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On the Power of Amortization in Secret Sharing: d-Uniform Secret Sharing and CDS with Constant Information Rate

Published:30 September 2020Publication History
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Abstract

Consider the following secret-sharing problem: A file s should be distributed between n servers such that (d-1)-subsets cannot recover the file, (d+1)-subsets can recover the file, and d-subsets should be able to recover s if and only if they appear in some pre-defined list L. The goal is to minimize the information ratio—that is, the number of bits stored on a server per each bit of the secret.

We show that for any constant d and any pre-defined list L, if the file is sufficiently long (exponential in nd), the problem can be solved with a constant asymptotic information ratio of cd that does not grow with the number of servers n. This result is based on a new construction of d-party conditional disclosure of secrets for arbitrary predicates over an n-size domain in which each party communicates at most four bits per secret bit.

In both settings, previous results achieved a non-constant information ratio that grows asymptotically with n, even for the simpler special case of d = 2. Moreover, our constructions yield the first example of an access structure whose amortized information ratio is constant, whereas its best-known non-amortized information ratio is sub-exponential, thus providing a unique evidence for the potential power of amortization in the context of secret sharing.

Our main result applies to exponentially long secrets, and so it should be mainly viewed as a barrier against amortizable lower-bound techniques. We also show that in some natural simple cases (e.g., low-degree predicates), amortization kicks in even for quasi-polynomially long secrets. Finally, we prove some limited lower bounds and point out some limitations of existing lower-bound techniques.

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      • Published in

        cover image ACM Transactions on Computation Theory
        ACM Transactions on Computation Theory  Volume 12, Issue 4
        December 2020
        156 pages
        ISSN:1942-3454
        EISSN:1942-3462
        DOI:10.1145/3427631
        Issue’s Table of Contents

        Copyright © 2020 ACM

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        Association for Computing Machinery

        New York, NY, United States

        Publication History

        • Published: 30 September 2020
        • Accepted: 1 August 2020
        • Revised: 1 July 2020
        • Received: 1 June 2019
        Published in toct Volume 12, Issue 4

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