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 bitlong 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 nonadaptive adversaries need different analysis and construction. For nonadaptive 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(rhotauo(1)), which we show to be optimal. For adaptive adversaries, we construct explicit schemes attaining a secret length Omega(N(rhotau)). We discuss our results and open questions.
BibTeX  Entry
@InProceedings{lin_et_al:LIPIcs:2018:10146,
author = {Fuchun Lin and Mahdi Cheraghchi and Venkatesan Guruswami and Reihaneh SafaviNaini and Huaxiong Wang},
title = {{Secret Sharing with Binary Shares}},
booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)},
pages = {53:153:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959770958},
ISSN = {18688969},
year = {2018},
volume = {124},
editor = {Avrim Blum},
publisher = {Schloss DagstuhlLeibnizZentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2018/10146},
URN = {urn:nbn:de:0030drops101461},
doi = {10.4230/LIPIcs.ITCS.2019.53},
annote = {Keywords: Secret sharing scheme, Wiretap channel}
}
Keywords: 

Secret sharing scheme, Wiretap channel 
Collection: 

10th Innovations in Theoretical Computer Science Conference (ITCS 2019) 
Issue Date: 

2018 
Date of publication: 

08.01.2019 