Paper 2013/105

Lossy Chains and Fractional Secret Sharing

Yuval Ishai, Eyal Kushilevitz, and Omer Strulovich

Abstract

Motivated by the goal of controlling the amount of work required to access a shared resource or to solve a cryptographic puzzle, we introduce and study the related notions of {\em lossy chains} and {\em fractional secret sharing}. Fractional secret sharing generalizes traditional secret sharing by allowing a fine-grained control over the amount of uncertainty about the secret. More concretely, a fractional secret sharing scheme realizes a fractional access structure $f:2^{[n]}\to [m]$ by guaranteeing that from the point of view of each set $T\subseteq [n]$ of parties, the secret is {\em uniformly} distributed over a set of $f(T)$ potential secrets. We show that every (monotone) fractional access structure can be realized. For {\em symmetric} structures, in which $f(T)$ depends only on the size of $T$, we give an efficient construction with share size $poly(n,\log m)$. Our construction of fractional secret sharing schemes is based on the new notion of {\em lossy chains} which may be of independent interest. A lossy chain is a Markov chain $(X_0,\ldots,X_n)$ which starts with a random secret $X_0$ and gradually loses information about it at a rate which is specified by a {\em loss function} $g$. Concretely, in every step $t$, the distribution of $X_0$ conditioned on the value of $X_t$ should always be uniformly distributed over a set of size $g(t)$. We show how to construct such lossy chains efficiently for any possible loss function $g$, and prove that our construction achieves an optimal asymptotic information rate.

Metadata
Available format(s)
PDF
Category
Foundations
Publication info
Published elsewhere. The 30th Symposium on Theoretical Aspects of Computer Science (STACS 2013)
Keywords
Secret sharingMarkov chains
Contact author(s)
yuvali @ cs technion ac il
History
2013-02-27: received
Short URL
https://ia.cr/2013/105
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2013/105,
      author = {Yuval Ishai and Eyal Kushilevitz and Omer Strulovich},
      title = {Lossy Chains and Fractional Secret Sharing},
      howpublished = {Cryptology {ePrint} Archive, Paper 2013/105},
      year = {2013},
      url = {https://eprint.iacr.org/2013/105}
}
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