Pseudorandomness and combinatorial constructions

International Congress of Mathematicians, ICM 2006

Volumn 3, Issue , 2006, Pages 1111-1136

  Trevisan, Luca ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

Computational complexity; Derandomization; Pseudorandomness; Randomness extraction

Indexed keywords

COMPUTATIONAL THEORIES; DERANDOMIZATION; DETERMINISTIC ALGORITHMS; DETERMINISTIC SIMULATION; EXPLICIT CONSTRUCTIONS; PROBABILISTIC ALGORITHM; PSEUDORANDOMNESS; RANDOMNESS EXTRACTIONS;

ALGORITHMS; COMPUTATIONAL COMPLEXITY;

RANDOM PROCESSES;

EID: 84874147333     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (87)

1. Agrawal, Manindra, Kayal, Neeraj, and Saxena, Nitin, PRIMES is in P. Ann. of Math. 160 (2) (2004), 781-793.
2. Aleliunas, Romas, Karp, Richard M., Lipton, Richard J., Lovász, László, and Rackoff, Charles, Random walks, universal traversal sequences, and the complexity of maze problems. In Proceedings of the 20th IEEE Symposium on Foundations of Computer Science, IEEE, New York 1979, 218-223.
3. Alon, Noga, and Capalbo, Michael R., Explicit unique-neighbor expanders. In Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, IEEE, New York 2002, 73-79.
4. Alon, Noga, and Spencer, Joel, The Probabilistic Method. Wiley-Intersci. Ser. Discrete Math. Optim., JohnWiley and Sons, New York 2000.
5. Arora, Sanjeev, and Sudan, Madhu, Improved low degree testing and its applications. Combinatorica 23 (3) (2003), 365-426.
6. Babai, László, Fortnow, Lance, Nisan, Noam, and Wigderson, Avi, BPP has subexponential time simulations unless EXPTIME has publishable proofs. Comput. Complexity 3 (4) (1993), 307-318.
7. Barak, Boaz, Impagliazzo, Russell, and Wigderson, Avi, Extracting randomness using few independent sources. In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, IEEE, New York 2004, 384-393.
8. Barak, Boaz, Kindler, Guy, Shaltiel, Ronen, Sudakov, Benny, and Wigderson, Avi, Simulating independence: new constructions of condensers, Ramsey graphs, dispersers, and extractors. In Proceedings of the 37th ACM Symposium on Theory of Computing, ACM Press, New York 2005, 1-10.
9. Blum, Manuel, and Micali, Silvio, How to generate cryptographically strong sequences of pseudorandom bits. SIAM J. Comput. 13 (4) (1984), 850-864.
10. Bogdanov, Andrej, and Trevisan, Luca, On wost-case to average-case reductions for NP problems. In Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, IEEE, New York 2003, 308-317.
11. Bourgain, Jean, More on the sum-product phenomenon in prime fields and its applications. Int. J. Number Theory 1 (1) (2005), 1-32.
12. Bourgain, Jean, Katz, Nets, and Tao, Terence, A sum-product estimate for finite fields, and applications. Geom. Funct. Anal. 14 (2004), 27-57.
13. Capalbo, Michael R., Reingold, Omer, Vadhan, Salil P., and Wigderson, Avi, Randomness conductors and constant-degree lossless expanders. In Proceedings of the 34th ACM Symposium on Theory of Computing, ACM Press, New York 2002, 659-668.
14. Chor, Benny, and Goldreich, Oded, Unbiased bits from sources of weak randomness and probabilistic communication complexity. SIAM J. Comput. 17 (2) (1988), 230-261.
15. Chung, Fan R. K., Graham, Ronald L., and Wilson, Richard M., Quasi-random graphs. Combinatorica 9 (4) (1989), 345-362.
16. Cohen, Aviad, and Wigderson, Avi, Dispersers, deterministic amplification, and weak random sources. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Science, IEEE, New York 1989, 14-19.
17. Elias, Peter, List decoding for noisy channels. Technical Report 335, Research Laboratory of Electronics, MIT, 1957.
