## CryptoDB

### Amos Beimel

#### Publications

Year
Venue
Title
2019
EUROCRYPT
A secret-sharing scheme allows some authorized sets of parties to reconstruct a secret; the collection of authorized sets is called the access structure. For over 30 years, it was known that any (monotone) collection of authorized sets can be realized by a secret-sharing scheme whose shares are of size $2^{n-o(n)}$ and until recently no better scheme was known. In a recent breakthrough, Liu and Vaikuntanathan (STOC 2018) have reduced the share size to $O(2^{0.994n})$. Our first contribution is improving the exponent of secret sharing down to 0.892. For the special case of linear secret-sharing schemes, we get an exponent of 0.942 (compared to 0.999 of Liu and Vaikuntanathan).Motivated by the construction of Liu and Vaikuntanathan, we study secret-sharing schemes for uniform access structures. An access structure is k-uniform if all sets of size larger than k are authorized, all sets of size smaller than k are unauthorized, and each set of size k can be either authorized or unauthorized. The construction of Liu and Vaikuntanathan starts from protocols for conditional disclosure of secrets, constructs secret-sharing schemes for uniform access structures from them, and combines these schemes in order to obtain secret-sharing schemes for general access structures. Our second contribution in this paper is constructions of secret-sharing schemes for uniform access structures. We achieve the following results:A secret-sharing scheme for k-uniform access structures for large secrets in which the share size is $O(k^2)$ times the size of the secret.A linear secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is $\tilde{O}(2^{h(k/n)n/2})$ (where h is the binary entropy function). By counting arguments, this construction is optimal (up to polynomial factors).A secret-sharing scheme for k-uniform access structures for a binary secret in which the share size is $2^{\tilde{O}(\sqrt{k \log n})}$. Our third contribution is a construction of ad-hoc PSM protocols, i.e., PSM protocols in which only a subset of the parties will compute a function on their inputs. This result is based on ideas we used in the construction of secret-sharing schemes for k-uniform access structures for a binary secret.
2018
EUROCRYPT
2018
ASIACRYPT
In a k-party CDS protocol, each party sends one message to a referee (without seeing the other messages) such that the referee will learn a secret held by the parties if and only if the inputs of the parties satisfy some condition (e.g., if the inputs are all equal). This simple primitive is used to construct attribute based encryption, symmetrically-private information retrieval, priced oblivious transfer, and secret-sharing schemes for any access structure. Motivated by these applications, CDS protocols have been recently studied in many papers.In this work, we study linear CDS protocols, where each of the messages of the parties is a linear function of the secret and random elements taken from some finite field. Linearity is an important property of CDS protocols as many applications of CDS protocols required it.Our main result is a construction of linear k-party CDS protocols for an arbitrary function $f:[N]^{k}\rightarrow \left\{ 0,1 \right\}$ with messages of size $O(N^{(k-1)/2})$ (a similar result was independently and in parallel proven by Liu et al. [27]). By a lower bound of Beimel et al. [TCC 2017], this message size is optimal. We also consider functions with few inputs that return 1, and design more efficient CDS protocols for them.CDS protocols can be used to construct secret-sharing schemes for uniform access structures, where for some k all sets of size less than k are unauthorized, all sets of size greater than k are authorized, and each set of size k can be either authorized or unauthorized. We show that our results imply that every k-uniform access structure with n parties can be realized by a linear secret-sharing scheme with share size $\min \left\{ (O(n/k))^{(k-1)/2},O(n \cdot 2^{n/2}) \right\}$. Furthermore, the linear k-party CDS protocol with messages of size $O(N^{(k-1)/2})$ was recently used by Liu and Vaikuntanathan [STOC 2018] to construct a linear secret-sharing scheme with share size $O(2^{0.999n})$ for any n-party access structure.
2017
EUROCRYPT
2017
TCC
2016
JOFC
2015
JOFC
2015
TCC
2014
CRYPTO
2014
TCC
2014
TCC
2012
CRYPTO
2011
CRYPTO
2010
TCC
2010
JOFC
2010
CRYPTO
2009
TCC
2008
TCC
2008
CRYPTO
2007
CRYPTO
2007
TCC
2007
TCC
2007
JOFC
2006
TCC
2005
TCC
2004
TCC
2004
JOFC
2003
JOFC
2001
EPRINT
A secret-sharing scheme enables a dealer to distribute a secret among n parties such that only some predefined authorized sets of parties will be able to reconstruct the secret from their shares. The (monotone) collection of authorized sets is called an access structure, and is freely identified with its characteristic monotone function f:{0,1}^n --> {0,1}. A family of secret-sharing schemes is called efficient if the total length of the n shares is polynomial in n. Most previously known secret-sharing schemes belonged to a class of linear schemes, whose complexity coincides with the monotone span program size of their access structure. Prior to this work there was no evidence that nonlinear schemes can be significantly more efficient than linear schemes, and in particular there were no candidates for schemes efficiently realizing access structures which do not lie in NC. The main contribution of this work is the construction of two efficient nonlinear schemes: (1) A scheme with perfect privacy whose access structure is conjectured not to lie in NC; (2) A scheme with statistical privacy whose access structure is conjectured not to lie in P/poly. Another contribution is the study of a class of nonlinear schemes, termed quasi-linear schemes, obtained by composing linear schemes over different fields. We show that while these schemes are possibly (super-polynomially) more powerful than linear schemes, they cannot efficiently realize access structures outside NC.
2000
CRYPTO
1999
CRYPTO
1995
CRYPTO
1993
CRYPTO
1992
CRYPTO

#### Program Committees

Eurocrypt 2019
TCC 2018 (Program chair)
TCC 2018
TCC 2016
TCC 2014
TCC 2012
TCC 2010
Crypto 2010
Crypto 2007
PKC 2006
TCC 2005
Crypto 2005