Query Complexity Lower Bounds for Local List-Decoding and Hard-Core Predicates (Even for Small Rate and Huge Lists)

Authors Noga Ron-Zewi, Ronen Shaltiel, Nithin Varma

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Noga Ron-Zewi
  • University of Haifa, Israel
Ronen Shaltiel
  • University of Haifa, Israel
Nithin Varma
  • University of Haifa, Israel


We are grateful to Ilan Newman for participating in early stages of this research and for many helpful discussions.

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Noga Ron-Zewi, Ronen Shaltiel, and Nithin Varma. Query Complexity Lower Bounds for Local List-Decoding and Hard-Core Predicates (Even for Small Rate and Huge Lists). In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


A binary code Enc:{0,1}^k → {0,1}ⁿ is (1/2-ε,L)-list decodable if for every w ∈ {0,1}ⁿ, there exists a set List(w) of size at most L, containing all messages m ∈ {0,1}^k such that the relative Hamming distance between Enc(m) and w is at most 1/2-ε. A q-query local list-decoder for Enc is a randomized procedure Dec that when given oracle access to a string w, makes at most q oracle calls, and for every message m ∈ List(w), with high probability, there exists j ∈ [L] such that for every i ∈ [k], with high probability, Dec^w(i,j) = m_i. We prove lower bounds on q, that apply even if L is huge (say L = 2^{k^{0.9}}) and the rate of Enc is small (meaning that n ≥ 2^{k}): - For ε = 1/k^{ν} for some constant 0 < ν < 1, we prove a lower bound of q = Ω(log(1/δ)/ε²), where δ is the error probability of the local list-decoder. This bound is tight as there is a matching upper bound by Goldreich and Levin (STOC 1989) of q = O(log(1/δ)/ε²) for the Hadamard code (which has n = 2^k). This bound extends an earlier work of Grinberg, Shaltiel and Viola (FOCS 2018) which only works if n ≤ 2^{k^ν} and the number of coins tossed by Dec is small (and therefore does not apply to the Hadamard code, or other codes with low rate). - For smaller ε, we prove a lower bound of roughly q = Ω(1/(√ε)). To the best of our knowledge, this is the first lower bound on the number of queries of local list-decoders that gives q ≥ k for small ε. Local list-decoders with small ε form the key component in the celebrated theorem of Goldreich and Levin that extracts a hard-core predicate from a one-way function. We show that black-box proofs cannot improve the Goldreich-Levin theorem and produce a hard-core predicate that is hard to predict with probability 1/2 + 1/𝓁^ω(1) when provided with a one-way function f:{0,1}^𝓁 → {0,1}^𝓁, where f is such that circuits of size poly(𝓁) cannot invert f with probability ρ = 1/2^√𝓁 (or even ρ = 1/2^Ω(𝓁)). This limitation applies to any proof by black-box reduction (even if the reduction is allowed to use nonuniformity and has oracle access to f).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Local list-decoding
  • Hard-core predicates
  • Black-box reduction
  • Hadamard code