18. Erdo″s, Paul, Some remarks on the theory of graphs. Bull. Amer. Math. Soc. 53 (1947), 292-294.
19. Feigenbaum, Joan, and Fortnow, Lance, Random-self-reducibility of complete sets. SIAM J. Comput. 22 (1993), 994-1005.
20. Frankl, Peter, and Wilson, Richard M., Intersection theoremswith geometric consequences. Combinatorica 1 (4) (1981), 357-368.
21. Goldreich, Oded, Foundations of Cryptography. Volume 1, Cambridge University Press, Cambridge 2001.
22. Goldwasser, Shafi, and Micali, Silvio, Probabilistic encryption. J. Comput. System Sci. 28 (2) (1984), 270-299.
23. Guruswami, Venkatesan, List Decoding of Error-Correcting Codes. PhD thesis, MIT, 2001.
24. Guruswami, Venkatesan, and Rudra, Atri, Explicit capacity-achieving list-decodable codes. Technical Report TR05-133, Electronic Colloquium on Computational Complexity, 2005.
25. Hamming, Richard, Error detecting and error correcting codes. Bell System Tech. J. 29 (1950), 147-160.
26. Håstad, Johan, Impagliazzo, Russell, Levin, Leonid, and Luby, Michael, A pseudorandom generator from any one-way function. SIAM J. Comput. 28 (4) (1999), 1364-1396.
27. Impagliazzo, Russell, Hard-core distributions for somewhat hard problems. In Proceedings of the 36th IEEE Symposium on Foundations of Computer Science, IEEE, New York 1995, 538-545.
28. Impagliazzo, Russell, Hardness as randomness: a survey of universal derandomization. Proceedings of the International Congress of Mathematicians (Beijing, 2002), Vol. III, Higher Ed. Press, Beijing 2002, 659-672.
29. Impagliazzo, Russell, Kabanets, Valentine, and Wigderson, Avi, In search of an easy witness: exponential time vs. probabilistic polynomial time. J. Comput. System Sci. 65 (4) (2002), 672-694.
30. Impagliazzo, Russell, Nisan, Noam, and Wigderson, Avi, Pseudorandomness for network algorithms. In Proceedings of the 26th ACM Symposium on Theory of Computing, ACM Press, New York 1994, 356-364.
31. Impagliazzo, Russell, and Wigderson, Avi, P = BPP unless E has sub-exponential circuits. In Proceedings of the 29th ACM Symposium on Theory of Computing, ACM Press, New York 1997, 220-229.
32. Iwama, Kazuo, and Morizumi, Hiroki, An explicit lower bound of 5n - o(n) for boolean circuits. In Proceedings of the 27th Symposium on Mathematical Foundations of Computer Science, Lecture Notes in Comput. Sci. 2420, Springer-Verlag, London 2002, 353-364.
33. Jerrum, Mark, Sinclair, Alistair, and Vigoda, Eric, A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM 51 (4) (2004), 671-697.
34. Kabanets, Valentine, and Impagliazzo, Russell, Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complexity 13 (1-2) (2004), 1-46.
35. Klivans, Adam, and van Melkebeek, Dieter, Graph nonisomorphism has subexponential size proofs unless the polynomial-time hierarchy collapses. SIAM J. Comput. 31 (5) (2002), 1501-1526.
36. Konyagin, Sergei, A sum-product estimate in fields of prime order. math.NT/0304217, 2003.
37. Krivelevich, Michael, and Sudakov, Benny, Pseudo-random graphs. Preprint, 2005.
38. Lachish, Oded, and Raz, Ran, Explicit lower bound of 4.5n - o(n) for boolean circuits. In Proceedings of the 33rd ACM Symposium on Theory of Computing, ACM Press, New York 2001, 399-408.
39. Lu, Chi-Jen, Encryption against storage-bounded adversaries from on-line strong extractors. J. Cryptology 17 (1) (2004), 27-42.
40. Lu, Chi-Jen, Reingold, Omer, Vadhana, Salil P., and Wigderson, Avi, Extractors: optimal up to constant factors. In Proceedings of the 35th ACM Symposium on Theory of Computing, ACM Press, New York, 602-611.
