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A Tight Lower Bound for Entropy Flattening

Authors Yi-Hsiu Chen, Mika Göös, Salil P. Vadhan, Jiapeng Zhang

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ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
Keywords
  • Entropy
  • One-way function

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Abstract

We study entropy flattening: Given a circuit C_X implicitly describing an n-bit source X (namely, X is the output of C_X on a uniform random input), construct another circuit C_Y describing a source Y such that (1) source Y is nearly flat (uniform on its support), and (2) the Shannon entropy of Y is monotonically related to that of X. The standard solution is to have C_Y evaluate C_X altogether Theta(n^2) times on independent inputs and concatenate the results (correctness follows from the asymptotic equipartition property). In this paper, we show that this is optimal among black-box constructions: Any circuit C_Y for entropy flattening that repeatedly queries C_X as an oracle requires Omega(n^2) queries.
Entropy flattening is a component used in the constructions of pseudorandom generators and other cryptographic primitives from one-way functions [Johan Håstad et al., 1999; John Rompel, 1990; Thomas Holenstein, 2006; Iftach Haitner et al., 2006; Iftach Haitner et al., 2009; Iftach Haitner et al., 2013; Iftach Haitner et al., 2010; Salil P. Vadhan and Colin Jia Zheng, 2012]. It is also used in reductions between problems complete for statistical zero-knowledge [Tatsuaki Okamoto, 2000; Amit Sahai and Salil P. Vadhan, 1997; Oded Goldreich et al., 1999; Vadhan, 1999]. The Theta(n^2) query complexity is often the main efficiency bottleneck. Our lower bound can be viewed as a step towards proving that the current best construction of pseudorandom generator from arbitrary one-way functions by Vadhan and Zheng (STOC 2012) has optimal efficiency.

Cite As Get BibTex

Yi-Hsiu Chen, Mika Göös, Salil P. Vadhan, and Jiapeng Zhang. A Tight Lower Bound for Entropy Flattening. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 23:1-23:28, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.CCC.2018.23

Author Details

Yi-Hsiu Chen
  • School of Engineering and Applied Sciences, Harvard University, USA
Mika Göös
  • School of Engineering and Applied Sciences, Harvard University, USA
Salil P. Vadhan
  • Computer Science and Applied Mathematics, Harvard University, USA
Jiapeng Zhang
  • University of California San Diego, USA

Funding

  • Chen, Yi-Hsiu: Supported by NSF grant CCF-1749750
  • Göös, Mika: Supported by Michael O. Rabin Postdoctoral Fellowship
  • Vadhan, Salil P.: Supported by NSF grant CCF-1749750
  • Zhang, Jiapeng: Supported by NSF CCF-1614023

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