin, 2001, pp. 70-82.
2. M. AGRAWAL, E. ALLENDER, R. IMPAGLIAZZO, R. PITASSI, AND S. RUDICH, Reducing the complexity of reductions, Comput. Complexity, 10 (2001), pp. 117-138.
3. M. AGRAWAL, N. KAYAL, AND N. SAXENA, PRIMES is in P, Ann. of Math. (2), 160 (2004), pp. 781-793.
4. M. AJTAI, Generating hard instances of lattice problems (extended abstract), in Proceedings of the ACM Symposium on Theory of Computing (STOC), ACM, New York, 1996, pp. 99-108.
5. M. AJTAI, Generating hard instances of lattice problems, in Complexity of Computations and Proofs, J. Krajíčiek, ed., Quad. Mat. 13, Aracne, Rome, 2004, pp. 1-32.
6. E. ALLENDER, When worlds collide: Derandomization, lower bounds, and Kolmogorov complexity, in FST TCS 2001: Conference on Foundations of Software Technology and Theoretical Computer Science, Lecture Notes in Comput. Sci. 2245, Springer-Verlag, Berlin, 2001, pp. 1-15.
7. E. ALLENDER, NL-printable sets and nondeterministic Kolmogorov complexity, Theoret. Comput. Sci., 355 (2006), pp. 127-138.
8. E. ALLENDER, H. BUHRMAN, AND M. KOUCKÝ, What can be efficiently reduced to the Kolmogorov random strings? Ann. Pure Appl. Logic, 138 (2006), pp. 2-19.
9. E. ALLENDER, H. BUHRMAN, M. KOUCKÝ, D. VAN MELKEBEEK, AND D. RONNEBURGER, Power from random strings, in Proceedings of the 43rd IEEE Annual Symposium on Foundations of Computer Science, IEEE Computer Society, Los Alamitos, CA, 2002, pp. 669-678.
10. E. ALLENDER AND V. GORE, Rudimentary reductions revisited, Inform. Process. Lett., 40 (1991), pp. 89-95.
11. E. ALLENDER, M. KOUCKÝ, D. RONNEBURGER, AND S. ROY, Derandomization and distinguishing complexity, in Proceedings of the 18th IEEE Conference on Computational Complexity, IEEE Computer Society, Los Alamitos, CA, 2003, pp. 209-220.
12. L. ANTUNES, L. FORTNOW, AND D. VAN MELKEBEEK, Computational depth, in Proceedings of the IEEE Conference on Computational Complexity, IEEE Computer Society, Los Alamitos, CA, 2001, pp. 266-273.
13. L. BABAI, L. FORTNOW, AND C. LUND, Non-deterministic exponential time has two-prover interactive protocols, Comput. Complexity, 1 (1991), pp. 3-40.
14. L. BABAI, L. FORTNOW, N. NISAN, AND A. WIGDERSON, BPP has subexponential time simulations unless EXPTIME has publishable proofs, Comput. Complexity, 3 (1993), pp. 307-318.
15. 1, J. Comput. System Sci., 41 (1990), pp. 274-306.
16. A. BEN-AMRAM AND N. JONES, Computational complexity via programming languages: Constant/actors do matter, Acta Inform., 37 (2000), pp. 83-120.
17. P. BERMAN, Relationship between density and deterministic complexity of NP-complete languages, in ICALP, Lecture Notes in Comput. Sci. 62, Springer-Verlag, Berlin, 1978, pp. 63-71.
18. M. SLUM AND S. MIOALI, How to generate cryptographically strong sequences of pseudo-random bits, SIAM J. Comput., 13 (1984), pp. 850-864.
19. H. BUHRMAN, L. FORTNOW, I. NEWMAN, AND N. VERESHCHAGIN, Increasing Kolmogorov complexity, in Proceedings of the Symposium on Theoretical Aspects of Computer Science (STAGS), Lecture Notes in Comput. Sci. 3404, Springer-Verlag, Berlin, 2005, pp. 412-421.