41. Lubotzky, Alexander, Phillips, R., and Sarnak, Peter, Ramanujan graphs. Combinatorica 8 (1988), 261-277.
42. Luby, Michael, andVelickovic, Boban, On deterministic approximation of DNF. Algorithmica 16 (4/5) (1996), 415-433.
43. Miltersen, Peter B., andVinodchandran, N.V., Derandomizing Arthur-Merlin games using hitting sets. In Proceedings of the 40th IEEE Symposium on Foundations of Computer Science, IEEE, New York 1999, 71-80.
44. Nisan, Noam, and Wigderson, Avi, Hardness vs randomness. J. Comput. System Sci. 49 (1994), 149-167.
45. Nisan, Noam, and Zuckerman, David, Randomness is linear in space. J. Comput. System Sci. 52 (1) (1996), 43-52.
46. Parvaresh, Farzad, and Vardy, Alexander, Correcting errors beyond the Guruswami-Sudan radius in polynomial time. In Proceedings of the 46th IEEE Symposium on Foundations of Computer Science, IEEE, New York 2005, 285-294.
47. Rabin, Michael, Probabilistic algorithm for testing primality. J. Number Theory 12 (1980), 128-138.
48. Rao, Anup, Extractors for a constant number of polynomially small min-entropy independent sources. In Proceedings of the 38th ACM Symposium on Theory of Computing, ACM Press, New York 2006, 497-506.
49. Raz, Ran, Multi-linear formulas for permanent and determinant are of super-polynomial size. In Proceedings of the 36th ACM Symposium on Theory of Computing, ACM Press, New York 2004, 633-641.
50. 2. In Proceedings of the 45th IEEE Symposium on Foundations of Computer Science, IEEE, New York 2004, 344-351.
51. Raz, Ran, and Reingold, Omer, On recycling randomness in space bounded computation. In Proceedings of the 31st ACM Symposium on Theory of Computing, ACM Press, New York 1999, 159-168.
52. Razborov, Alexander A., and Rudich, Steven, Natural proofs. J. Comput. System Sci. 55 (1) (1997), 24-35.
53. Reingold, Omer, Undirected ST-connectivity in log-space. In Proceedings of the 37th ACM Symposium on Theory of Computing, ACM Press, New York 2005, 376-385.
54. Reingold, Omer, Shaltiel, Ronen, and Wigderson, Avi, Extracting randomness by repeated condensing. In Proceedings of the 41st IEEE Symposium on Foundations of Computer Science, IEEE, New York 2000, 22-31.
55. Santha, Miklos, and Vazirani, Umesh, Generating quasi-random sequences from slightly random sources. J. Comput. System Sci. 33 (1986), 75-87.
56. Schwartz, Jacob T., Fast probabilistic algorithms for verification of polynomial identities. J. ACM 27 (1980), 701-717.
57. Shaltiel, Ronen, Recent developments in explicit constructions of extractors. Bull. Eur. Assoc. Theor. Comput. Sci. EATCS 77 (2002), 67-95.
58. Shaltiel, Ronen, and Umans, Christopher, Simple extractors for all min-entropies and a new pseudorandom generator. J. ACM 52 (2) (2005), 172-216.
59. Shannon, Claude, A mathematical theory of communications. Bell System Tech. J. 27 (1948), 379-423, 623-656, 1948.
60. Shannon, Claude, The synthesis of two-terminal switching circuits. Bell System Tech. J. 28 (1949), 59-98.
61. Sinclair, Alistair, and Jerrum, Mark, Approximate counting, uniform generation and rapidly mixing Markov chains. Inform. and Comput. 82 (1) (1989), 93-133.
62. Sipser, Michael, Borel sets and circuit complexity. In Proceedings of the 15th ACM Symposium on Theory of Computing, ACM Press, New York 1983, 61-69.
63. Sipser, Michael, Atopological viewof some problems in complexity theory. In Proceedings of the Symposium on Mathematical Foundations of Computer Science, IEEE, New York 1984, 567-572.
64. Sipser, Michael, Introduction to the Theory of Computation. PWS Publishing Co., Boston, MA, 1997.
