Affine Extractors and AC0-Parity

Authors Xuangui Huang, Peter Ivanov, Emanuele Viola

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Author Details

Xuangui Huang
  • Northeastern University, Boston, MA, USA
Peter Ivanov
  • Northeastern University, Boston, MA, USA
Emanuele Viola
  • Northeastern University, Boston, MA, USA


We are grateful to the anonymous reviewers for helpful feedback.

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Xuangui Huang, Peter Ivanov, and Emanuele Viola. Affine Extractors and AC0-Parity. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 9:1-9:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)


We study a simple and general template for constructing affine extractors by composing a linear transformation with resilient functions. Using this we show that good affine extractors can be computed by non-explicit circuits of various types, including AC0-Xor circuits: AC0 circuits with a layer of parity gates at the input. We also show that one-sided extractors can be computed by small DNF-Xor circuits, and separate these circuits from other well-studied classes. As a further motivation for studying DNF-Xor circuits we show that if they can approximate inner product then small AC0-Xor circuits can compute it exactly - a long-standing open problem.

Subject Classification

ACM Subject Classification
  • Theory of computation → Circuit complexity
  • affine extractor
  • resilient function
  • constant-depth circuit
  • parity gate
  • inner product


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  1. Miklós Ajtai. Σ^1_1-formulae on finite structures. Annals of Pure and Applied Logic, 24(1):1-48, 1983. Google Scholar
  2. Miklós Ajtai. Approximate counting with uniform constant-depth circuits. In Advances in computational complexity theory, pages 1-20. Amer. Math. Soc., Providence, RI, 1993. Google Scholar
  3. Miklos Ajtai and Nathan Linial. The influence of large coalitions. Combinatorica, 13:129-145, 1993. Google Scholar
  4. Miklos Ajtai and Avi Wigderson. Deterministic simulation of probabilistic constant-depth circuits. Advances in Computing Research - Randomness and Computation, 5:199-223, 1989. Google Scholar
  5. Adi Akavia, Andrej Bogdanov, Siyao Guo, Akshay Kamath, and Alon Rosen. Candidate weak pseudorandom functions in AC0 MOD 2. In Moni Naor, editor, ACM Innovations in Theoretical Computer Science conf. (ITCS), pages 251-260. ACM, 2014. URL:
  6. Louay M. J. Bazzi. Polylogarithmic independence can fool DNF formulas. SIAM J. Comput., 38(6):2220-2272, 2009. Google Scholar
  7. Michael Ben-Or and Nathan Linial. Collective coin flipping, robust voting schemes and minima of Banzhaf values. In 26th Symposium on Foundations of Computer Science, pages 408-416, Portland, Oregon, 21-23 October 1985. IEEE. Google Scholar
  8. Andrej Bogdanov and Emanuele Viola. Pseudorandom bits for polynomials. SIAM J. on Computing, 39(6):2464-2486, 2010. Google Scholar
  9. Mark Braverman. Polylogarithmic independence fools AC^0 circuits. J. of the ACM, 57(5), 2010. Google Scholar
  10. Mark Bun, Robin Kothari, and Justin Thaler. Quantum algorithms and approximating polynomials for composed functions with shared inputs. In Timothy M. Chan, editor, ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 662-678. SIAM, 2019. URL:
  11. Eshan Chattopadhyay, Jesse Goodman, and Jyun-Jie Liao. Affine extractors for almost logarithmic entropy. In 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022, pages 622-633. IEEE, 2021. URL:
  12. Eshan Chattopadhyay and David Zuckerman. Explicit two-source extractors and resilient functions. In ACM Symp. on the Theory of Computing (STOC), pages 670-683, 2016. Google Scholar
  13. Mahdi Cheraghchi, Elena Grigorescu, Brendan Juba, Karl Wimmer, and Ning Xie. AC^0∘mod_2 lower bounds for the boolean inner product. J. Comput. Syst. Sci., 97:45-59, 2018. Google Scholar
  14. Gil Cohen and Igor Shinkar. The complexity of DNF of parities. In ACM Innovations in Theoretical Computer Science conf. (ITCS), pages 47-58, 2016. Google Scholar
  15. Gil Cohen and Avishay Tal. Two structural results for low degree polynomials and applications. In Naveen Garg, Klaus Jansen, Anup Rao, and José D. P. Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2015, August 24-26, 2015, Princeton, NJ, USA, volume 40 of LIPIcs, pages 680-709. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. URL:
  16. Ilias Diakonikolas, Parikshit Gopalan, Ragesh Jaiswal, Rocco A. Servedio, and Emanuele Viola. Bounded independence fools halfspaces. SIAM J. on Computing, 39(8):3441-3462, 2010. Google Scholar
  17. Guy Even, Oded Goldreich, Michael Luby, Noam Nisan, and Boban Velickovic. Approximations of general independent distributions. In ACM Symp. on the Theory of Computing (STOC), pages 10-16, 1992. Google Scholar
