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Communication Complexity of Collision

Authors Mika Göös, Siddhartha Jain

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Author Details

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


We thank anonymous RANDOM reviewers for their helpful comments.

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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)


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
  • Collision
  • Communication complexity
  • Lifting


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  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:
  2. Scott Aaronson. Impossibility of succinct quantum proofs for collision-freeness. Quantum Information and Computation, 12(1-2):21-28, 2012. URL:
  3. Scott Aaronson. The collision lower bound after 12 years, 2013. QStart talk. URL:
  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:
  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:
  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:
  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:
  8. Manuel Blum, Alfredo De Santis, Silvio Micali, and Giuseppe Persiano. Noninteractive zero-knowledge. SIAM Journal on Computing, 20(6):1084-1118, 1991. URL:
  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:
  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:
  12. Mark Bun and Justin Thaler. Dual polynomials for collision and element distinctness. Theory Comput., 12(1):1-34, 2016. URL:
  13. Mika Göös and Toniann Pitassi. Communication lower bounds via critical block sensitivity. SIAM Journal on Computing, 47(5):1778-1806, 2018. URL:
  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:
  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:
  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:
  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:
  18. Dmitry Itsykson and Dmitry Sokolov. Resolution over linear equations modulo two. Annals of Pure and Applied Logic, 171(1):1-31, 2020. URL:
  19. Eyal Kushilevitz and Noam Nisan. Communication Complexity. Cambridge University Press, 1997. URL:
  20. Samuel Kutin. Quantum lower bound for the collision problem with small range. Theory of Computing, 1(2):29-36, 2005. URL:
  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:
  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:
  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:
  24. Anup Rao and Amir Yehudayoff. Communication Complexity: And Applications. Cambridge University Press, 2020. URL:
  25. Ran Raz and Avi Wigderson. Monotone circuits for matching require linear depth. Journal of the ACM, 39(3):736-744, July 1992. URL:
  26. Alexander Sherstov. The pattern matrix method. SIAM Journal on Computing, 40(6):1969-2000, 2011. URL:
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