On the Complexity of Modulo-q Arguments and the Chevalley - Warning Theorem

Authors Mika Göös, Pritish Kamath, Katerina Sotiraki, Manolis Zampetakis

Thumbnail PDF


  • Filesize: 0.87 MB
  • 42 pages

Document Identifiers

Author Details

Mika Göös
  • Stanford University, CA, USA
Pritish Kamath
  • Toyota Technological Institute at Chicago, IL, USA
Katerina Sotiraki
  • Massachusetts Institute of Technology, Cambridge, MA, USA
Manolis Zampetakis
  • Massachusetts Institute of Technology, Cambridge, MA, USA


We thank Christos Papadimitriou, Robert Robere, Dmitry Sokolov and Noah Stephens-Davidowitz for helpful discussions. We also thank anonymous referees for valuable suggestions.

Cite AsGet BibTex

Mika Göös, Pritish Kamath, Katerina Sotiraki, and Manolis Zampetakis. On the Complexity of Modulo-q Arguments and the Chevalley - Warning Theorem. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 19:1-19:42, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


We study the search problem class PPA_q defined as a modulo-q analog of the well-known polynomial parity argument class PPA introduced by Papadimitriou (JCSS 1994). Our first result shows that this class can be characterized in terms of PPA_p for prime p. Our main result is to establish that an explicit version of a search problem associated to the Chevalley - Warning theorem is complete for PPA_p for prime p. This problem is natural in that it does not explicitly involve circuits as part of the input. It is the first such complete problem for PPA_p when p ≥ 3. Finally we discuss connections between Chevalley-Warning theorem and the well-studied short integer solution problem and survey the structural properties of PPA_q.

Subject Classification

ACM Subject Classification
  • Theory of computation → Complexity classes
  • Total NP Search Problems
  • Modulo-q arguments
  • Chevalley - Warning Theorem


