Norms, XOR lemmas, and lower bounds for GF(2) polynomials and multiparty protocols

Proceedings of the Annual IEEE Conference on Computational Complexity

Volumn , Issue , 2007, Pages 141-154

  Viola, Emanuele ^a   Wigderson, Avi ^a  

^a INSTITUTE FOR ADVANCED STUDY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL COMPLEXITY; CORRELATION METHODS; LOGIC GATES; MATHEMATICAL MODELS; POLYNOMIAL APPROXIMATION;

COMPUTATIONAL MODELS; MULTIPARTY COMMUNICATION COMPLEXITY; MULTIPARTY PROTOCOLS; XOR LEMMAS;

BOOLEAN FUNCTIONS;

EID: 34748897950     PISSN: 10930159     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1109/CCC.2007.15     Document Type: Conference Paper

References (43)

1. N. Alon, O. Goldreich, J. Håstad, and R. Peralta. Simple constructions of almost k-wise independent random variables. Random Structures & Algorithms, 3(3):289-304, 1992.
2. N. Alon, T. Kaufman, M. Krivelevich, S. Litsyn, and D. Ron. Testing low-degree polynomials over GF (2). In Approximation, randomization, and combinatorial optimization, volume 2764 of Lecture Notes in Comput. Sci., pages 188-199. Springer, Berlin, 2003.
3. L. Babai, N. Nisan, and M. Szegedy. Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. J Comput. System Sci., 45(2):204-232, 1992. Twenty-first Symposium on the Theory of Computing (Seattle, WA, 1989).
4. J. Bourgain. Estimation of certain exponential sums arising in complexity theory. C. R. Math. Acad. Sci. Paris, 340(9):627-631, 2005.
5. A. K. Chandra, M. L. Furst, and R. J. Lipton. Multi-party protocols. In Proceedings of the Fifteenth Annual ACM Symposium on Theory of Computing, Boston, Massachusetts, 25-27 Apr. 1983.
6. q. Electronic Colloquium on Computational Complexity, Technical Report TR06-107, 2006. http://www.eccc.uni-trier.de/eccc.
7. F. R. K. Chung and P. Tetali. Communication complexity and quasi randomness. SIAM J. Discrete Math., 6(1):110-123, 1993.
8. U. Feige. Error reduction by parallel repetition-the state of the art. Technical report, Jerusalem, Israel, Israel, 1995.
9. O. Goldreich and L. A. Levin. A hard-core predicate for all one-way functions. In Proceedings of the Twenty First Annual ACM Symposium on Theory of Computing, pages 25-32, Seattle, Washington, 15-17 May 1989.
10. O. Goldreich, N. Nisan, and A. Wigderson. On Yao's XOR lemma. Technical Report TR95-050, Electronic Colloquium on Computational Complexity, March 1995. http://www.eccc.uni-trier.de/eccc.
11. W. T. Gowers. A new proof of Szemerédi's theorem for arithmetic progressions of length four. Geom. Funct. Anal., 8(3):529-551, 1998.
12. W. T. Gowers. A new proof of Szemerédi's theorem. Geom. Funct. Anal., 11(3):465-588, 2001.
13. 3 norm, 2005. arXiv.org:math/0503014.
14. F. Green, A. Roy, and H. Straubing. Bounds on an exponential sum arising in Boolean circuit complexity. C. R. Math. Acad. Sci. Paris, 341(5):279-282, 2005.
15. V. Grolmusz. Separating the communication complexities of MOD m and MOD p circuits. J. Comput. System Sci., 51(2):307-313, 1995.
16. D. Gutfreund and E. Viola. Fooling parity tests with parity gates. In Proceedings of the Eight International Workshop on Randomization and Computation (RANDOM), Lecture Notes in Computer Science, Volume 3122, pages 381-392. Springer-Verlag, 2004.
17. A. Hajnal, W. Maass, P. Pudlák, M. Szegedy, and G. Turán. Threshold circuits of bounded depth. J. Comput. System Sci., 46(2):129-154, 1993.
18. J. Håstad and M. Goldmann. On the power of small-depth threshold circuits. Comput. Complexity, 1(2):113-129, 1991.
