Derandomizing polynomial identity tests means proving circuit lower bounds

Conference Proceedings of the Annual ACM Symposium on Theory of Computing

Volumn , Issue , 2003, Pages 355-364

  Kabanets, Valentine ^a   Impagliazzo, Russell ^a  

^a UNIVERSITY OF CALIFORNIA   (United States)

Author keywords

BPP; Circuit lower bounds; Derandomization; NEXP; Polynomial identity testing

Indexed keywords

DIGITAL ARITHMETIC; RANDOM PROCESSES; SET THEORY;

ARITHMETIC CIRCUITS;

POLYNOMIALS;

EID: 0038107689     PISSN: 07349025     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1145/780591.780595     Document Type: Conference Paper

References (59)

1. M. Agrawal and S. Biswas. Primality and identity testing via Chinese remaindering. In Proceedings of the Fortieth Annual IEEE Symposium on Foundations of Computer Science, pages 202-209, 1999.
2. M. Agrawal, N. Kayal, and N. Saxena. PRIMES is in P. Manuscript, 2002.
3. A. Andreev, A. Clementi, and J. Rolim. A new general derandomization method. Journal of the Association for Computing Machinery, 45(1):179-213, 1998. (preliminary version in ICALP'96).
4. S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and the hardness of approximation problems. Journal of the Association for Computing Machinery, 45 (3): 501-555, 1998. (preliminary version in FOCS'92).
5. S. Arora and S. Safra. Probabilistic checking of proofs: A new characterization of NP. Journal of the Association for Computing Machinery, 45(1):70-122, 1998. (preliminary version in FOCS'92).
6. L. Babai. Trading group theory for randomness. In Proceedings of the Seventeenth Annual ACM Symposium on Theory of Computing, pages 421-429, 1985.
7. L. Babai, L. Fortnow, and C. Lund. Non-deterministic exponential time has two-prover interactive protocols. Computational Complexity, 1:3-40, 1991.
8. L. Babai, L. Fortnow, N. Nisan, and A. Wigderson. BPP has subexponential time simulations unless EXPTIME has publishable proofs. Complexity, 3:307-318, 1993.
9. L. Babai and S. Moran. Arthur-Merlin games: A randomized proof system, and a hierarchy of complexity classes. Journal of Computer and System Sciences, 36:254-276, 1988.
10. M. Blum, A. Chandra, and M. Wegman. Equivalence of free Boolean graphs can be tested in polynomial time. Information Processing Letters, 10:80-82, 1980.
11. M. Blum and S. Kannan. Designing programs that check their work. Journal of the Association for Computing Machinery, 42:269-291, 1995.
12. R. Boppana and R. Hirschfeld. Pseudo-random generators and complexity classes. In S. Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 1-26. JAI Press, Greenwich, CT, 1989.
13. H. Buhrman, L. Fortnow, and L. Thierauf. Nonrelativizing separations. In Proceedings of the Thirteenth Annual IEEE Conference on Computational Complexity, pages 8-12, 1998.
14. S. Chari, P. Rohatgi, and A. Srinivasan. Randomness-optimal unique element isolation with applications to perfect matching and related problems. SIAM Journal on Computing, 24(5):1036-1050, 1995.
15. Z. Chen and M. Kao. Reducing randomness via irrational numbers. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 200-209, 1997.
16. A. Chistov. Fast parallel evaluation of the rank of matrices over a field of arbitrary characteristic. In Fundamentals of Computation Theory, volume 199 of Lecture Notes in Computer Science, pages 63-79. Springer Verlag, 1985.
17. M. Clausen, A. Dress. J. Grabmeier, and M. Karpinsky. On zero-testing and interpolation of k-sparse multivariate polynomials over finite fields. Theoretical Computer Science, 84(2):151-164, 1991.
18. R. DeMillo and R. Lipton. A probabilistic remark on algebraic program testing. Information Processing Letters, 7:193-195, 1978.
19. J. Gathen. Feasible arithmetic computations: Valiant's hypothesis. Journal of Symbolic Computation, 4:137-172, 1987.
20. O. Goldreich and D. Zuckerman. Another proof that BPP⊆PH (and more). Electronic Colloquium, on Computational Complexity, TR97-045, 1997.
21. D. Grigoriev, M. Karpinsky, and M. Singer. Fast parallel algorithms for sparse multivariate polynomial interpolation over finite fields. SIAM Journal on Computing, 19(6):1059-1063, 1990.
22. O. Ibarra and S. Moran. Probabilistic algorithms for deciding equivalence of straight-line programs. Journal of the Association for Computing Machinery, 30(1):217-228, 1983.
23. 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. (preliminary version in CCC'01).
24. R. Impagliazzo, R. Shaltiel, and A. Wigderson. Near-optimal conversion of hardness into pseudo-randomness. In Proceedings of the Fortieth Annual IEEE Symposium on Foundations of Computer Science, pages 181-190, 1999.
25. R. Impagliazzo, R. Shaltiel, and A. Wigderson. Extractors and pseudorandom generators with optimal seed length. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 1-10, 2000.
26. 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, 1997.
27. R. Impagliazzo and A. Wigderson. Randomness vs. time: De-randomization under a uniform assumption. In Proceedings of the Thirty-Ninth Annual IEEE Symposium on Foundations of Computer Science, pages 734-743, 1998.
