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Secret Sharing with Binary Shares

Authors Fuchun Lin, Mahdi Cheraghchi, Venkatesan Guruswami, Reihaneh Safavi-Naini, Huaxiong Wang



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Author Details

Fuchun Lin
  • Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, SG
Mahdi Cheraghchi
  • Department of Computing, Imperial College London, UK
Venkatesan Guruswami
  • Computer Science Department, Carnegie Mellon University, USA
Reihaneh Safavi-Naini
  • Department of Computer Science, University of Calgary, CA
Huaxiong Wang
  • Division of Mathematical Sciences, School of Physical and Mathematical Sciences, Nanyang Technological University, SG

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Fuchun Lin, Mahdi Cheraghchi, Venkatesan Guruswami, Reihaneh Safavi-Naini, and Huaxiong Wang. Secret Sharing with Binary Shares. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 53:1-53:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)
https://doi.org/10.4230/LIPIcs.ITCS.2019.53

Abstract

Shamir's celebrated secret sharing scheme provides an efficient method for encoding a secret of arbitrary length l among any N <= 2^l players such that for a threshold parameter t, (i) the knowledge of any t shares does not reveal any information about the secret and, (ii) any choice of t+1 shares fully reveals the secret. It is known that any such threshold secret sharing scheme necessarily requires shares of length l, and in this sense Shamir's scheme is optimal. The more general notion of ramp schemes requires the reconstruction of secret from any t+g shares, for a positive integer gap parameter g. Ramp secret sharing scheme necessarily requires shares of length l/g. Other than the bound related to secret length l, the share lengths of ramp schemes can not go below a quantity that depends only on the gap ratio g/N. In this work, we study secret sharing in the extremal case of bit-long shares and arbitrarily small gap ratio g/N, where standard ramp secret sharing becomes impossible. We show, however, that a slightly relaxed but equally effective notion of semantic security for the secret, and negligible reconstruction error probability, eliminate the impossibility. Moreover, we provide explicit constructions of such schemes. One of the consequences of our relaxation is that, unlike standard ramp schemes with perfect secrecy, adaptive and non-adaptive adversaries need different analysis and construction. For non-adaptive adversaries, we explicitly construct secret sharing schemes that provide secrecy against any tau fraction of observed shares, and reconstruction from any rho fraction of shares, for any choices of 0 <= tau < rho <= 1. Our construction achieves secret length N(rho-tau-o(1)), which we show to be optimal. For adaptive adversaries, we construct explicit schemes attaining a secret length Omega(N(rho-tau)). We discuss our results and open questions.

Subject Classification

ACM Subject Classification
  • Security and privacy → Information-theoretic techniques
  • Theory of computation → Expander graphs and randomness extractors
  • Theory of computation → Error-correcting codes
Keywords
  • Secret sharing scheme
  • Wiretap channel

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