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Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness

Authors Markus Bläser, Anurag Pandey

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ACM Subject Classification
  • Theory of computation → Algebraic complexity theory
Keywords
  • Algebraic Complexity theory
  • Polynomial Identity Testing
  • Hitting Set
  • Pseudorandomness

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Abstract

We give a randomized polynomial time algorithm for polynomial identity testing for the class of n-variate poynomials of degree bounded by d over a field 𝔽, in the blackbox setting.
Our algorithm works for every field 𝔽 with | 𝔽 | ≥ d+1, and uses only d log n + log (1/ ε) + O(d log log n) random bits to achieve a success probability 1 - ε for some ε > 0. In the low degree regime that is d ≪ n, it hits the information theoretic lower bound and differs from it only in the lower order terms. Previous best known algorithms achieve the number of random bits (Guruswami-Xing, CCC'14 and Bshouty, ITCS'14) that are constant factor away from our bound. Like Bshouty, we use Sidon sets for our algorithm. However, we use a new construction of Sidon sets to achieve the improved bound.
We also collect two simple constructions of hitting sets with information theoretically optimal size against the class of n-variate, degree d polynomials. Our contribution is that we give new, very simple proofs for both the constructions.

Cite As Get BibTex

Markus Bläser and Anurag Pandey. Polynomial Identity Testing for Low Degree Polynomials with Optimal Randomness. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 8:1-8:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.8

Author Details

Markus Bläser
  • Department of Computer Science, Saarland University, Saarland Informatics Campus, Saarbrücken, Germany
Anurag Pandey
  • Max Planck Institut für Informatik, Saarland Informatics Campus, Saarbrücken, Germany

Funding

  • Pandey, Anurag: Supported by the Chair of Raimund Seidel, Department of Computer Science, Saarland University, Saarbrücken, Germany.

Acknowledgements

We thank Rohit Gurjar, Mrinal Kumar and Raimund Seidel for insightful discussions. We thank the Simons Institute for the Theory of Computing (Berkeley) and Schloss Dagstuhl - Leibniz-Zentrum für Informatik (Dagstuhl), for hosting us during certain phases of this research.

