Weak Secrecy in the Multi-Way Untrusted Relay Channel with Compute-and-Forward

We investigate the problem of secure communications in a Gaussian multi-way relay channel applying the compute-and-forward scheme using nested lattice codes. All nodes employ half-duplex operation and can exchange confidential messages only via an untrusted relay. The relay is assumed to be honest but curious, i.e., an eavesdropper that conforms to the system rules and applies the intended relaying scheme. We start with the general case of the single-input multiple-output (SIMO) L-user multi-way relay channel and provide an achievable secrecy rate region under a weak secrecy criterion. We show that the securely achievable sum rate is equivalent to the difference between the computation rate and the multiple access channel (MAC) capacity. Particularly, we show that all nodes must encode their messages such that the common computation rate tuple falls outside the MAC capacity region of the relay. We provide results for the single-input single-output (SISO) and the multiple-input single-input (MISO) L-user multi-way relay channel as well as the two-way relay channel. We discuss these results and show the dependency between channel realization and achievable secrecy rate. We further compare our result to available results in the literature for different schemes and show that the proposed scheme operates close to the compute-and-forward rate without secrecy.Comment: submitted to JSAC Special Issue on Fundamental Approaches to Network Coding in Wireless Communication System

Über die Modellierung und Simulation zufälliger Phasenfluktuationen

Nachrichtentechnische Systeme werden stets durch unvermeidbare zufällige Störungen beeinflußt. Neben anderen Komponenten sind davon besonders Oszillatoren betroffen. Die durch die Störungen verursachten zufälligen Schwankungen in der Oszillatorausgabe können als Amplituden- und Phasenabweichungen modelliert werden. Dabei zeigt sich, daß vor allem zufällige Phasenfluktuationen von Bedeutung sind. Zufällige Phasenfluktuationen können unter Verwendung stochastischer Prozesse zweiter Ordnung mit kurzem oder langem Gedächtnis modelliert werden. Inhalt der Dissertation ist die Herleitung eines Verfahrens zur Simulation zufälliger Phasenfluktuationen von Oszillatoren mit kurzem Gedächtnis unter Berücksichtigung von Datenblattangaben

Quantum superalgebras at roots of unity and non-abelian symmetries of integrable models

We consider integrable vertex models whose Boltzmann weights (R-matrices) are trigonometric solutions to the graded Yang-Baxter equation. As is well known the latter can be generically constructed from quantum affine superalgebras $U_{q}(\hat g)$. These algebras do not form a symmetry algebra of the model for generic values of the deformation parameter $q$ when periodic boundary conditions are imposed. If $q$ is evaluated at a root of unity we demonstrate that in certain commensurate sectors one can construct non-abelian subalgebras which are translation invariant and supercommute with the transfer matrix and therefore with all charges of the model. In the line of argument we introduce the restricted quantum superalgebra $U^{res}_q(\hat g)$ and investigate its root of unity limit. We prove several new formulas involving supercommutators of arbitrary powers of the Chevalley-Serre generators and derive higher order quantum Serre relations as well as an analogue of Lustzig's quantum Frobenius theorem for superalgebras.Comment: 31 pages, tcilatex (minor typos corrected

Über die Modellierung und Simulation zufälliger Phasenfluktuationen

Nachrichtentechnische Systeme werden stets durch unvermeidbare zufällige Störungen beeinflußt. Neben anderen Komponenten sind davon besonders Oszillatoren betroffen. Die durch die Störungen verursachten zufälligen Schwankungen in der Oszillatorausgabe können als Amplituden- und Phasenabweichungen modelliert werden. Dabei zeigt sich, daß vor allem zufällige Phasenfluktuationen von Bedeutung sind. Zufällige Phasenfluktuationen können unter Verwendung stochastischer Prozesse zweiter Ordnung mit kurzem oder langem Gedächtnis modelliert werden. Inhalt der Dissertation ist die Herleitung eines Verfahrens zur Simulation zufälliger Phasenfluktuationen von Oszillatoren mit kurzem Gedächtnis unter Berücksichtigung von Datenblattangaben

Using LTI Dynamics to Identify the Influential Nodes in a Network.

Networks are used for modeling numerous technical, social or biological systems. In order to better understand the system dynamics, it is a matter of great interest to identify the most important nodes within the network. For a large set of problems, whether it is the optimal use of available resources, spreading information efficiently or even protection from malicious attacks, the most important node is the most influential spreader, the one that is capable of propagating information in the shortest time to a large portion of the network. Here we propose the Node Imposed Response (NiR), a measure which accurately evaluates node spreading power. It outperforms betweenness, degree, k-shell and h-index centrality in many cases and shows the similar accuracy to dynamics-sensitive centrality. We utilize the system-theoretic approach considering the network as a Linear Time-Invariant system. By observing the system response we can quantify the importance of each node. In addition, our study provides a robust tool set for various protective strategies

Generated and extracted networks.

<p>Four networks are generated using Barabási-Albert and Watts-Strogatz models for <i>scale-free</i> and <i>small-world</i> networks respectively. The rest are the real world networks of various sizes and characteristics taken from: <i>SNAP—Stanford Large Network Dataset Collection</i>, <i>UCLA’s Beyond BGP:Internet Topology Project</i> and <i>The Internet Topology Zoo</i>. All data sets are available online. Column <i>nodes</i> represents the number of nodes in the original network. Columns <i>diameter</i>, <i>density</i> and <i>clust. coeff.</i> represent the mean values calculated from the set of sampled networks. The <i>avg. degree</i> is the same for both the original and sampled networks.</p

Simple graph and corresponding system response.

<p>(a) An example of a small tree graph where the node <i>i</i> is a source of the infection. The signal in the form of the unit step or impulse function enters the network at node <i>i</i>. Here we observe the state of the system in each time step by measuring the signal strength in all nodes and adding them together. The resulting measurements are step and impulse response respectively. (b) Step and impulse response of a corresponding LTI system with the single input in the position of node <i>i</i>. The response corresponds to the spreading dynamics over time which originates in the source node <i>i</i>. This is the simple case of almost certain infection of the neighboring nodes in each time step over the tree graph. The step response reveals the number of infected nodes over time. Impulse response shows the number of infected nodes in each time step.</p

The correlation between degree and the <i>NiR</i>.

<p>There is a strong correlation between degree and <i>NiR</i> value for all types of the networks observed. The correlation against the node degree for all networks takes a value of 0.94 ± 0.02. The sum of all degrees of the node’s neighbors (degree of the distance one) correlates even more with the <i>NiR</i> value when the correlation coefficient is 0.98 ± 0.01. Random scale-free network, as well as the General Relativity collaboration network (<i>ca-GrQc</i>) show the expected pattern on the graph clearly demonstrating the presence of the small number of hubs, compared to the relatively large number of non-central nodes. On the other hand, the small-world model generates approximately the same number of nodes which could be grouped by the importance.</p

Expected time of infection and step response: small networks example.

<p>For three small networks the expected time of full infection, <i>E</i> [<i>X</i>(<i>p</i>)], is calculated. For all networks the source of the infection is the parent node (top). At each time step the parent node tries to infect neighboring susceptible nodes with the probability <i>p</i>. All nodes will be eventually infected and the time of full infection is presented with a certain distribution (i.e. the distribution of the expected number of trials in discrete time for the infection to reach all nodes). The <i>E</i> [<i>X</i>(<i>p</i>)] is the mean of the distribution for each network (the expected number of trials before the success.) The <i>S</i>_<i>max</i>(<i>p</i>) is the maximum step response value of the corresponding LTI system.</p