5,288 research outputs found
The Share Size of Secret-Sharing Schemes for Almost All Access Structures and Graphs
The share size of general secret-sharing schemes is poorly understood. The gap between the best known upper bound on the total share size per party of (Applebaum and Nir, CRYPTO 2021) and the best known lower bound of (Csirmaz, J. of Cryptology 1997) is huge (where is the number of parties in the scheme). To gain some understanding on this problem, we study the share size of secret-sharing schemes of almost all access structures, i.e., of almost all collections of authorized sets. This is motivated by the fact that in complexity, many times almost all objects are hardest (e.g., most Boolean functions require exponential size circuits). All previous constructions of secret-sharing schemes were for the worst access structures (i.e., all access structures) or for specific families of access structures.
We prove upper bounds on the share size for almost all access structures. We combine results on almost all monotone Boolean functions (Korshunov, Probl. Kibern. 1981) and a construction of (Liu and Vaikuntanathan, STOC 2018) and conclude that almost all access structures have a secret-sharing scheme with share size .
We also study graph secret-sharing schemes. In these schemes, the parties are vertices of a graph and a set can reconstruct the secret if and only if it contains an edge. Again, for this family there is a huge gap between the upper bounds - (Erdös and Pyber, Discrete Mathematics 1997) - and the lower bounds - (van Dijk, Des. Codes Crypto. 1995). We show that for almost all graphs, the share size of each party is . This result is achieved by using robust 2-server conditional disclosure of secrets protocols, a new primitive introduced and constructed in (Applebaum et al., STOC 2020), and the fact that the size of the maximal independent set in a random graph is small. Finally, using robust conditional disclosure of secrets protocols, we improve the total share size for all very dense graphs
Exact information ratios for secret sharing on small graphs with girth at least 5
In a secret-sharing scheme, a piece of information – the secret – is distributed among a finite set of participants in such a way that only some predefined coalitions can recover it.
The efficiency of the scheme is measured by the amount of information the most heavily loaded participant must remember.
This amount is called information ratio, and one of the most interesting problems of this topic is to calculate the exact information ratio of given structures.
In this paper, the information ratios of all but one graph-based schemes on 8 or 9 vertices with a girth at least 5 and all graph-based schemes on 10 vertices and 10 edges with a girth at least 5 are determined using two polyhedral combinatoric tools: the entropy method and covering with stars.
Beyond the investigation of new graphs, the paper contains a few improvements and corrections of recent results on graphs with 9 vertices.
Furthermore, we determine the exact information ratio of a large class of generalized sunlet graphs consisting of some pendant paths attached to a cycle of length at least 5
Entropy of Closure Operators
The entropy of a closure operator has been recently proposed for the study of
network coding and secret sharing. In this paper, we study closure operators in
relation to their entropy. We first introduce four different kinds of rank
functions for a given closure operator, which determine bounds on the entropy
of that operator. This yields new axioms for matroids based on their closure
operators. We also determine necessary conditions for a large class of closure
operators to be solvable. We then define the Shannon entropy of a closure
operator, and use it to prove that the set of closure entropies is dense.
Finally, we justify why we focus on the solvability of closure operators only.Comment: arXiv admin note: substantial text overlap with arXiv:1209.655
LDPC coded transmissions over the Gaussian broadcast channel with confidential messages
We design and assess some practical low-density parity-check (LDPC) coded
transmission schemes for the Gaussian broadcast channel with confidential
messages (BCC). This channel model is different from the classical wiretap
channel model as the unauthorized receiver (Eve) must be able to decode some
part of the information. Hence, the reliability and security targets are
different from those of the wiretap channel. In order to design and assess
practical coding schemes, we use the error rate as a metric of the performance
achieved by the authorized receiver (Bob) and the unauthorized receiver (Eve).
We study the system feasibility, and show that two different levels of
protection against noise are required on the public and the secret messages.
This can be achieved in two ways: i) by using LDPC codes with unequal error
protection (UEP) of the transmitted information bits or ii) by using two
classical non-UEP LDPC codes with different rates. We compare these two
approaches and show that, for the considered examples, the solution exploiting
UEP LDPC codes is more efficient than that using non-UEP LDPC codes.Comment: 5 pages, 5 figures, to be presented at IEEE ICT 201
Practical LDPC coded modulation schemes for the fading broadcast channel with confidential messages
The broadcast channel with confidential messages is a well studied scenario
from the theoretical standpoint, but there is still lack of practical schemes
able to achieve some fixed level of reliability and security over such a
channel. In this paper, we consider a quasi-static fading channel in which both
public and private messages must be sent from the transmitter to the receivers,
and we aim at designing suitable coding and modulation schemes to achieve such
a target. For this purpose, we adopt the error rate as a metric, by considering
that reliability (security) is achieved when a sufficiently low (high) error
rate is experienced at the receiving side. We show that some conditions exist
on the system feasibility, and that some outage probability must be tolerated
to cope with the fading nature of the channel. The proposed solution exploits
low-density parity-check codes with unequal error protection, which are able to
guarantee two different levels of protection against noise for the public and
the private information, in conjunction with different modulation schemes for
the public and the private message bits.Comment: 6 pages, 4 figures, to be presented at IEEE ICC'14 - Workshop on
Wireless Physical Layer Securit
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