Entropy

Optimal Channel Design: A Game Theoretical Analysis

M. H. R. Khouzani, Pasquale Malacaria

Published Sep 5, 2018• 8 citations

Abstract

This paper studies the problem of optimal channel design. For a given input probability distribution and for hard and soft design constraints, the aim here is to design a (probabilistic) channel whose output leaks minimally from its input. To analyse this problem, general notions of entropy and information leakage are introduced. It can be shown that, for all notions of leakage here defined, the optimal channel design problem can be solved using convex programming with zero duality gap. Subsequently, the optimal channel design problem is studied in a game-theoretical framework: games allow for analysis of optimal strategies of both the defender and the adversary. It is shown that all channel design problems can be studied in this game-theoretical framework, and that the defender's Bayes-Nash equilibrium strategies are equivalent to the solutions of the convex programming problem. Moreover, the adversary's equilibrium strategies correspond to a robust inference problem.

DOI: 10.3390/e20090675

Related papers

Relative Perfect Secrecy: Universally Optimal Strategies and Channel Design

M. H. R. Khouzani, Pasquale Malacaria

Perfect secrecy describes cases where an adversary cannot learn anything about the secret beyond its prior distribution. A classical result by Shannon shows that a necessary condition for perfect secrecy is that the adversary should not be able to eliminate any of the possible secrets. In this paper we answer the following fundamental question: What is the lowest leakage of information that can be achieved when some of the secrets have to be eliminated? We address this question by deriving the minimum leakage in closed-form, and explicitly providing "universally optimal" randomized strategies, in the sense that they guarantee the minimum leakage irrespective of the measure of entropy used to quantify the leakage. We then introduce a generalization of Rényi family of asymmetric measures of leakage which generalizes the g-leakage and show that a slight modification of our strategies are optimal with respect to an important class of such measures. Subsequently, we show that our schemes constitute the Nash Equilibria of closely related two-person zero sum games. This game perspective provides implicit solutions for a wider set of structural constraints and asymmetric entropies. Finally we demonstrate how this work can also be seen as designing a universally optimal channel given a specified prior.

IEEE Computer Security Foundations Workshop (CSFW)Jun 1, 201620 citations

Optimization and Game-Theoretic Approaches for Defensive Resource Allocation in Communication and Network Security

Zhifan Xu

This dissertation investigates the physical layer security of wireless communication networks using game-theoretic and optimization models. Considered is a wireless network of parallel channels that may be under eavesdropping attack. Game models are played between a friendly interferer (he) who can broadcast artificial noise and a strategic eavesdropper (she). The friendly interferer wants to maximize the secrecy level of the wireless communication network by decreasing the eavesdropper's listening capability with intentionally generated noise signals, while the eavesdropper wants to maximize her own capability and avoid interference. The game-theoretic models reflect the strategic confrontation between the friendly interferer and the eavesdropper in different communication environments. These models address some of the limitations in literature. Most existing resource allocation strategies for friendly interference adopt the conservative assumption that all parallel channels are under attack. Game-theoretic approaches provide more efficient strategies to the friendly interferer by anticipating the channels to be attacked by the eavesdropper. The first model considers the case in which the friendly interferer can flexibly allocate cooperative jamming power to all communication channels in the network under a total power constraint. When all channels used for secret communication are under the threat of eavesdropping attacks, the optimal power allocation policy results from solving a convex optimization problem. In this case, the optimal policy is unique and can be obtained via a water-filling scheme. When the eavesdropper can not attack all channels, the eavesdropper should behave strategically and may select the targets probabilistically. In this case, a non-zero-sum game helps the friendly interferer predict and concentrate on the targets selected by the eavesdropper. When the eavesdropper can only attack a single channel, an analytical form of the NE strategies is presented; while an iterative numerical method that guarantees the convergence to NE strategies is proposed if the eavesdropper can attack multiple channels. Compared to naive power allocation strategies, such as splitting the power equally to each channel, the game-theoretic power allocation strategy always delivers better expected total secrecy capacity. Next, an extension of the power allocation game for cooperative jamming models the case in which the state information of the eavesdropping channels may not be known completely. New utility functions for the friendly interferer and the eavesdropper are designed to include the distributional information of eavesdropping channels' state information. A non-zero sum game using the new utility functions is analyzed between the friendly interferer and the strategic eavesdropper who can only attack a limited number of channels. Note that the friendly interferer may not have the capability to send cooperative jamming signals to multiple channels simultaneously. For such a friendly interferer, a probabilistic cooperative jamming scheme is devised such that the defender focuses on protecting randomly selected channels. In the mean time, the eavesdropper selects the most vulnerable channels to attack. A linearly constrained non-zero-sum bimatrix game models the competition between the defender and the eavesdropper. The Nash equilibrium is shown to be equivalent to the solution of a quadratic programming problem. A closed form solution is derived for the special case in which the eavesdropper can only attack a single channel. We show that there exists a cut-off index determining the set of channels that will be covered in the equilibrium strategies. Furthermore, the adverse effect of the interference signal plays an important role in the value of the cut-off index and it represents the vulnerability of wireless communication networks against eavesdroppers Wireless communication networks of parallel channels actually share a lot of similarities with general cyber networks. Thus, the final model in this dissertation is a cost-efficient network security game for defensive resource allocation in a heterogeneous cyber network of N nodes. A defender, by allocating defensive resources to a networked node, can decrease the node's vulnerabilities. At the same time, the cost of using defensive resources is taken into consideration. Two non-zero-sum Nash games model the competition between the defender and two types of attackers, respectively. The first type tries to maximize damage based on the value of security assets related to networked nodes, while the second type aims at infiltrating the network. Consideration of the case that the defender does not know the attacker's type leads to a Bayesian game. For all three games, we derive conditions for the existence of a unique Nash equilibrium (NE), and obtain the equilibria and the value functions in closed form.