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


  1. James Aisenberg, Maria Luisa Bonet, and Sam Buss. 2-d tucker is PPA complete. Electronic Colloquium on Computational Complexity (ECCC), 22:163, 2015. URL: http://eccc.hpi-web.de/report/2015/163.
  2. Noga Alon. Splitting necklaces. Advances in Mathematics, 63(3):247-253, 1987. URL: https://doi.org/10.1016/0001-8708(87)90055-7.
  3. Noga Alon, Shmuel Friedland, and Gil Kalai. Regular subgraphs of almost regular graphs. Journal of Combinatorial Theory, Series B, 37(1):79-91, 1984. Google Scholar
  4. Noga Alon and Douglas B. West. The Borsuk-Ulam theorem and bisection of necklaces. Proceedings of the American Mathematical Society, 98(4):623-628, 1986. URL: https://doi.org/10.1090/S0002-9939-1986-0861764-9.
  5. Paul Beame, Stephen A. Cook, Jeff Edmonds, Russell Impagliazzo, and Toniann Pitassi. The relative complexity of NP search problems. J. Comput. Syst. Sci., 57(1):3-19, 1998. URL: https://doi.org/10.1006/jcss.1998.1575.
  6. Paul Beame and Søren Riis. More on the relative strength of counting principles. In Proceedings of the DIMACS Workshop on Proof Complexity and Feasible Arithmetics, volume 39, pages 13-35, 1998. Google Scholar
  7. Richard Beigel and John Gill. Counting classes: Thresholds, parity, mods, and fewness. Theor. Comput. Sci., 103(1):3-23, 1992. URL: https://doi.org/10.1016/0304-3975(92)90084-S.
  8. Aleksandrs Belovs, Gábor Ivanyos, Youming Qiao, Miklos Santha, and Siyi Yang. On the polynomial parity argument complexity of the combinatorial nullstellensatz. In 32nd Computational Complexity Conference, CCC 2017, July 6-9, 2017, Riga, Latvia, pages 30:1-30:24, 2017. URL: https://doi.org/10.4230/LIPIcs.CCC.2017.30.
  9. Nir Bitansky, Omer Paneth, and Alon Rosen. On the cryptographic hardness of finding a nash equilibrium. In 2015 IEEE 56th Annual Symposium on Foundations of Computer Science, pages 1480-1498. IEEE, 2015. Google Scholar
  10. Josh Buresh-Oppenheim and Tsuyoshi Morioka. Relativized NP search problems and propositional proof systems. In 19th Annual IEEE Conference on Computational Complexity (CCC 2004), 21-24 June 2004, Amherst, MA, USA, pages 54-67, 2004. URL: https://doi.org/10.1109/CCC.2004.1313795.
  11. Joshua Buresh-Oppenheim. On the TFNP complexity of factoring. Manuscript, 2006. URL: http://www.cs.toronto.edu/~bureshop/factor.pdf.
  12. Samuel R. Buss, Dima Grigoriev, Russell Impagliazzo, and Toniann Pitassi. Linear gaps between degrees for the polynomial calculus modulo distinct primes. J. Comput. Syst. Sci., 62(2):267-289, 2001. URL: https://doi.org/10.1006/jcss.2000.1726.
  13. Samuel R. Buss and Alan S. Johnson. Propositional proofs and reductions between NP search problems. Ann. Pure Appl. Logic, 163(9):1163-1182, 2012. URL: https://doi.org/10.1016/j.apal.2012.01.015.
  14. I. Bárány, S. B. Shlosman, and A. Szücs. On a topological generalization of a theorem of tverberg. Journal of the London Mathematical Society, s2-23(1):158-164, 1981. URL: https://doi.org/10.1112/jlms/s2-23.1.158.
  15. Claude Chevalley. Démonstration d'une hypothèse de m. artin. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg, 11(1):73-75, December 1935. URL: https://doi.org/10.1007/BF02940714.
  16. Arka Rai Choudhuri, Pavel Hubácek, Chethan Kamath, Krzysztof Pietrzak, Alon Rosen, and Guy N Rothblum. Finding a nash equilibrium is no easier than breaking fiat-shamir. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, pages 1103-1114. ACM, 2019. Google Scholar
  17. Constantinos Daskalakis, Paul W. Goldberg, and Christos H. Papadimitriou. The complexity of computing a nash equilibrium. SIAM J. Comput., 39(1):195-259, 2009. URL: https://doi.org/10.1137/070699652.
  18. Constantinos Daskalakis and Christos H. Papadimitriou. Continuous local search. In Proceedings of the Twenty-Second Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2011, San Francisco, California, USA, January 23-25, 2011, pages 790-804, 2011. URL: https://doi.org/10.1137/1.9781611973082.62.
  19. Xiaotie Deng, Jack R. Edmonds, Zhe Feng, Zhengyang Liu, Qi Qi, and Zeying Xu. Understanding PPA-Completeness. In Ran Raz, editor, 31st Conference on Computational Complexity (CCC 2016), volume 50 of Leibniz International Proceedings in Informatics (LIPIcs), pages 23:1-23:25, Dagstuhl, Germany, 2016. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik. URL: https://doi.org/10.4230/LIPIcs.CCC.2016.23.
  20. Aris Filos-Ratsikas and Paul W. Goldberg. Consensus halving is ppa-complete. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2018, Los Angeles, CA, USA, June 25-29, 2018, pages 51-64, 2018. URL: https://doi.org/10.1145/3188745.3188880.
  21. Aris Filos-Ratsikas and Paul W. Goldberg. The complexity of splitting necklaces and bisecting ham sandwiches. In STOC (to appear), 2019. URL: http://arxiv.org/abs/1805.12559.
  22. C. Goldberg and D. West. Bisection of circle colorings. SIAM Journal on Algebraic Discrete Methods, 6(1):93-106, 1985. URL: https://doi.org/10.1137/0606010.
  23. Mika Göös, Pritish Kamath, Robert Robere, and Dmitry Sokolov. Adventures in monotone complexity and TFNP. In 10th Innovations in Theoretical Computer Science Conference, ITCS 2019, January 10-12, 2019, San Diego, California, USA, pages 38:1-38:19, 2019. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.38.
  24. Michelangelo Grigni. A sperner lemma complete for ppa. Information Processing Letters, 77(5-6):255-259, 2001. Google Scholar
  25. Alexandros Hollender. The classes PPA-k: Existence from arguments modulo k. In Ioannis Caragiannis, Vahab Mirrokni, and Evdokia Nikolova, editors, Web and Internet Economics, pages 214-227, Cham, 2019. Springer International Publishing. Google Scholar
  26. Emil Jerábek. Integer factoring and modular square roots. J. Comput. Syst. Sci., 82(2):380-394, 2016. URL: https://doi.org/10.1016/j.jcss.2015.08.001.
  27. Alan S. Johnson. Reductions and propositional proofs for total NP search problems. UC San Diego Electronic Theses and Dissertations, 2011. URL: https://escholarship.org/uc/item/89r774x7.
  28. David S. Johnson, Christos H. Papadimitriou, and Mihalis Yannakakis. How easy is local search? J. Comput. Syst. Sci., 37(1):79-100, 1988. URL: https://doi.org/10.1016/0022-0000(88)90046-3.
  29. Ilan Komargodski, Moni Naor, and Eylon Yogev. White-box vs. black-box complexity of search problems: Ramsey and graph property testing. Journal of the ACM (JACM), 66(5):34, 2019. Google Scholar
  30. Pravesh K Kothari and Ruta Mehta. Sum-of-squares meets nash: lower bounds for finding any equilibrium. In Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, pages 1241-1248. ACM, 2018. Google Scholar
  31. Donald L. Kreher and Douglas R. Stinson. Combinatorial Algorithms: Generation, Enumeration, and Search, volume 7 of Discrete Mathematics and Its Applications. CRC Press, 1998. Google Scholar
  32. Nimrod Megiddo and Christos H. Papadimitriou. On total functions, existence theorems and computational complexity. Theor. Comput. Sci., 81(2):317-324, 1991. URL: https://doi.org/10.1016/0304-3975(91)90200-L.
  33. Christos H. Papadimitriou. On the complexity of the parity argument and other inefficient proofs of existence. J. Comput. Syst. Sci., 48(3):498-532, 1994. URL: https://doi.org/10.1016/S0022-0000(05)80063-7.
  34. Christian Reiher. On kemnitz’conjecture concerning lattice-points in the plane. The Ramanujan Journal, 13(1-3):333-337, 2007. Google Scholar
  35. Aviad Rubinstein. Settling the complexity of computing approximate two-player nash equilibria. In IEEE 57th Annual Symposium on Foundations of Computer Science, FOCS 2016, 9-11 October 2016, Hyatt Regency, New Brunswick, New Jersey, USA, pages 258-265, 2016. Google Scholar
  36. Katerina Sotiraki, Manolis Zampetakis, and Giorgos Zirdelis. Ppp-completeness with connections to cryptography. In 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 148-158, 2018. URL: https://doi.org/10.1109/FOCS.2018.00023.
  37. Ewald Warning. Bemerkung zur vorstehenden arbeit von herrn chevalley. Abh. Math. Sem. Univ. Hamburg, 11:76-83, 1936. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail