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On the Power of Adaptivity for Function Inversion

Authors Karthik Gajulapalli, Alexander Golovnev, Samuel King

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LIPIcs.ITC.2024.5.pdf
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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.

Cite AsGet BibTex

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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