65. Solovay, Robert, and Strassen, Volker, A fast Monte-Carlo test for primality. SIAM J. Comput. 6 (1) (1977), 84-85.
66. Sudan, Madhu, Decoding of Reed-Solomon codes beyond the error-correction bound. J. Complexity 13 (1) (1997), 180-193.
67. Sudan, Madhu, List decoding: Algorithms and applications. SIGACT News 31 (1) (2000), 16-27.
68. Sudan, Madhu, Trevisan, Luca, and Vadhan, Salil, Pseudorandom generators without the XOR lemma. J. Comput. System Sci. 62 (2) (2001), 236-266, .
69. Ta-Shma, Amnon, Umans, Christopher, and Zuckerman, David, Loss-less condensers, unbalanced expanders, and extractors. In Proceedings of the 33rd ACMSymposium on Theory of Computing, ACM Press, New York 2001, 143-152.
70. Trevisan, Luca, Extractors and pseudorandom generators. J. ACM 48 (4) (2001), 860-879.
71. Trevisan, Luca, List-decoding using the XOR Lemma. In Proceedings of the 44th IEEE Symposium on Foundations of Computer Science, IEEE, New York 2003, 126-135.
72. Trevisan, Luca, Some applications of coding theory in computational complexity. Quad. Mat. 13 (2004), 347-424.
73. Trevisan, Luca, On uniform amplification of hardness in NP. In Proceedings of the 37th ACM Symposium on Theory of Computing, ACM Press, New York 2005, 31-38.
74. Trevisan, Luca, and Vadhan, Salil, Pseudorandomness and average-case complexity via uniform reductions. In Proceedings of the 17th IEEE Conference on Computational Complexity, IEEE, New York 2002, 129-138.
75. Umans, Christopher, Pseudo-random generators for all hardnesses. J. Comput. System Sci. 67 (2) (2003), 419-440.
76. Vadhan, Salil, Randomness extractors and their many guises. In Proceedings of the 43rd IEEE Symposium on Foundations of Computer Science, IEEE, New York 2002, 9-10.
77. Vadhan, Salil P., Constructing locally computable extractors and cryptosystems in the bounded-storage model. J. Cryptology 17 (1) (2004), 43-77.
78. Vazirani, Umesh, Randomness, Adversaries and Computation. PhD thesis, University of California, Berkeley, 1986.
79. Vazirani, Umesh, andVazirani, Vijay, Random polynomial time is equal to slightly random polynomial time. In Proceedings of the 26th IEEE Symposium on Foundations of Computer Science, IEEE, New York 1985, 417-428.
80. Vazirani, Umesh V., Strong communication complexity or generating quasirandom sequences form two communicating semi-random sources. Combinatorica 7 (4) (1987), 375-392.
81. Viola, Emanuele, The complexity of constructing pseudorandom generators from hard functions. Comput. Complexity 13 (3-4) (2004), 147-188.
82. von Neumann, John, Various techniques used in connection with random digits. J. Res. Nat. Bur. Standards App. Math. Ser. 12 (1951), 36-38.
83. Wigderson, Avi, and Zuckerman, David, Expanders that beat the eigenvalue bound: Explicit construction and applications. Combinatorica 19 (1) (1999), 125-138.
84. Yao, Andrew C., Theory and applications of trapdoor functions. In Proceedings of the 23th IEEE Symposium on Foundations of Computer Science, IEEE, New York 1982, 80-91.
85. Zippel, Richard, Probabilistic algorithms for sparse polynomials. In Symbolic and algebraic computation (ed. by Edward W. Ng), Lecture Notes in Comput. Sci. 72, Springer-Verlag, Berlin 1979, 216-226.
86. Zuckerman, David, General weak random sources. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science, IEEE, New York 1990, 534-543.
87. Zuckerman, David, Linear degree extractors and the inapproximability of max clique and chromatic number. In Proceedings of the 38th ACM Symposium on Theory of Computing, ACM Press, New York 2006, 681-690.