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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. URL: https://doi.org/10.1016/0168-0072(83)90038-6.
  2. Benny Applebaum, Sergei Artemenko, Ronen Shaltiel, and Guang Yang. Incompressible functions, relative-error extractors, and the power of nondeterministic reductions. Comput. Complex., 25(2):349-418, 2016. URL: https://doi.org/10.1007/s00037-016-0128-9.
  3. Sergei Artemenko and Ronen Shaltiel. Lower bounds on the query complexity of non-uniform and adaptive reductions showing hardness amplification. In Leslie Ann Goldberg, Klaus Jansen, R. Ravi, and José D. P. Rolim, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques - 14th International Workshop, APPROX 2011, and 15th International Workshop, RANDOM 2011, Princeton, NJ, USA, August 17-19, 2011. Proceedings, volume 6845 of Lecture Notes in Computer Science, pages 377-388. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-22935-0_32.
  4. Gil Cohen, Anat Ganor, and Ran Raz. Two sides of the coin problem. In Klaus Jansen, José D. P. Rolim, Nikhil R. Devanur, and Cristopher Moore, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2014, September 4-6, 2014, Barcelona, Spain, volume 28 of LIPIcs, pages 618-629. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2014. URL: https://doi.org/10.4230/LIPIcs.APPROX-RANDOM.2014.618.
  5. Yevgeniy Dodis, Abhishek Jain, Tal Moran, and Daniel Wichs. Counterexamples to hardness amplification beyond negligible. In Ronald Cramer, editor, Theory of Cryptography - 9th Theory of Cryptography Conference, TCC 2012, Taormina, Sicily, Italy, March 19-21, 2012. Proceedings, volume 7194 of Lecture Notes in Computer Science, pages 476-493. Springer, 2012. URL: https://doi.org/10.1007/978-3-642-28914-9_27.
  6. Oded Goldreich and Leonid A. Levin. A hard-core predicate for all one-way functions. In David S. Johnson, editor, Proceedings of the 21st Annual ACM Symposium on Theory of Computing, May 14-17, 1989, Seattle, Washigton, USA, pages 25-32. ACM, 1989. URL: https://doi.org/10.1145/73007.73010.
  7. Aryeh Grinberg, Ronen Shaltiel, and Emanuele Viola. Indistinguishability by adaptive procedures with advice, and lower bounds on hardness amplification proofs. In Mikkel Thorup, editor, 59th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2018, Paris, France, October 7-9, 2018, pages 956-966. IEEE Computer Society, 2018. URL: https://doi.org/10.1109/FOCS.2018.00094.
  8. Venkatesan Guruswami. Algorithmic results in list decoding. Foundations and Trends in Theoretical Computer Science, 2(2), 2006. URL: https://doi.org/10.1561/0400000007.
  9. Dan Gutfreund and Guy N. Rothblum. The complexity of local list decoding. In Ashish Goel, Klaus Jansen, José D. P. Rolim, and Ronitt Rubinfeld, editors, Approximation, Randomization and Combinatorial Optimization. Algorithms and Techniques, 11th International Workshop, APPROX 2008, and 12th International Workshop, RANDOM 2008, Boston, MA, USA, August 25-27, 2008. Proceedings, volume 5171 of Lecture Notes in Computer Science, pages 455-468. Springer, 2008. URL: https://doi.org/10.1007/978-3-540-85363-3_36.
  10. Nutan Limaye, Karteek Sreenivasaiah, Srikanth Srinivasan, Utkarsh Tripathi, and S. Venkitesh. A fixed-depth size-hierarchy theorem for AC^0[⊕] via the coin problem. In Moses Charikar and Edith Cohen, editors, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019, Phoenix, AZ, USA, June 23-26, 2019, pages 442-453. ACM, 2019. URL: https://doi.org/10.1145/3313276.3316339.
  11. Sofya Raskhodnikova, Noga Ron-Zewi, and Nithin Varma. Erasures versus errors in local decoding and property testing. Electronic Colloquium on Computational Complexity (ECCC), 25:195, 2018. URL: https://doi.org/10.4230/LIPIcs.ITCS.2019.63.
  12. Omer Reingold, Luca Trevisan, and Salil P. Vadhan. Notions of reducibility between cryptographic primitives. In Moni Naor, editor, Theory of Cryptography, First Theory of Cryptography Conference, TCC 2004, Cambridge, MA, USA, February 19-21, 2004, Proceedings, volume 2951 of Lecture Notes in Computer Science, pages 1-20. Springer, 2004. URL: https://doi.org/10.1007/978-3-540-24638-1_1.
  13. Noga Ron-Zewi, Ronen Shaltiel, and Nithin Varma. Query complexity lower bounds for local list-decoding and hard-core predicates (even for small rate and huge lists). Electron. Colloquium Comput. Complex., 27:133, 2020. URL: https://eccc.weizmann.ac.il/report/2020/133.
  14. Ronen Shaltiel. Is it possible to improve Yao’s XOR lemma using reductions that exploit the efficiency of their oracle? In Jaroslaw Byrka and Raghu Meka, editors, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2020, August 17-19, 2020, Virtual Conference, volume 176 of LIPIcs, pages 10:1-10:20. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2020.10.
  15. Ronen Shaltiel and Emanuele Viola. Hardness amplification proofs require majority. SIAM J. Comput., 39(7):3122-3154, 2010. URL: https://doi.org/10.1137/080735096.
  16. Amnon Ta-Shma and David Zuckerman. Extractor codes. IEEE Trans. Inf. Theory, 50(12):3015-3025, 2004. URL: https://doi.org/10.1109/TIT.2004.838377.
  17. Emanuele Viola. The complexity of hardness amplification and derandomization, 2006. Google Scholar
  18. Sergey Yekhanin. Locally decodable codes. Found. Trends Theor. Comput. Sci., 6(3):139-255, 2012. URL: https://doi.org/10.1561/0400000030.
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