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Lower Bound on Average-Case Complexity of Inversion of Goldreich’s Function by Drunken Backtracking Algorithms

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Abstract

We prove an exponential lower bound on the average time of inverting Goldreich’s function by drunken backtracking algorithms; this resolves the open question stated in Cook et al. (Proceedings of TCC, pp. 521–538, 2009). The Goldreich’s function has n binary inputs and n binary outputs. Every output depends on d inputs and is computed from them by the fixed predicate of arity d. Our Goldreich’s function is based on an expander graph and on the nonlinear predicates that are linear in Ω(d) variables. Drunken algorithm is a backtracking algorithm that somehow chooses a variable for splitting and randomly chooses the value for the variable to be investigated at first.

After the submission to the journal we found out that the same result was independently obtained by Rachel Miller.

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Acknowledgements

The author thanks Ilya Mironov for bringing attention to [5], Edward A. Hirsch and Dmitry Sokolov for fruitful discussions and also thanks Sonya Alexandrova, Anya Luter, Ilya Posov and anonymous reviewers for useful comments that improved the readability of the paper.

This work was partially supported by the grant MK-4108.2012.1 from the President of RF, by RFBR grants 12-01-31239-mol-a and 11-01-00760-a, by the Programme of Fundamental Research of RAS and by the Ministry of education and science of Russian Federation, project 8216.

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  1. Steklov Institute of Mathematics at St. Petersburg, 27 Fontanka, St. Petersburg, 191023, Russia

    Dmitry Itsykson

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  1. Dmitry Itsykson
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Itsykson, D. Lower Bound on Average-Case Complexity of Inversion of Goldreich’s Function by Drunken Backtracking Algorithms. Theory Comput Syst 54, 261–276 (2014). https://doi.org/10.1007/s00224-013-9514-8

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