28. V. Kabanets. Easiness assumptions and hardness tests: Trading time for zero error. Journal of Computer and System Sciences, 63(2):236-252, 2001. (preliminary version in CCC'00).
29. E. Kaltofen. Greatest common divisors of polynomials given by straight-line programs. Journal of the Association for Computing Machinery, 35(1):231-264, 1988.
30. E. Kaltofen. Factorization of polynomials given by straight-line programs. In S. Micali, editor, Randomness and Computation, volume 5 of Advances in Computing Research, pages 375-412. JAI Press, Greenwich, CT, 1989.
31. A. Klivans and D. Melkebeek. Graph nonisomorphism has subexponential size proofs unless the polynomial hierarchy collapses. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 659-667, 1999.
32. A. Klivans and D. Spielman. Randomness efficient identity testing of multivariate polynomials. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pages 216-223, 2001.
33. D. Lewin and S. Vadhan. Checking polynomial identities over any field: Towards a derandomization? In Proceedings of the Thirtieth Annual ACM Symposium on Theory of Computing, pages 438-447, 1998.
34. N. Linial, Y. Mansour, and N. Nisan. Constant depth circuits, Fourier transform and learnability. Journal of the Association for Computing Machinery, 40(3):607-620, 1993.
35. L. Lovasz. On determinants, matchings and random algorithms. In L. Budach, editor, Fundamentals of Computing Theory. Akademia-Verlag, Berlin, 1979.
36. C. Lund, L. Fortnow, H. Karloff, and N. Nisan. Algebraic methods for interactive proof systems. Journal of the Association for Computing Machinery, 39(4):859-868, 1992.
37. K. Mulmuley, U. Vazirani, and V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105-113, 1987.
38. N. Nisan and A. Wigderson. Hardness vs. randomness. Journal of Computer and System Sciences, 49:149-167, 1994.
39. C. Papadimitriou. Computational Complexity, Addison-Wesley, Reading, Massachusetts, 1994.
40. R. Paturi, P. Pudlák, M. Saks, and F. Zane. An improved exponential-time algorithm for k-SAT. In Proceedings of the Thirty-Ninth Annual IEEE Symposium on Foundations of Computer Science, pages 628-637, 1998.
41. R. Paturi, P. Pudlák, and F. Zane. Satisfiability coding lemma. Chicago Journal of Theoretical Computer Science, 1999. (preliminary version in FOCS'97).
42. R. Paturi, M. Saks, and F. Zane. Exponential lower bounds for depth 3 Boolean circuits. In Proceedings of the Twenty-Ninth Annual ACM Symposium on Theory of Computing, pages 86-91, 1997.
43. V. Pratt. Every prime has a succinct certificate. SIAM Journal on Computing, 4:214-220, 1975.
44. R. Raz, O. Reingold, and S. Vadhan. Extracting all the randomness and reducing the error in Trevisan's extractors. In Proceedings of the Thirty-First Annual ACM Symposium on Theory of Computing, pages 149-158, 1999.
45. A. Razborov and S. Rudich. Natural proofs. Journal of Computer and System Sciences, 55:24-35, 1997.
46. R. Roth and G. Benedek. Interpolation and approximation of sparse multivariate polynomials. SIAM Journal on Computing, 20(2):291-314, 1991.
47. J. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. Journal of the Association for Computing Machinery, 27(4):701-717, 1980.
48. R. Shaltiel and C. Umans. Simple extractors for all min-entropies and a new pseudo-random generator. In Proceedings of the Forty-Second Annual IEEE Symposium on Foundations of Computer Science, pages 648-657, 2001.
49. A. Shamir. IP=PSPACE. Journal of the Association for Computing Machinery, 39(4):869-877, 1992.
50. V. Strassen. Vermeidung von Divisionen. Journal für die Reine und Angewandte Mathematik, 264:182-202, 1973.
51. M. Sudan, L. Trevisan, and S. Vadhan. Pseudorandom generators without the XOR lemma. Journal of Computer and System Sciences, 62(2):236-266, 2001. (preliminary version in STOC'99).
52. S. Toda. PP is as hard as the polynomial-time hierarchy. SIAM Journal on Computing, 20(5):865-877, 1991.
53. C. Umans. Pseudo-random generators for all hardnesses. In Proceedings of the Thirty-Fourth Annual ACM Symposium on Theory of Computing, pages 127-134, 2002.
54. L. Valiant. Completeness classes in algebra. In Proceedings of the Eleventh Annual ACM Symposium on Theory of Computing, pages 249-261, 1979.
55. L. Valiant. The complexity of computing the permanent. Theoretical Computer Science. 8:189-201, 1979.
56. L. Valiant and V. Vazirani. NP is as easy as detecting unique solutions. Theoretical Computer Science, 47:85-93, 1986.
57. A. Yao. Theory and applications of trapdoor functions. In Proceedings of the Twenty-Third Annual IEEE Symposium on Foundations of Computer Science, pages 80-91, 1982.
58. F. Zane. Circuits, CNFs, and Satisfiability. PhD thesis, UCSD, 1998.
59. R. Zippel. Probabilistic algorithms for sparse polynomials. In Proceedings of an International Symposium on Symbolic and Algebraic Manipulation (EUROSAM'79), Lecture Notes in Computer Science, pages 216-226, 1979.