  18. Michael Ezra and Ron D. Rothblum. Small circuits imply efficient arthur-merlin protocols. In Mark Braverman, editor, 13th Innovations in Theoretical Computer Science Conference, ITCS 2022, January 31 - February 3, 2022, Berkeley, CA, USA, volume 215 of LIPIcs, pages 67:1-67:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL:
  19. Magnus Gausdal Find, Alexander Golovnev, Edward A. Hirsch, and Alexander S. Kulikov. A better-than-3n lower bound for the circuit complexity of an explicit function. In Irit Dinur, editor, IEEE Symp. on Foundations of Computer Science (FOCS), pages 89-98. IEEE Computer Society, 2016. URL:
  20. Merrick L. Furst, James B. Saxe, and Michael Sipser. Parity, circuits, and the polynomial-time hierarchy. Mathematical Systems Theory, 17(1):13-27, 1984. Google Scholar
  21. Oded Goldreich, Emanuele Viola, and Avi Wigderson. On randomness extraction in AC0. In IEEE Conf. on Computational Complexity (CCC), 2015. Google Scholar
  22. András Hajnal, Wolfgang Maass, Pavel Pudlák, Márió Szegedy, and György Turán. Threshold circuits of bounded depth. J. of Computer and System Sciences, 46(2):129-154, 1993. Google Scholar
  23. Johan Håstad. Computational limitations of small-depth circuits. MIT Press, 1987. Google Scholar
  24. Johan Håstad. On the correlation of parity and small-depth circuits. SIAM J. on Computing, 43(5):1699-1708, 2014. Google Scholar
  25. Johan Håstad, Benjamin Rossman, Rocco A. Servedio, and Li-Yang Tan. An average-case depth hierarchy theorem for boolean circuits. J. of the ACM, 64(5):35:1-35:27, 2017. URL:
  26. Russell Impagliazzo, William Matthews, and Ramamohan Paturi. A satisfiability algorithm for AC^0. In ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 961-972, 2012. Google Scholar
  27. Stasys Jukna. On graph complexity. Comb. Probab. Comput., 15(6):855-876, 2006. URL:
  28. Nathan Linial and Noam Nisan. Approximate inclusion-exclusion. Combinatorica, 10(4):349-365, 1990. Google Scholar
  29. Raghu Meka. Explicit resilient functions matching ajtai-linial. In ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 1132-1148, 2017. Google Scholar
  30. Ryan O'Donnell. Analysis of Boolean Functions. Cambridge University Press, 2014. Google Scholar
  31. A. A. Razborov. Formulas of bounded depth in the basis &,⊕ and some combinatorial problems. Voprosy Kibernet. (Moscow), 134:149-166, 1988. Google Scholar
  32. Alexander Razborov. Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Akademiya Nauk SSSR. Matematicheskie Zametki, 41(4):598-607, 1987. English translation in Mathematical Notes of the Academy of Sci. of the USSR, 41(4):333-338, 1987. Google Scholar
  33. Alexander A. Razborov. A simple proof of Bazzi’s theorem. ACM Transactions on Computation Theory (TOCT), 1(1), 2009. Google Scholar
  34. Petr Savický. Improved boolean formulas for the ramsey graphs. Random Struct. Algorithms, 6(4):407-416, 1995. URL:
  35. Rocco A. Servedio and Emanuele Viola. On a special case of rigidity. Available at, 2012.
  36. Roman Smolensky. Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In 19th ACM Symp. on the Theory of Computing (STOC), pages 77-82. ACM, 1987. Google Scholar
  37. Avishay Tal. Shrinkage of de morgan formulae by spectral techniques. In IEEE Symp. on Foundations of Computer Science (FOCS), pages 551-560. IEEE Computer Society, 2014. URL:
  38. Luca Trevisan. Some applications of coding theory in computational complexity. In Complexity of computations and proofs, volume 13 of Quad. Mat., pages 347-424. Dept. Math., Seconda Univ. Napoli, Caserta, 2004. Google Scholar
  39. Emanuele Viola. Pseudorandom bits for constant-depth circuits with few arbitrary symmetric gates. SIAM J. on Computing, 36(5):1387-1403, 2007. Google Scholar
  40. Emanuele Viola. On the power of small-depth computation. Foundations and Trends in Theoretical Computer Science, 5(1):1-72, 2009. Google Scholar
  41. Emanuele Viola. Extractors for circuit sources. SIAM J. on Computing, 43(2):355-972, 2014. Google Scholar
  42. Emanuele Viola. Quadratic maps are hard to sample. ACM Trans. Computation Theory, 8(4), 2016. Google Scholar
  43. Emanuele Viola. Special topics in complexity theory. ECCC lecture notes. Also available at, 2017.
  44. Emanuele Viola. Sampling lower bounds: boolean average-case and permutations. SIAM J. on Computing, 49(1), 2020. Available at URL:
  45. Emanuele Viola. AC0 unpredictability. ACM Trans. Computation Theory, 13(1), 2021. Google Scholar
  46. Jake Wellens. Assorted results in boolean function complexity, uniform sampling and clique partitions of graphs. PhD thesis, Massachusetts Institute of Technology, 2020. Google Scholar