20. H. BUHRMAN AND E. MAYORDOMO, An excursion to the Kolmogorov random strings, J. Comput. System Sci., 54 (1997), pp. 393-399.
21. H. BUHRMAN AND L. TORENVLIRT, Randomness is hard, SIAM J. Comput., 30 (2000), pp. 1485-1501.
22. J.-Y. CAI, Some recent progress on the complexity of lattice problems, in Proceedings of the IEEE Conference on Computational Complexity, IEEE Computer Society, Los Alamitos, CA, 1999, pp. 158-179.
23. J.-Y. CAI, On the average-case hardness of CVP, in Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, Los Alamitos, CA, 2001, pp. 308-319.
24. J.-Y. CAI AND A. NERURKAR, An improved worst-case to average-case connection for lattice problems, in Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, Los Alamitos, CA, 1997, pp. 468-477.
25. J. FEIGENBAUM AND L. FORTNOW. Random-self-reducibility of complete sets, SIAM J. Comput., 22 (1993), pp. 994-1005.
26. M. FÜRER, The tight deterministic time hierarchy, in Proceedings of the ACM Symposium on Theory of Computing (STOC), ACM, New York, 1982, pp. 8-16.
27. M. FÜRER, Data structures for distributed counting, J. Comput. System Sci., 28 (1984), pp. 231-243.
28. J. GESKE, D. HUYNH, AND J. SEIFERAS, A note on almost-everywhere-complex sets and separating deterministic-time-complexity classes, Inform. Comput., 92 (1991), pp. 97-104.
29. O. GOLDREICH, S. GOLDWASSER, AND S. MICALI, How to construct random functions, J. ACM, 33 (1986), pp. 792-807.
30. J. HÅSTAD, R. IMPAGLIAZZO, L. A. LEVIN, AND M. LUBY, A pseudorandom generator from any one-way function, SIAM J. Comput., 28 (1999), pp. 1364-1396.
31. F. HENNIE AND R. STEARNS, Two-tape simulation of multitape Turing machines, J. ACM, 13 (1966), pp. 533-546.
32. R. IMPAGLIAZZO, V. KABANETS, AND A. WIGDERSON, In search of an easy witness: Exponential time vs. probabilistic polynomial time, J. Comput. System Sci., 65 (2002), pp. 672-694.
33. R. IMPAGLIAZZO AND A. WIGDERSON, P = BPP if E requires exponential circuits: Derandomizing the XOR lemma, in Proceedings of the ACM Symposium on Theory of Computing (STOC), ACM, New York, 1997, pp. 220-229.
34. R. IMPAGLIAZZO AND A. WIGDERSON, Randomness vs. time: De-randomization under a uniform ansumption, J. Comput. System Sci., 63 (2001), pp. 672-688.
35. V. KABANETS, Easiness assumptions and hardness tests: Trading time for zero error, J. Cornput. System Sci., 63 (2001), pp. 236-252.
36. V. KABANETS AND J.-Y. CAI, Circuit minimization problem, in Proceedings of the ACM Symposium on Theory of Computing (STOC), ACM, New York, 2000, pp. 73-79.
37. A. R. KLIVANS AND D. VAN MELKEBEEK, Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses, SIAM J. Comput., 31 (2002), pp. 1501-1526.
38. K.-I. KO, On the complexity of learning minimum time-bounded Turing machines, SIAM J. Comput., 20 (1991), pp. 962-986.
39. M. KOUCKÝ, On Traversal Sequences, Exploration Sequences, and Completeness of Kolmogorov Random Strings, Ph.D. thesis, Rutgers University, New Brunswick, NJ, 2003.
40. M. KUMMER, On the complexity of random strings, in Proceedings of the Symposium on Theoretical Aspects of Computer Science (STAGS), Lecture Notes in Comput. Sci. 1046, Springer-Verlag, Berlin, 1996, pp. 25-36.
