Abstract
Secretsharing allows splitting a piece of secret information among a group of shareholders, so that it takes a large enough subset of them to recover it. In weighted secretsharing, each shareholder has an integer weight, and it takes a subset of largeenough weight to recover the secret. Schemes in the literature for weighted threshold secret sharing either have share sizes that grow linearly with the total weight, or ones that depend on huge public information (essentially a garbled circuit) of size (quasi)polynomial in the number of parties.
To do better, we investigate a relaxation, (α, β)ramp weighted secret sharing, where subsets of weight β W can recover the secret (with W the total weight), but subsets of weight α W or less cannot learn anything about it. These can be constructed from standard secretsharing schemes, but known constructions require long shares even for short secrets, achieving share sizes of max(W,secret/ε), where ε = βα. In this note we first observe that simple rounding let us replace the total weight W by N/ε, where N is the number of parties. Combined with known constructions, this yields share sizes of O(max(N,secret)/ε).
Our main contribution is a novel connection between weighted secret sharing and wiretap channels, that improves or even eliminates the dependence on N, at a price of increased dependence on 1/ε. We observe that for certain additivenoise (ℛ,?) wiretap channels, any semantically secure scheme can be naturally transformed into an (α,β)ramp weighted secretsharing, where α,β are essentially the respective capacities of the channels ?,ℛ. We present two instantiations of this type of construction, one using Binary Symmetric wiretap Channels, and the other using additive Gaussian Wiretap Channels. Depending on the parameters of the underlying wiretap channels, this gives rise to (α, β)ramp schemes with share sizes secret⋅log N/poly(ε) or even just secret/poly(ε).
BibTeX  Entry
@InProceedings{benhamouda_et_al:LIPIcs.ITC.2023.8,
author = {Benhamouda, Fabrice and Halevi, Shai and Stambler, Lev},
title = {{Weighted Secret Sharing from Wiretap Channels}},
booktitle = {4th Conference on InformationTheoretic Cryptography (ITC 2023)},
pages = {8:18:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772716},
ISSN = {18688969},
year = {2023},
volume = {267},
editor = {Chung, KaiMin},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2023/18336},
URN = {urn:nbn:de:0030drops183365},
doi = {10.4230/LIPIcs.ITC.2023.8},
annote = {Keywords: Secret sharing, ramp weighted secret sharing, wiretap channel}
}
Keywords: 

Secret sharing, ramp weighted secret sharing, wiretap channel 
Collection: 

4th Conference on InformationTheoretic Cryptography (ITC 2023) 
Issue Date: 

2023 
Date of publication: 

21.07.2023 