**Secret Sharing with Binary Shares**

*Fuchun Lin and Mahdi Cheraghchi and Venkatesan Guruswami and Reihaneh Safavi-Naini and Huaxiong Wang*

**Abstract: **Shamir's celebrated secret sharing scheme provides an efficient method for
encoding a secret of arbitrary length \ell among any N \leq 2^\ell 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 \ell, 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
\ell/g. Other than the bound related to secret length \ell, 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 \leq \tau < \rho \leq 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)).

**Category / Keywords: **Secret Sharing Scheme, Stochastic Code, Randomness Extractors, Wiretap Channel

**Original Publication**** (with minor differences): **ITCS2019

**Date: **received 10 Aug 2018, last revised 12 Dec 2018

**Contact author: **fclin at ntu edu sg

**Available format(s): **PDF | BibTeX Citation

**Version: **20181213:020635 (All versions of this report)

**Short URL: **ia.cr/2018/746

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