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On the Complexity of Decomposable Randomized Encodings, Or: How Friendly Can a Garbling-Friendly PRF Be?

Authors Marshall Ball, Justin Holmgren, Yuval Ishai, Tianren Liu, Tal Malkin

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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Randomized Encoding
  • Private Simultaneous Messages

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Abstract

Garbling schemes, also known as decomposable randomized encodings (DRE), have found many applications in cryptography. However, despite a large body of work on constructing such schemes, very little is known about their limitations.
We initiate a systematic study of the DRE complexity of Boolean functions, obtaining the following main results: 
- Near-quadratic lower bounds. We use a classical lower bound technique of Nečiporuk [Dokl. Akad. Nauk SSSR '66] to show an Ω(n²/log n) lower bound on the size of any DRE for many explicit Boolean functions. For some natural functions, we obtain a corresponding upper bound, thus settling their DRE complexity up to polylogarithmic factors. Prior to our work, no superlinear lower bounds were known, even for non-explicit functions. 
- Garbling-friendly PRFs. We show that any exponentially secure PRF has Ω(n²/log n) DRE size, and present a plausible candidate for a "garbling-optimal" PRF that nearly meets this bound. This candidate establishes a barrier for super-quadratic DRE lower bounds via natural proof techniques. In contrast, we show a candidate for a weak PRF with near-exponential security and linear DRE size. 
 Our results establish several qualitative separations, including near-quadratic separations between computational and information-theoretic DRE size of Boolean functions, and between DRE size of weak vs. strong PRFs.

Cite As Get BibTex

Marshall Ball, Justin Holmgren, Yuval Ishai, Tianren Liu, and Tal Malkin. On the Complexity of Decomposable Randomized Encodings, Or: How Friendly Can a Garbling-Friendly PRF Be?. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 86:1-86:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ITCS.2020.86

Author Details

Marshall Ball
  • Columbia University, New York, NY, USA
Justin Holmgren
  • Simons Institute, Berkeley, CA, USA
Yuval Ishai
  • Technion - Israel Institute of Technology, Haifa, Israel
Tianren Liu
  • University of Washington, Seattle, WA, USA
Tal Malkin
  • Columbia University, New York, NY, USA

Funding

The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either express or implied, of ODNI, IARPA, or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.
  • Ball, Marshall: Supported by an IBM Research PhD Fellowship. This work is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) via Contract No. 2019-1902070006. Part of this work was performed while this author was visiting the third author at Technion in Haifa, Israel.
  • Holmgren, Justin: Research done in part while at IBM Research and in part while at Princeton University supported by the Simons Collaboration on Algorithms and Geometry.
  • Ishai, Yuval: Supported by ERC Project NTSC (742754), NSF-BSF grant 2015782, BSF grant 2018393, and a grant from the Ministry of Science and Technology, Israel and Department of Science and Technology, Government of India.
  • Liu, Tianren: This work was mostly done in Massachusetts Institute of Technology. Research supported by NSF Grants CNS-1350619, CNS-1414119 and CNS-1718161, an MIT-IBM grant and Vinod Vaikuntanathan’s DARPA Young Faculty Award.
  • Malkin, Tal: This research is based upon work supported in part by the Office of the Director of National Intelligence (ODNI), Intelligence Advanced Research Projects Activity (IARPA) via Contract No. 2019-1902070006.

Acknowledgements

We thank Igor Shparlinski for helpful discussions and pointers to the literature on the hidden shift problem. We also thank Siyao Guo, Lucas Kowlaczyk, Ron Rothblum, Jonathan Ullman, and Vinod Vaikuntanathan for related discussions and collaborations.

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