Secret Sharing Schemes with a large number of players from Toric Varieties

A general theory for constructing linear secret sharing schemes over a finite field \Fq from toric varieties is introduced. The number of players can be as large as $(q-1)^r-1$ for $r\geq 1$. We present general methods for obtaining the reconstruction and privacy thresholds as well as conditions for multiplication on the associated secret sharing schemes. In particular we apply the method on certain toric surfaces. The main results are ideal linear secret sharing schemes where the number of players can be as large as $(q-1)^2-1$. We determine bounds for the reconstruction and privacy thresholds and conditions for strong multiplication using the cohomology and the intersection theory on toric surfaces.Comment: 15 pages, 4 figures. arXiv admin note: text overlap with arXiv:1203.454

Improving Quantum Secret-Sharing Schemes

We propose a protocol that enables a dealer to share a quantum secret with n players using less than n quantum shares for several access structures. For threshold schemes we derived an expression that shows how many quantum shares can be saved in this scheme. Also, several features that are available for classical secret-sharing schemes (and previously not known to be possible for quantum secret-sharing) become available with this protocol

Efficient data intensive secure computation : fictional or real

Secure computation has the potential to completely reshape the cybersecruity landscape, but this will happen only if we can make it practical. Despite significant improvements recently, secure computation is still orders of magnitude slower than computation in the clear. Even with the latest technology, running the killer apps, which are often data intensive, in secure computation is still a mission impossible. In this paper, I present two approaches that could lead to practical data intensive secure computation. The first approach is by designing data structures. Traditionally, data structures have been widely used in computer science to improve performance of computation. However, in secure computation they have been largely overlooked in the past. I will show that data structures could be effective performance boosters in secure computation. Another approach is by using fully homomorphic encryption (FHE). A common belief is that FHE is too inefficient to have any practical applications for the time being. Contrary to this common belief, I will show that in some cases FHE can actually lead to very efficient secure computation protocols. This is due to the high degree of internal parallelism in recent FHE schemes. The two approaches are explained with Private Set Intersection (PSI) as an example. I will also show the performance figures measured from prototype implementations

An Epitome of Multi Secret Sharing Schemes for General Access Structure

Secret sharing schemes are widely used now a days in various applications, which need more security, trust and reliability. In secret sharing scheme, the secret is divided among the participants and only authorized set of participants can recover the secret by combining their shares. The authorized set of participants are called access structure of the scheme. In Multi-Secret Sharing Scheme (MSSS), k different secrets are distributed among the participants, each one according to an access structure. Multi-secret sharing schemes have been studied extensively by the cryptographic community. Number of schemes are proposed for the threshold multi-secret sharing and multi-secret sharing according to generalized access structure with various features. In this survey we explore the important constructions of multi-secret sharing for the generalized access structure with their merits and demerits. The features like whether shares can be reused, participants can be enrolled or dis-enrolled efficiently, whether shares have to modified in the renewal phase etc., are considered for the evaluation

Efficient Multi-Point Local Decoding of Reed-Muller Codes via Interleaved Codex

Reed-Muller codes are among the most important classes of locally correctable codes. Currently local decoding of Reed-Muller codes is based on decoding on lines or quadratic curves to recover one single coordinate. To recover multiple coordinates simultaneously, the naive way is to repeat the local decoding for recovery of a single coordinate. This decoding algorithm might be more expensive, i.e., require higher query complexity. In this paper, we focus on Reed-Muller codes with usual parameter regime, namely, the total degree of evaluation polynomials is $d=\Theta({q})$, where $q$ is the code alphabet size (in fact, $d$ can be as big as $q/4$ in our setting). By introducing a novel variation of codex, i.e., interleaved codex (the concept of codex has been used for arithmetic secret sharing \cite{C11,CCX12}), we are able to locally recover arbitrarily large number $k$ of coordinates of a Reed-Muller code simultaneously at the cost of querying $O(q^2k)$ coordinates. It turns out that our local decoding of Reed-Muller codes shows ({\it perhaps surprisingly}) that accessing $k$ locations is in fact cheaper than repeating the procedure for accessing a single location for $k$ times. Our estimation of success error probability is based on error probability bound for $t$-wise linearly independent variables given in \cite{BR94}