Jul 29, 20260 citations

Power Allocation for Cooperative Jamming Against a Strategic Eavesdropper Over Parallel Channels

Zhifan Xu, Melike Baykal‐Gürsoy

This paper considers a friendly interferer allocating jamming power to eavesdropping channels to increase the level of secrecy of a wireless network. The friendly interferer has access to limited power, while the eavesdropper may not have the ability to attack all channels simultaneously. When all channels used for secret communication are under the threat of eavesdropping attacks, the optimal power allocation policy results from solving a convex optimization problem. In this case, the optimal policy is unique and can be obtained via a water-filling scheme. When the eavesdropper can not attack all channels, the eavesdropper should behave strategically and may select targets probabilistically. We propose a non-zero-sum game that helps the friendly interferer predict and concentrate on the targets selected by the eavesdropper. Under certain conditions, we prove that there exists a unique Nash equilibrium (NE) strategy pair, which has a threshold type structure. We provide conditions under which the eavesdropper’s equilibrium strategy is deterministic. We devise a strategy iteration algorithm to compute an equilibrium power allocation strategy. We present examples showing that the game-theoretic power allocation strategy performs better than the conservative power allocation strategy that assumes every channel to be under attack.

IEEE Transactions on Information Forensics and SecurityDec 12, 202212 citations

The role of game theory in key agreement over a public channel

Nebojsa Milosavljevic, Michael Gastpar, Kannan Ramchandran

In this work we study the problem of key agreement over a public noiseless channel when Alice, Bob and Charlie observe discrete memoryless sources of an unknown distribution. Alice and Bob want to agree on a key KABthat is protected from Charlie. At the same time, Alice and Charlie want to agree on a key KACthat is protected from Bob. In order to construct codebooks for the key agreement, Alice has to know how the sources are distributed. Therefore, she requests Bob and Charlie to send her sufficient information about their observations. We also assume that Bob and Charlie, besides agreeing with Alice on the keys, want to learn as much as possible about the other user's key: we call this quantity the leakage. We model these reports by having Bob and Charlie select discrete memoryless channels and passing their true observations through them. We approach this problem from a game-theoretic point of view. For a class of Bob and Charlie's objective functions which are linear in the key rate and the leakage rate, we characterize a Nash equilibrium. Also, we propose a strategy that Alice can apply in order to ensure that Bob and Charlie's honest reporting is a Nash equilibrium.

IEEE International Symposium on Information Theory (ISIT)Jun 1, 20100 citations

Competing for Secrecy in the MISO Interference Channel

S. Ali. A. Fakoorian, Arnold Lee Swindlehurst

A secure communication game is considered for the two-user MISO Gaussian interference channel with confidential messages, where each transmitter aims to maximize the difference between its secrecy rate and the secrecy rate of the other transmitter. In this novel problem, the weaker link tries to minimize the extra secret information obtained by its adversary, while the stronger link attempts to maximize it. We provide an information theoretic formulation for this non-cooperative zero-sum game scenario and determine that, under the assumption of Gaussian signaling at the transmitters, there exists a unique Nash equilibrium solution for the proposed problem. Moreover, we obtain in closed-form the optimal strategies (beamformers) at the transmitters that result in the Nash equilibrium. While the NE strategies are achievable in one shot when full channel state information is available at the transmitters (CSIT), we also propose an iterative step-by-step algorithm that converges to the NE point, but only requires limited CSIT. Numerical results are presented to illustrate the analytical findings and to study the role of different channel parameters on the NE strategies.

IEEE Transactions on Signal ProcessingOct 9, 201221 citations

Guaranteed Payoffs Resource Allocation Problem With Application to Countering Reinforced Eavesdropping

Andrey Garnaev, Wade K. Trappe

Wireless communication between a transmitter (e.g., an operator) and agents (e.g., drones) may be vulnerable to eavesdropping attacks on some or all agents. A drawback of the traditional communication approach that aims to maximize the total secrecy capacity is that it may neglect communication with specific agents, thereby leading to mission failure for those agents. To address this drawback, we develop and solve a novel guaranteed-payoffs problem that generalizes max-min optimization in a resource-restricted setting and prove that its solution combines egalitarian and water-filling properties. The developed generalization is a convenient tool for researchers to apply across different research areas. Based on this, we design an anti-eavesdropping strategy that maximizes the minimal secrecy capacity of communication with agents facing an adversary capable of reinforcing eavesdropping devices in a game-theoretic framework. Robustness of the derived anti-eavesdropping strategy is proven, and strategies are illustrated.

IEEE Communications LettersJan 1, 20260 citations