References

  1. Manindra Agrawal. Proving lower bounds via pseudo-random generators. In FSTTCS 2005: Foundations of software technology and theoretical computer science, volume 3821 of Lecture Notes in Comput. Sci., pages 92-105. Springer, Berlin, 2005. URL: https://doi.org/10.1007/11590156_6.
  2. Manindra Agrawal and Somenath Biswas. Primality and identity testing via Chinese remaindering. J. ACM, 50(4):429-443, 2003. URL: https://doi.org/10.1145/792538.792540.
  3. Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. PRIMES is in P. Ann. of Math. (2), 160(2):781-793, 2004. URL: https://doi.org/10.4007/annals.2004.160.781.
  4. Manindra Agrawal, Neeraj Kayal, and Nitin Saxena. Errata: PRIMES is in P. Ann. of Math. (2), 189(1):317-318, 2019. URL: https://doi.org/10.4007/annals.2019.189.1.6.
  5. NOGA ALON. Combinatorial nullstellensatz. Combinatorics, Probability and Computing, 8(1-2):7–29, 1999. URL: https://doi.org/10.1017/S0963548398003411.
  6. Matthew Anderson, Michael A. Forbes, Ramprasad Saptharishi, Amir Shpilka, and Ben Lee Volk. Identity testing and lower bounds for read-k oblivious algebraic branching programs. In 31st Conference on Computational Complexity, volume 50 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 30, 25. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2016. Google Scholar
  7. Sanjeev Arora, Carsten Lund, Rajeev Motwani, Madhu Sudan, and Mario Szegedy. Proof verification and the hardness of approximation problems. J. ACM, 45(3):501-555, 1998. URL: https://doi.org/10.1145/278298.278306.
  8. Sanjeev Arora and Shmuel Safra. Probabilistic checking of proofs: a new characterization of NP. J. ACM, 45(1):70-122, 1998. URL: https://doi.org/10.1145/273865.273901.
  9. László Babai, Lance Fortnow, and Carsten Lund. Nondeterministic exponential time has two-prover interactive protocols. In 31st Annual Symposium on Foundations of Computer Science, Vol. I, II (St. Louis, MO, 1990), pages 16-25. IEEE Comput. Soc. Press, Los Alamitos, CA, 1990. URL: https://doi.org/10.1109/FSCS.1990.89520.
  10. Markus Bläser and Christian Engels. Randomness efficient testing of sparse black box identities of unbounded degree over the reals. In 28th International Symposium on Theoretical Aspects of Computer Science, volume 9 of LIPIcs. Leibniz Int. Proc. Inform., pages 555-566. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2011. Google Scholar
  11. Markus Bläser, Moritz Hardt, Richard J. Lipton, and Nisheeth K. Vishnoi. Deterministically testing sparse polynomial identities of unbounded degree. Inform. Process. Lett., 109(3):187-192, 2009. URL: https://doi.org/10.1016/j.ipl.2008.09.029.
  12. Markus Bläser, Moritz Hardt, and David Steurer. Asymptotically optimal hitting sets against polynomials. In Automata, languages and programming. Part I, volume 5125 of Lecture Notes in Comput. Sci., pages 345-356. Springer, Berlin, 2008. URL: https://doi.org/10.1007/978-3-540-70575-8_29.
  13. Manuel Blum, Ashok K. Chandra, and Mark N. Wegman. Equivalence of free Boolean graphs can be decided probabilistically in polynomial time. Inform. Process. Lett., 10(2):80-82, 1980. URL: https://doi.org/10.1016/S0020-0190(80)90078-2.
  14. Manuel Blum and Sampath Kannan. Designing programs that check their work. J. ACM, 42(1):269–291, January 1995. URL: https://doi.org/10.1145/200836.200880.
  15. Andrej Bogdanov. Pseudorandom generators for low degree polynomials. In STOC'05: Proceedings of the 37th Annual ACM Symposium on Theory of Computing, pages 21-30. ACM, New York, 2005. URL: https://doi.org/10.1145/1060590.1060594.
  16. Nader H. Bshouty. Testers and their applications. Electronic Colloquium on Computational Complexity (ECCC), 19:11, 2012. URL: http://eccc.hpi-web.de/report/2012/011.
  17. Suresh Chari, Pankaj Rohatgi, and Aravind Srinivasan. Randomness-optimal unique element isolation with applications to perfect matching and related problems. SIAM J. Comput., 24(5):1036-1050, 1995. URL: https://doi.org/10.1137/S0097539793250330.
  18. Zhi-Zhong Chen and Ming-Yang Kao. Reducing randomness via irrational numbers. In STOC '97 (El Paso, TX), pages 200-209. ACM, New York, 1999. Google Scholar
  19. Gil Cohen and Amnon Ta-Shma. Pseudorandom generators for low degree polynomials from algebraic geometry codes. Electronic Colloquium on Computational Complexity (ECCC), 20:155, 2013. URL: http://eccc.hpi-web.de/report/2013/155.
  20. Shagnik Das. A brief note on estimates of binomial coefficients. http://page.mi.fu-berlin.de/shagnik/notes/binomials.pdf. Accessed on 2020-02-18.
  21. Richard A. DeMillo and Richard J. Lipton. A probabilistic remark on algebraic program testing. Inf. Process. Lett., 7(4):193-195, 1978. URL: https://doi.org/10.1016/0020-0190(78)90067-4.
  22. Michael A. Forbes, Sumanta Ghosh, and Nitin Saxena. Towards blackbox identity testing of log-variate circuits. In 45th International Colloquium on Automata, Languages, and Programming, volume 107 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 54, 16. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2018. Google Scholar
  23. Michael A. Forbes and Amir Shpilka. Explicit Noether normalization for simultaneous conjugation via polynomial identity testing. In Approximation, randomization, and combinatorial optimization, volume 8096 of Lecture Notes in Comput. Sci., pages 527-542. Springer, Heidelberg, 2013. URL: https://doi.org/10.1007/978-3-642-40328-6_37.
  24. Michael A. Forbes and Amir Shpilka. Quasipolynomial-time identity testing of non-commutative and read-once oblivious algebraic branching programs. In 2013 IEEE 54th Annual Symposium on Foundations of Computer Science - FOCS 2013, pages 243-252. IEEE Computer Soc., Los Alamitos, CA, 2013. URL: https://doi.org/10.1109/FOCS.2013.34.
