Communication Complexity of Collision

Authors Mika Göös, Siddhartha Jain



PDF
Thumbnail PDF

File

LIPIcs.APPROX-RANDOM.2022.19.pdf
  • Filesize: 0.69 MB
  • 9 pages

Document Identifiers

Author Details

Mika Göös
  • EPFL, Lausanne, Switzerland
Siddhartha Jain
  • EPFL, Lausanne, Switzerland

Acknowledgements

We thank anonymous RANDOM reviewers for their helpful comments.

Cite AsGet BibTex

Mika Göös and Siddhartha Jain. Communication Complexity of Collision. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 19:1-19:9, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2022.19

Abstract

The Collision problem is to decide whether a given list of numbers (x_1,…,x_n) ∈ [n]ⁿ is 1-to-1 or 2-to-1 when promised one of them is the case. We show an n^Ω(1) randomised communication lower bound for the natural two-party version of Collision where Alice holds the first half of the bits of each x_i and Bob holds the second half. As an application, we also show a similar lower bound for a weak bit-pigeonhole search problem, which answers a question of Itsykson and Riazanov (CCC 2021).

Subject Classification

ACM Subject Classification
  • Theory of computation → Communication complexity
Keywords
  • Collision
  • Communication complexity
  • Lifting

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Scott Aaronson. Quantum lower bound for the collision problem. In Proceedings of the 34th Symposium on Theory of Computing (STOC), pages 635-642. ACM, 2002. URL: https://doi.org/10.1145/509907.509999.
  2. Scott Aaronson. Impossibility of succinct quantum proofs for collision-freeness. Quantum Information and Computation, 12(1-2):21-28, 2012. URL: https://doi.org/10.26421/QIC12.1-2-3.
  3. Scott Aaronson. The collision lower bound after 12 years, 2013. QStart talk. URL: https://scottaaronson.blog/?p=1458.
  4. Scott Aaronson, Robin Kothari, William Kretschmer, and Justin Thaler. Quantum lower bounds for approximate counting via laurent polynomials. In Shubhangi Saraf, editor, 35th Computational Complexity Conference, CCC 2020, July 28-31, 2020, Saarbrücken, Germany (Virtual Conference), volume 169 of LIPIcs, pages 7:1-7:47. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.CCC.2020.7.
  5. Scott Aaronson and Yaoyun Shi. Quantum lower bounds for the collision and the element distinctness problems. Journal of the ACM, 51(4):595-605, July 2004. URL: https://doi.org/10.1145/1008731.1008735.
  6. Andris Ambainis. Polynomial degree and lower bounds in quantum complexity: collision and element distinctness with small range. Theory Comput., 1:37-46, 2005. URL: https://doi.org/10.4086/toc.2005.v001a003.
  7. Anurag Anshu, Shalev Ben-David, and Srijita Kundu. On query-to-communication lifting for adversary bounds. In Proceedings of the 36th Computational Complexity Conference (CCC), volume 200, pages 30:1-30:39. Schloss Dagstuhl, 2021. URL: https://doi.org/10.4230/LIPICS.CCC.2021.30.
  8. Manuel Blum, Alfredo De Santis, Silvio Micali, and Giuseppe Persiano. Noninteractive zero-knowledge. SIAM Journal on Computing, 20(6):1084-1118, 1991. URL: https://doi.org/10.1137/0220068.
  9. Adam Bouland, Lijie Chen, Dhiraj Holden, Justin Thaler, and Prashant Nalini Vasudevan. On the power of statistical zero knowledge. SIAM Journal on Computing, 49(4):FOCS17-1-FOCS17-58, 2019. URL: https://doi.org/10.1137/17m1161749.
  10. Gilles Brassard, Peter Høyer, and Alain Tapp. Quantum cryptanalysis of hash and claw-free functions. In Proceedings of the 3rd Latin American Symposium on Theoretical Informatics (LATIN), pages 163-169. Springer, 1998. Google Scholar
  11. Sergey Bravyi, Aram Harrow, and Avinatan Hassidim. Quantum algorithms for testing properties of distributions. IEEE Transactions on Information Theory, 57(6):3971-3981, 2011. URL: https://doi.org/10.1109/TIT.2011.2134250.
  12. Mark Bun and Justin Thaler. Dual polynomials for collision and element distinctness. Theory Comput., 12(1):1-34, 2016. URL: https://doi.org/10.4086/toc.2016.v012a016.
  13. Mika Göös and Toniann Pitassi. Communication lower bounds via critical block sensitivity. SIAM Journal on Computing, 47(5):1778-1806, 2018. URL: https://doi.org/10.1137/16M1082007.
  14. Lov K. Grover and Terry Rudolph. How significant are the known collision and element distinctness quantum algorithms? Quantum Inf. Comput., 4(3):201-206, 2004. URL: https://doi.org/10.26421/QIC4.3-5.
  15. Pavel Hrubeš and Pavel Pudlák. Random formulas, monotone circuits, and interpolation. In Proceedings of the 58th Symposium on Foundations of Computer Science (FOCS), pages 121-131, 2017. URL: https://doi.org/10.1109/FOCS.2017.20.
  16. Trinh Huynh and Jakob Nordström. On the virtue of succinct proofs: Amplifying communication complexity hardness to time-space trade-offs in proof complexity. In Proceedings of the 44th Symposium on Theory of Computing (STOC), pages 233-248. ACM, 2012. URL: https://doi.org/10.1145/2213977.2214000.
  17. Dmitry Itsykson and Artur Riazanov. Proof complexity of natural formulas via communication arguments. In Proceedings of 36th Computational Complexity Conference (CCC), volume 200, pages 3:1-3:34. Schloss Dagstuhl, 2021. URL: https://doi.org/10.4230/LIPIcs.CCC.2021.3.
  18. Dmitry Itsykson and Dmitry Sokolov. Resolution over linear equations modulo two. Annals of Pure and Applied Logic, 171(1):1-31, 2020. URL: https://doi.org/10.1016/j.apal.2019.102722.
  19. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997. URL: https://doi.org/10.1017/CBO9780511574948.
  20. Samuel Kutin. Quantum lower bound for the collision problem with small range. Theory of Computing, 1(2):29-36, 2005. URL: https://doi.org/10.4086/toc.2005.v001a002.
  21. Shachar Lovett and Jiapeng Zhang. On the impossibility of entropy reversal, and its application to zero-knowledge proofs. In Proceedings of the 15th Theory of Cryptography Conference (TCC), pages 31-55. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-70500-2_2.
  22. Frédéric Magniez, Miklos Santha, and Mario Szegedy. Quantum algorithms for the triangle problem. SIAM Journal on Computing, 37(2):413-424, January 2007. URL: https://doi.org/10.1137/050643684.
  23. Gatis Midrijānis. A polynomial quantum query lower bound for the set equality problem. In Proceedings of the 31st International Conference on Automata, Languages and Programming (ICALP), volume 3142, pages 996-1005. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-27836-8_83.
  24. Anup Rao and Amir Yehudayoff. Communication Complexity: And Applications. Cambridge University Press, 2020. URL: https://doi.org/10.1017/9781108671644.
  25. Ran Raz and Avi Wigderson. Monotone circuits for matching require linear depth. Journal of the ACM, 39(3):736-744, July 1992. URL: https://doi.org/10.1145/146637.146684.
  26. Alexander Sherstov. The pattern matrix method. SIAM Journal on Computing, 40(6):1969-2000, 2011. URL: https://doi.org/10.1137/080733644.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail