Paper 2018/001

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

Benny Applebaum and Barak Arkis

Abstract

Consider the following secret-sharing problem. Your goal is to distribute a long file s between n servers such that (d1)-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 predefined list L. How small can the information ratio (i.e., the number of bits stored on a server per each bit of the secret) be? We initiate the study of such d-uniform access structures, and view them as a useful scaled-down version of general access structures. Our main result shows that, for constant , any -uniform access structure admits a secret sharing scheme with a *constant* asymptotic information ratio of that does not grow with the number of servers . This result is based on a new construction of -party Conditional Disclosure of Secrets (Gertner et al., JCSS '00) for arbitrary predicates over -size domain in which each party communicates at most four bits per secret bit. In both settings, previous results achieved non-constant information ratio which grows asymptotically with even for the simpler (and widely studied) special case of . Moreover, our results provide a unique example for a natural class of access structures that can be realized with information rate smaller than its bit-representation length (i.e., for -uniform access structures) showing that amortization can beat the representation size barrier. 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, point out some limitations of existing lower-bound techniques, and describe some applications to the setting of private simultaneous messages.

Metadata
Available format(s)
PDF
Category
Foundations
Publication info
Preprint. MINOR revision.
Keywords
secret sharinginformation-theoretic cryptography
Contact author(s)
benny applebaum @ gmail com
History
2018-09-23: revised
2018-01-02: received
See all versions
Short URL
https://ia.cr/2018/001
License
Creative Commons Attribution
CC BY

BibTeX

@misc{cryptoeprint:2018/001,
      author = {Benny Applebaum and Barak Arkis},
      title = {On the Power of Amortization in Secret Sharing: $d$-Uniform Secret Sharing and {CDS} with Constant Information Rate},
      howpublished = {Cryptology {ePrint} Archive, Paper 2018/001},
      year = {2018},
      url = {https://eprint.iacr.org/2018/001}
}
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