  25. Shafi Goldwasser and Ofer Grossman. Bipartite perfect matching in pseudo-deterministic NC. In 44th International Colloquium on Automata, Languages, and Programming, volume 80 of LIPIcs. Leibniz Int. Proc. Inform., pages Art. No. 87, 13. Schloss Dagstuhl. Leibniz-Zent. Inform., Wadern, 2017. Google Scholar
  26. Timothy Gowers. Gowers weblog: What are dense sidon subsets of 1,2,…,nlike? https://gowers.wordpress.com/2012/07/13/what-are-dense-sidon-subsets-of-12-n-like/. Published on 2012-07-13.
  27. Rohit Gurjar, Arpita Korwar, and Nitin Saxena. Identity testing for constant-width, and any-order, read-once oblivious arithmetic branching programs. Theory Comput., 13:Paper No. 2, 21, 2017. URL: https://doi.org/10.4086/toc.2017.v013a002.
  28. Rohit Gurjar, Arpita Korwar, Nitin Saxena, and Thomas Thierauf. Deterministic identity testing for sum of read-once oblivious arithmetic branching programs. Comput. Complexity, 26(4):835-880, 2017. URL: https://doi.org/10.1007/s00037-016-0141-z.
  29. Venkatesan Guruswami and Chaoping Xing. Hitting sets for low-degree polynomials with optimal density. In IEEE 29th Conference on Computational Complexity, CCC 2014, Vancouver, BC, Canada, June 11-13, 2014, pages 161-168. IEEE Computer Society, 2014. URL: https://doi.org/10.1109/CCC.2014.24.
  30. Russell Impagliazzo and Avi Wigderson. P = BPP if E requires exponential circuits: derandomizing the XOR lemma. In STOC '97 (El Paso, TX), pages 220-229. ACM, New York, 1999. Google Scholar
  31. Valentine Kabanets and Russell Impagliazzo. Derandomizing polynomial identity tests means proving circuit lower bounds. In Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, pages 355-364. ACM, New York, 2003. URL: https://doi.org/10.1145/780542.780595.
  32. Adam R. Klivans and Daniel Spielman. Randomness efficient identity testing of multivariate polynomials. In Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, pages 216-223. ACM, New York, 2001. URL: https://doi.org/10.1145/380752.380801.
  33. Mrinal Kumar, Ramprasad Saptharishi, and Anamay Tengse. Near-optimal bootstrapping of hitting sets for algebraic circuits. In Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 639-646. SIAM, Philadelphia, PA, 2019. URL: https://doi.org/10.1137/1.9781611975482.40.
  34. Mrinal Kumar and Ben Lee Volk. Lower bounds for matrix factorization. CoRR, abs/1904.01182, 2019. URL: http://arxiv.org/abs/1904.01182.
  35. Daniel Lewin and Salil Vadhan. Checking polynomial identities over any field: towards a derandomization? In STOC '98 (Dallas, TX), pages 438-447. ACM, New York, 1999. Google Scholar
  36. L. Lovász. On determinants, matchings, and random algorithms. In Fundamentals of computation theory (Proc. Conf. Algebraic, Arith. and Categorical Methods in Comput. Theory, Berlin/Wendisch-Rietz, 1979), volume 2 of Math. Res., pages 565-574. Akademie-Verlag, Berlin, 1979. Google Scholar
  37. C. Lu. Hitting set generators for sparse polynomials over any finite fields. In 2012 IEEE 27th Conference on Computational Complexity, pages 280-286, June 2012. URL: https://doi.org/10.1109/CCC.2012.20.
  38. Carsten Lund, Lance Fortnow, Howard Karloff, and Noam Nisan. Algebraic methods for interactive proof systems. In 31st Annual Symposium on Foundations of Computer Science, Vol. I, II (St. Louis, MO, 1990), pages 2-10. IEEE Comput. Soc. Press, Los Alamitos, CA, 1990. URL: https://doi.org/10.1109/FSCS.1990.89518.
  39. Ketan Mulmuley, Umesh V. Vazirani, and Vijay V. Vazirani. Matching is as easy as matrix inversion. Combinatorica, 7(1):105-113, 1987. URL: https://doi.org/10.1007/BF02579206.
  40. Kevin O'Bryant. A complete annotated bibliography of work related to sidon sequences. The Electronic Journal of Combinatorics [electronic only], DS11:39 p., electronic only-39 p., electronic only, 2004. URL: http://eudml.org/doc/129129.
  41. Imre Z. Ruzsa. An infinite Sidon sequence. J. Number Theory, 68(1):63-71, 1998. URL: https://doi.org/10.1006/jnth.1997.2192.
  42. Nitin Saxena. Progress on polynomial identity testing. Bulletin of the EATCS, 99:49-79, 2009. Google Scholar
  43. Nitin Saxena. Progress on polynomial identity testing-II. In Perspectives in computational complexity, volume 26 of Progr. Comput. Sci. Appl. Logic, pages 131-146. Birkhäuser/Springer, Cham, 2014. Google Scholar
  44. J. T. Schwartz. Fast probabilistic algorithms for verification of polynomial identities. J. Assoc. Comput. Mach., 27(4):701-717, 1980. URL: https://doi.org/10.1145/322217.322225.
  45. Adi Shamir. IP = PSPACE. In 31st Annual Symposium on Foundations of Computer Science, Vol. I, II (St. Louis, MO, 1990), pages 11-15. IEEE Comput. Soc. Press, Los Alamitos, CA, 1990. URL: https://doi.org/10.1109/FSCS.1990.89519.
  46. Amir Shpilka and Ilya Volkovich. Improved polynomial identity testing for read-once formulas. In Approximation, randomization, and combinatorial optimization, volume 5687 of Lecture Notes in Comput. Sci., pages 700-713. Springer, Berlin, 2009. URL: https://doi.org/10.1007/978-3-642-03685-9_52.
  47. Amir Shpilka and Amir Yehudayoff. Arithmetic circuits: a survey of recent results and open questions. Found. Trends Theor. Comput. Sci., 5(3-4):207-388 (2010), 2009. URL: https://doi.org/10.1561/0400000039.
  48. Itaï Ben Yaacov. The vandermonde determinant identity in higher dimension, 2014. URL: http://arxiv.org/abs/1405.0993.
  49. Richard Zippel. Probabilistic algorithms for sparse polynomials. In Symbolic and algebraic computation (EUROSAM '79, Internat. Sympos., Marseille, 1979), volume 72 of Lecture Notes in Comput. Sci., pages 216-226. Springer, Berlin-New York, 1979. Google Scholar
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