Asymptotically Optimal Hitting Sets Against Polynomials

  • Markus Bläser
  • Moritz Hardt
  • David Steurer
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5125)


Our main result is an efficient construction of a hitting set generator against the class of polynomials of degree d i in the i-th variable. The seed length of this generator is Open image in new window . Here, logD = ∑  i log(d i  + 1) is a lower bound on the seed length of any hitting set generator against this class. Our construction is the first to achieve asymptotically optimal seed length for every choice of the parameters d i . In fact, we present a nearly linear time construction with this asymptotic guarantee. Furthermore, our results extend to classes of polynomials parameterized by upper bounds on the number of nonzero terms in each variable. Underlying our constructions is a general and novel framework that exploits the product structure common to the classes of polynomials we consider. This framework allows us to obtain efficient and asymptotically optimal hitting set generators from primitives that need not be optimal or efficient by themselves.

As our main corollary, we obtain the first blackbox polynomial identity tests with an asymptotically optimal randomness consumption.


Characteristic Zero Polynomial Identity Seed Length Univariate Polynomial Arithmetic Circuit 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Alon, N.: Combinatorial Nullstellensatz. Comb. Probab. Comput. 8(1-2), 7–29 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    De Millo, R.A., Lipton, R.J.: A probabilistic remark on algebraic program testing. IPL 7 (1978)Google Scholar
  3. 3.
    Zippel, R.: Probabilistic algorithms for sparse polynomials. In: Proc. ISSAC, pp. 216–226. Springer, Berlin (1979)Google Scholar
  4. 4.
    Schwartz, J.: Fast probabilistic algorithms for verification of polynomial identities. Journal of the ACM 27, 701–717 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chari, S., Rohatgi, P., Srinivasan, A.: Randomness-optimal unique element isolation with applications to perfect matching and related problems. SIAM J. Comput. 24(5), 1036–1050 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lovász, L.: On determinants, matchings, and random algorithms. In: FCT, pp. 565–574 (1979)Google Scholar
  7. 7.
    Mulmuley, K., Vazirani, U.V., Vazirani, V.V.: Matching is as easy as matrix inversion. Combinatorica 7(1), 105–113 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Blum, M., Chandra, A.K., Wegman, M.N.: Equivalence of free boolean graphs can be decided probabilistically in polynomial time. IPL 10, 80–82 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Blum, M., Kannan, S.: Designing programs that check their work. J. ACM 42(1), 269–291 (1995)CrossRefzbMATHGoogle Scholar
  10. 10.
    Agrawal, M., Biswas, S.: Primality and identity testing via chinese remaindering. J. ACM 50(4), 429–443 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Agrawal, M., Kayal, N., Saxena, N.: PRIMES is in P. Ann. of Math (2) 160(2), 781–793 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Shamir, A.: IP = PSPACE. J. ACM 39(4), 869–877 (1992)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Lund, C., Fortnow, L., Karloff, H., Nisan, N.: Algebraic methods for interactive proof systems. J. ACM 39(4), 859–868 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Arora, S., Safra, S.: Probabilistic checking of proofs: a new characterization of NP. J. ACM 45(1), 70–122 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Arora, S., Lund, C., Motwani, R., Sudan, M., Szegedy, M.: Proof verification and the hardness of approximation problems. J. ACM 45(3), 501–555 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Agrawal, M.: Proving lower bounds via pseudo-random generators. In: Proc. 25th FSTTCS, pp. 92–105. Springer, Heidelberg (2005)Google Scholar
  17. 17.
    Dvir, Z., Shpilka, A.: Locally decodable codes with two queries and polynomial identity testing for depth 3 circuits. SIAM J. Comput. 36(5), 1404–1434 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Kayal, N., Saxena, N.: Polynomial identity testing for depth 3 circuits. In: Proc. 21st CCC, pp. 9–17. IEEE, Los Alamitos (2006)Google Scholar
  19. 19.
    Shpilka, A.: Interpolation of depth-3 arithmetic circuits with two multiplication gates. In: Proc. 39th STOC, pp. 284–293. ACM, New York (2007)Google Scholar
  20. 20.
    Kabanets, V., Impagliazzo, R.: Derandomizing polynomial identity tests means proving circuit lower bounds. Comput. Complex. 13(1/2), 1–46 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Chen, Z.Z., Kao, M.Y.: Reducing randomness via irrational numbers. SIAM J. Comput. 29(4), 1247–1256 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Lewin, D., Vadhan, S.: Checking polynomial identities over any field: Towards a derandomization? In: Proc. 30th STOC, pp. 437–438. ACM, New York (1998)Google Scholar
  23. 23.
    Klivans, A., Spielman, D.A.: Randomness efficient identity testing of multivariate polynomials. In: Proc. 33th STOC, pp. 216–223. ACM, New York (2001)Google Scholar
  24. 24.
    Bogdanov, A.: Pseudorandom generators for low degree polynomials. In: Proc. 37th STOC, pp. 21–30. ACM, New York (2005)Google Scholar
  25. 25.
    Ibarra, O.H., Moran, S.: Probabilistic algorithms for deciding equivalence of straight-line programs. J. ACM 30(1), 217–228 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Lipton, R., Vishnoi, N.: Deterministic identity testing for multivariate polynomials. In: Proc. SODA, pp. 756–760. ACM, New York (2003)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Markus Bläser
    • 1
  • Moritz Hardt
    • 2
  • David Steurer
    • 2
  1. 1.Saarland UniversitySaarbrückenGermany
  2. 2.Princeton UniversityPrinceton

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