19. 1. In Proceedings of the 10th International Workshop on Randomization and Computation (RANDOM), Lecture Notes in Computer Science., 2006.
20. 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 Oct. 1995. IEEE.
21. 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.
22. C. S. Jutla, A. C. Patthak, A. Rudra, and D. Zuckerman. Testing low-degree polynomials over prime fields. foes, 00:423-432, 2004.
23. E. Kushilevitz and N. Nisan. Communication complexity. Cambridge University Press, Cambridge, 1997.
24. L. A. Levin. One way functions and pseudorandom generators. Combinatorica, 7(4):357-363, 1987.
25. M. Luby, B. Velickovic, and A. Wigderson. Deterministic approximate counting of depth-2 circuits. In Proceedings of the 2nd Israeli Symposium on Theoretical Computer Science (ISTCS), pages 18-24, 1993.
26. J. Naor and M. Naor. Small-bias probability spaces: efficient constructions and applications. In Proceedings of the 22nd Annual ACM Symposium on the Theory of Computing, pages 213-223, 1990.
27. N. Nisan. Pseudorandom bits for constant depth circuits. Combinatorica, 11(1):63-70, 1991.
28. N. Nisan, S. Rudich, and M. Saks. Products and help bits in decision trees. SIAM J. Comput., 28(3):1035-1050, 1999.
29. N. Nisan and A. Wigderson. Hardness vs randomness. J. Computer & Systems Sciences, 49(2):149-167, Oct. 1994.
30. I. Parnafes, R. Raz, and A. Wigderson. Direct product results and the GCD problem, in old and new communication models. In STOC '97 (El Paso, TX), pages 363-372 (electronic). ACM, New York, 1999.
31. R. Raz. A parallel repetition theorem. SIAM J. Comput., 27(3):763-803 (electronic), 1998.
32. R. Raz. The BNS-Chung criterion for multi-party communication complexity. Comput. Complexity, 9(2):113-122, 2000.
33. A. A. Razborov. Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Mat. Zametki, 41(4):598-607, 623, 1987.
34. A. Samorodnitsky. Low-degree tests at large distances. In Proceedings of the 39th Annual ACM Symposium on Theory of Computing 2007, San Diego, CA USA, June 2007, 2007.
35. A. Samorodnitsky and L. Trevisan. Gowers uniformity, influence of variables, and PCPs. In Proceedings of the 38th Annual ACM Symposium on Theory of Computing 2006, Seattle, WA, USA, May 21-23, 2006, pages 11-20, 2006.
36. R. Shaltiel. Towards proving strong direct product theorems. Comput Complexity, 12(1-2):1-22, 2003.
37. R. Shaltiel and C. Umans. Simple extractors for all minentropies and a new pseudorandom generator. J ACM, 52(2):172-216, 2005.
38. R. Smolensky. Algebraic methods in the theory of lower bounds for boolean circuit complexity. In Proceedings of the Nineteenth Annual ACM Symposium on Theory of Computing, pages 77-82, New York City, 25-27 May 1987.
39. E. Viola. The Complexity of Hardness Amplification and Derandomization. PhD thesis, Harvard University, 2006. http://www.eccc.uni- trier.de/eccc.
40. E. Viola. New correlation bounds for gf(2) polynomials using gowers uniformity. Electronic Colloquium on Computational Complexity, Technical Report TR06-097, 2006. http://www.eccc.uni-trier.de/eccc.
41. E. Viola. Pseudorandom bits for constant-depth circuits with few arbitrary symmetric gates. SIAM Journal on Computing, 36(5):1387-1403, 2007.
42. 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 Nov. 1982. IEEE.
43. A. C.-C. Yao. Some complexity questions related to distributive computing. In STOC '79: Proceedings of the eleventh annual ACM symposium on Theory of computing, pages 209-213, New York, NY, USA, 1979. ACM Press.