41. R. LADNER, On the structure of polynomial time reducibility, J. ACM, 22 (1975), pp. 155-171.
42. R. E. LADNER AND N. A. LYNCH, Relativization of questions about log space computability, Math. Systems Theory, 10 (1976), pp. 19-32.
43. L. LEVIN, Randomness conservation inequalities: Information and independence in mathematical theories, Inform, and Control, 61 (1984), pp. 15-37.
44. M. LI AND P. VITANYI, Introduction to Kolmogorov Complexity and Its Applications, Springer-Verlag, New York, 1993.
45. L. LONGPRÉ, Resource Bounded Kolmogorov Complexity, A Link between Computational Complexity and Information Theory, Ph.D. thesis, Cornell University, Ithaca, New York, 1986.
46. D. MARTIN, Completeness, the recursion theorem and effectively simple sets, Proc. Amer. Math. Soc., 17 (1966), pp. 838-842.
47. D. MICCIANCIO, Improved cryptographic hash functions with worst-case/average-case connection, in Proceedings of the ACM Symposium on Theory of Computing (STOC), ACM, New York, 2002, pp. 609-618.
48. N. NISAN, Pseudorandom bits for constant depth circuits, Combinatorica, 11 (1991), pp. 63-70.
49. N. NISAN AND A. WIGDERSON, Hardness vs. randomness, J. Comput. System Sci., 49 (1994), pp. 149-167.
50. M. RABIN, Digitalized Signatures and Public-Key Functions as Intractible as Factorization, MIT Tech. report TR-212, MIT, Cambridge, MA, 1979.
51. A. RAZBOROV AND S. RUDICH, Natural proofs, J. Comput. System Sci., 55 (1997), pp. 24-35.
52. D. RONNEBURGER, Kolmogorov Complexity and Derandomization, Ph.D. thesis, Rutgers University, New Brunswick, NJ, 2004.
53. S. RUDICH, Super-bits, demi-bits, and NP/qpoly-natural proofs, in RANDOM, Lecture Notes in Comput. Sci. 1269, Springer-Verlag, Berlin, 1997.
54. M. SIPSER, A complexity theoretic approach to randomness, in Proceedings of the ACM Symposium on Theory of Computing (STOC), ACM, New York, 1983, pp. 330-335.
55. M. SUDAN, L. TREVISAN, AND S. VADHAN, Pseudorandom generators without the XOR lemma, J. Comput. System Sci., 62 (2001), pp. 236-266.
56. L. TREVISAN, Construction of extractors using pseudo-random generators, J. ACM, 48 (2001), pp. 860-879.
57. L. TREVISAN AND S. VADHAN, Pseudorandomness and average-case complexity via uniform reductions, in Proceedings of the IEEE Conference on Computational Complexity, IEEE Computer Society, Los Alamitos, CA, 2002, pp. 129-138.
58. U. VAZIRANI AND V. VAZIRANI, A natural encoding scheme proved probabilistic polynomial complete, Theoret. Comput. Sci., 24 (1983), pp. 291-300.
59. E. VIOLA, The complexity of constructing pseudorandom generators from hard functions, Comput. Complexity, 13 (2004), pp. 147-188.
60. H. VOLLMER, Introduction to Circuit Complexity, Springer-Verlag, New York, 1999.
61. O. WATANABE, A comparison of polynomial time completeness notions, Theoret. Comput. Sci., 54 (1987), pp. 249-265.
62. O. WATANABE AND S. TANG, On polynomial-time Turing and many-one completeness in PSPACE, Theoret. Comput. Sci., 97 (1992), pp. 199-215.
63. A. YAO, Theory and applications of trapdoor functions, in Proceedings of the IEEE Symposium on Foundations of Computer Science (FOCS), IEEE Computer Society, Los Alamitos, CA, 1982, pp. 80-91.