Infinite Secret Sharing -- Examples

The motivation for extending secret sharing schemes to cases when either the set of players is infinite or the domain from which the secret and/or the shares are drawn is infinite or both, is similar to the case when switching to abstract probability spaces from classical combinatorial probability. It might shed new light on old problems, could connect seemingly unrelated problems, and unify diverse phenomena. Definitions equivalent in the finitary case could be very much different when switching to infinity, signifying their difference. The standard requirement that qualified subsets should be able to determine the secret has different interpretations in spite of the fact that, by assumption, all participants have infinite computing power. The requirement that unqualified subsets should have no, or limited information on the secret suggests that we also need some probability distribution. In the infinite case events with zero probability are not necessarily impossible, and we should decide whether bad events with zero probability are allowed or not. In this paper, rather than giving precise definitions, we enlist an abundance of hopefully interesting infinite secret sharing schemes. These schemes touch quite diverse areas of mathematics such as projective geometry, stochastic processes and Hilbert spaces. Nevertheless our main tools are from probability theory. The examples discussed here serve as foundation and illustration to the more theory oriented companion paper

Probabilistic Infinite Secret Sharing

The study of probabilistic secret sharing schemes using arbitrary probability spaces and possibly infinite number of participants lets us investigate abstract properties of such schemes. It highlights important properties, explains why certain definitions work better than others, connects this topic to other branches of mathematics, and might yield new design paradigms. A probabilistic secret sharing scheme is a joint probability distribution of the shares and the secret together with a collection of secret recovery functions for qualified subsets. The scheme is measurable if the recovery functions are measurable. Depending on how much information an unqualified subset might have, we define four scheme types: perfect, almost perfect, ramp, and almost ramp. Our main results characterize the access structures which can be realized by schemes of these types. We show that every access structure can be realized by a non-measurable perfect probabilistic scheme. The construction is based on a paradoxical pair of independent random variables which determine each other. For measurable schemes we have the following complete characterization. An access structure can be realized by a (measurable) perfect, or almost perfect scheme if and only if the access structure, as a subset of the Sierpi\'nski space $\{0,1\}^P$, is open, if and only if it can be realized by a span program. The access structure can be realized by a (measurable) ramp or almost ramp scheme if and only if the access structure is a $G_\delta$ set (intersection of countably many open sets) in the Sierpi\'nski topology, if and only if it can be realized by a Hilbert-space program

Demonstrating Continuous Variable EPR Steering in spite of Finite Experimental Capabilities using Fano Steering Bounds

We show how one can demonstrate continuous-variable Einstein-Podolsky-Rosen (EPR) steering without needing to characterize entire measurement probability distributions. To do this, we develop a modified Fano inequality useful for discrete measurements of continuous variables, and use it to bound the conditional uncertainties in continuous-variable entropic EPR-steering inequalities. With these bounds, we show how one can hedge against experimental limitations including a finite detector size, dead space between pixels, and any such factors that impose an incomplete sampling of the true measurement probability distribution. Furthermore, we use experimental data from the position and momentum statistics of entangled photon pairs in parametric downconversion to show that this method is sufficiently sensitive for practical use.Comment: 7 pages, 2 figure

An Expressive Model for the Web Infrastructure: Definition and Application to the BrowserID SSO System

The web constitutes a complex infrastructure and as demonstrated by numerous attacks, rigorous analysis of standards and web applications is indispensable. Inspired by successful prior work, in particular the work by Akhawe et al. as well as Bansal et al., in this work we propose a formal model for the web infrastructure. While unlike prior works, which aim at automatic analysis, our model so far is not directly amenable to automation, it is much more comprehensive and accurate with respect to the standards and specifications. As such, it can serve as a solid basis for the analysis of a broad range of standards and applications. As a case study and another important contribution of our work, we use our model to carry out the first rigorous analysis of the BrowserID system (a.k.a. Mozilla Persona), a recently developed complex real-world single sign-on system that employs technologies such as AJAX, cross-document messaging, and HTML5 web storage. Our analysis revealed a number of very critical flaws that could not have been captured in prior models. We propose fixes for the flaws, formally state relevant security properties, and prove that the fixed system in a setting with a so-called secondary identity provider satisfies these security properties in our model. The fixes for the most critical flaws have already been adopted by Mozilla and our findings have been rewarded by the Mozilla Security Bug Bounty Program.Comment: An abridged version appears in S&P 201

Complexity and Unwinding for Intransitive Noninterference

The paper considers several definitions of information flow security for intransitive policies from the point of view of the complexity of verifying whether a finite-state system is secure. The results are as follows. Checking (i) P-security (Goguen and Meseguer), (ii) IP-security (Haigh and Young), and (iii) TA-security (van der Meyden) are all in PTIME, while checking TO-security (van der Meyden) is undecidable, as is checking ITO-security (van der Meyden). The most important ingredients in the proofs of the PTIME upper bounds are new characterizations of the respective security notions, which also lead to new unwinding proof techniques that are shown to be sound and complete for these notions of security, and enable the algorithms to return simple counter-examples demonstrating insecurity. Our results for IP-security improve a previous doubly exponential bound of Hadj-Alouane et al

Quantum Security for the Physical Layer

The physical layer describes how communication signals are encoded and transmitted across a channel. Physical security often requires either restricting access to the channel or performing periodic manual inspections. In this tutorial, we describe how the field of quantum communication offers new techniques for securing the physical layer. We describe the use of quantum seals as a unique way to test the integrity and authenticity of a communication channel and to provide security for the physical layer. We present the theoretical and physical underpinnings of quantum seals including the quantum optical encoding used at the transmitter and the test for non-locality used at the receiver. We describe how the envisioned quantum physical sublayer senses tampering and how coordination with higher protocol layers allow quantum seals to influence secure routing or tailor data management methods. We conclude by discussing challenges in the development of quantum seals, the overlap with existing quantum key distribution cryptographic services, and the relevance of a quantum physical sublayer to the future of communication security.Comment: 7 pages, 6 figure