**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 $(d-1)$-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 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 $d$, any $d$-uniform access structure admits a secret sharing scheme with a *constant* asymptotic information ratio of $c_d$ 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 (Gertner et al., JCSS '00) for arbitrary predicates over $n$-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 $n$ even for the simpler (and widely studied) special case of $d=2$. Moreover, our results provide a unique example for a natural class of access structures $F$ that can be realized with information rate smaller than its bit-representation length $\log |F|$ (i.e., $\Omega( d \log n)$ for $d$-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.

**Category / Keywords: **foundations / secret sharing, information-theoretic cryptography

**Date: **received 25 Dec 2017, last revised 23 Sep 2018

**Contact author: **benny applebaum at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20180923:142732 (All versions of this report)

**Short URL: **ia.cr/2018/001

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