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On the Power of Adaptivity for Function Inversion

Authors Karthik Gajulapalli, Alexander Golovnev, Samuel King

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Author Details

Karthik Gajulapalli
  • Georgetown University, Washington, DC, USA
Alexander Golovnev
  • Georgetown University, Washington, DC, USA
Samuel King
  • Georgetown University, Washington, DC, USA

Funding

This research is supported by the National Science Foundation CAREER award (grant CCF-2338730).

Acknowledgements

We would like to thank Spencer Peters for fruitful discussions on this topic. We are also grateful to the anonymous reviewers for their helpful comments.

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Karthik Gajulapalli, Alexander Golovnev, and Samuel King. On the Power of Adaptivity for Function Inversion. In 5th Conference on Information-Theoretic Cryptography (ITC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 304, pp. 5:1-5:10, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ITC.2024.5

Abstract

We study the problem of function inversion with preprocessing where, given a function f : [N] → [N] and a point y in its image, the goal is to find an x such that f(x) = y using at most T oracle queries to f and S bits of preprocessed advice that depend on f.
The seminal work of Corrigan-Gibbs and Kogan [TCC 2019] initiated a line of research that shows many exciting connections between the non-adaptive setting of this problem and other areas of theoretical computer science. Specifically, they introduced a very weak class of algorithms (strongly non-adaptive) where the points queried by the oracle depend only on the inversion point y, and are independent of the answers to the previous queries and the S bits of advice. They showed that proving even mild lower bounds on strongly non-adaptive algorithms for function inversion would imply a breakthrough result in circuit complexity.
We prove that every strongly non-adaptive algorithm for function inversion (and even for its special case of permutation inversion) must have ST = Ω(N log (N) log (T)). This gives the first improvement to the long-standing lower bound of ST = Ω(N log N) due to Yao [STOC 90]. As a corollary, we conclude the first separation between strongly non-adaptive and adaptive algorithms for permutation inversion, where the adaptive algorithm by Hellman [TOIT 80] achieves the trade-off ST = O(N log N).
Additionally, we show equivalence between lower bounds for strongly non-adaptive data structures and the one-way communication complexity of certain partial functions. As an example, we recover our lower bound on function inversion in the communication complexity framework.

Subject Classification

ACM Subject Classification
  • Theory of computation → Cryptographic primitives
Keywords
  • Function Inversion
  • Non-Adaptive lower bounds
  • Communication Complexity

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