Improving the Linear Programming Technique in the Search for Lower Bounds in Secret Sharing

We present a new improvement in the linear programming technique to derive lower bounds on the information ratio of secret sharing schemes. We obtain non-Shannon-type bounds without using information inequalities explicitly. Our new technique makes it possible to determine the optimal information ratio of linear secret sharing schemes for all access structures on 5 participants and all graph-based access structures on 6 participants. In addition, new lower bounds are presented also for some small matroid ports and, in particular, the optimal information ratios of the linear secret sharing schemes for the ports of the Vamos matroid are determined

Secret sharing schemes: Optimizing the information ratio

Secret sharing refers to methods used to distribute a secret value among a set of participants. This work deals with the optimization of two parameters regarding the efficiency of a secret sharing scheme: the information ratio and average information ratio. Only access structures (a special family of sets) on 5 and 6 participants will be considered. First, access structures with 5 participants will be studied, followed by the ones on 6 participants that are based on graphs. The main goal of the paper is to check existing lower bounds (and improve some of them) by using linear programs with the sage solver. Shannon information inequalities have been used to translate the polymatroid axioms into linear constraints

Access structures for finding characteristic-dependent linear rank inequalities

summary:Determining information ratios of access structures is an important problem in secret sharing. Information inequalities and linear rank inequalities play an important role for proving bounds on these ratios. Characteristic-dependent linear rank inequalities are rank inequalities which are true over vector spaces with specific field characteristic. In this paper, using ideas of secret sharing, we show a theorem that produces characteristic-dependent linear rank inequalities. These inequalities are then used for getting lower bounds on information ratios of some access structures in linear secret sharing

On the information ratio of non-perfect secret sharing schemes

The final publication is available at Springer via http://dx.doi.org/10.1007/s00453-016-0217-9A secret sharing scheme is non-perfect if some subsets of players that cannot recover the secret value have partial information about it. The information ratio of a secret sharing scheme is the ratio between the maximum length of the shares and the length of the secret. This work is dedicated to the search of bounds on the information ratio of non-perfect secret sharing schemes and the construction of efficient linear non-perfect secret sharing schemes. To this end, we extend the known connections between matroids, polymatroids and perfect secret sharing schemes to the non-perfect case. In order to study non-perfect secret sharing schemes in all generality, we describe their structure through their access function, a real function that measures the amount of information on the secret value that is obtained by each subset of players. We prove that there exists a secret sharing scheme for every access function. Uniform access functions, that is, access functions whose values depend only on the number of players, generalize the threshold access structures. The optimal information ratio of the uniform access functions with rational values has been determined by Yoshida, Fujiwara and Fossorier. By using the tools that are described in our work, we provide a much simpler proof of that result and we extend it to access functions with real values.Peer ReviewedPostprint (author's final draft

How to Use Undiscovered Information Inequalities: Direct Applications of the Copy Lemma

We discuss linear programming techniques that help to deduce corollaries of non classic inequalities for Shannon's entropy. We focus on direct applications of the copy lemma. These applications involve implicitly some (known or unknown) non-classic universal inequalities for Shannon's entropy, though we do not derive these inequalities explicitly. To reduce the computational complexity of these problems, we extensively use symmetry considerations. We present two examples of usage of these techniques: we provide a reduced size formal inference of the best known bound for the Ingleton score (originally proven by Dougherty \emph{et al.} with explicitly derived non Shannon type inequalities), and improve the lower bound for the optimal information ratio of the secret sharing scheme for an access structure based on the Vamos matroid.Comment: An extended version of the paper accepted to IEEE ISIT 201

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

Bounds on Guessing Numbers and Secret Sharing Combining Information Theory Methods

This paper is on developing some computer-assisted proof methods involving non-classical inequalities for Shannon entropy. Two areas of the applications of information inequalities are studied: Secret sharing schemes and hat guessing games. In the former a random secret value is transformed into shares distributed among several participants in such a way that only the qualified groups of participants can recover the secret value. In the latter each participant is assigned a hat colour and they try to guess theirs while seeing only some of the others'. The aim is to maximize the probability that every player guesses correctly, the optimal probability depends on the underlying sight graph. We use for both problems the method of non-Shannon-type information inequalities going back to Z. Zhang and R. W. Yeung. We employ the linear programming technique that allows to apply new information inequalities indirectly, without even writing them down explicitly. To reduce the complexity of the problems of linear programming involved in the bounds we extensively use symmetry considerations. Using these tools, we improve lower bounds on the ratio of key size to secret size for the former problem and an upper bound for one of the ten vertex graphs related to an open question by Riis for the latter problem.Comment: A preliminary version of the results presented in section 4 (bounds on the information ratio of access structures for secret sharing schemes) was published in proceedings of IEEE ISIT, the text of which is available as arXiv:2201.1165

Quantum Cryptography Beyond Quantum Key Distribution

Quantum cryptography is the art and science of exploiting quantum mechanical effects in order to perform cryptographic tasks. While the most well-known example of this discipline is quantum key distribution (QKD), there exist many other applications such as quantum money, randomness generation, secure two- and multi-party computation and delegated quantum computation. Quantum cryptography also studies the limitations and challenges resulting from quantum adversaries---including the impossibility of quantum bit commitment, the difficulty of quantum rewinding and the definition of quantum security models for classical primitives. In this review article, aimed primarily at cryptographers unfamiliar with the quantum world, we survey the area of theoretical quantum cryptography, with an emphasis on the constructions and limitations beyond the realm of QKD.Comment: 45 pages, over 245 reference