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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Additive non-Gaussian noise attacks on the scalar Costa scheme (SCS)

Roman Tzschoppe, Robert Bäuml, Robert F. H. Fischer, Johannes B. Huber, André Kaup

Published Mar 21, 2005• 9 citations
Publisher

Abstract

The additive attack public mutual information game is explicitly solved for one of the simplest quantization based watermarking schemes, the scalar Costa scheme (SCS). It is a zero-sum game played between the embedder and the attacker, and the payoff function is the mutual information. The solution of the game, a subgame perfect nash equilibrium, is found by backward induction. Therefore, the Blahut-Arimoto algorithm is employed for numerically optimizing the mutual information over noise distributions. Although the worst case distribution is in general strongly non-Gaussian, the capacity degradation compared to a suboptimal Gaussian noise attack is quite small. The loss, if the embedder optimizes SCS for a Gaussian attack but the worst case attack is employed, is negligible.

DOI
10.1117/12.587492

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The additive attack public mutual information game is explicitly solved for one of the simplest quantization based watermarking schemes, the scalar Costa scheme (SCS). It is a zero-sum game played between the embedder and the attacker, and the payoff function is the mutual information. The solution of the game, a subgame perfect nash equilibrium, is found by backward induction. Therefore, the Blahut-Arimoto algorithm is employed for numerically optimizing the mutual information over noise distributions. Although the worst case distribution is in general strongly non-Gaussian, the capacity degradation compared to a suboptimal Gaussian noise attack is quite small. The loss, if the embedder optimizes SCS for a Gaussian attack but the worst case attack is employed, is negligible.
We generalize alternating optimization algorithms of Blahut-Arimoto type to the quantum setting. In particular, we give iterative algorithms to compute the mutual information of quantum channels, the Holevo quantity of classical-quantum channels, the coherent information of less noisy quantum channels, and the thermodynamic capacity of quantum channels. Our convergence analysis is based on quantum entropy inequalities and leads to a priori additive ε\varepsilonε-approximations after O(ε−1log⁡N)\mathcal{O}\left(\varepsilon^{-1}\log N\right)O(ε−1logN) iterations, where NNN denotes the input dimension of the channel. We complement our analysis with an a posteriori stopping criterion which allows us to terminate the algorithm after significantly fewer iterations compared to the a priori criterion in numerical examples. Finally, we discuss heuristics to accelerate the convergence.
We investigate the behavior of two users and one jammer in an additive white Gaussian noise (AWGN) channel with and without fading when they participate in a noncooperative zero-sum game, with the channel's input/output mutual information as the objective function. We assume that the jammer can eavesdrop on the channel and can use the information obtained to perform correlated jamming. We also differentiate between the availability of perfect and noisy information about the user signals at the jammer. Under various assumptions on the channel characteristics, and the extent of information available at the users and the jammer, we show the existence, or otherwise nonexistence of a simultaneously optimal set of strategies for the users and the jammer, and characterize those strategies whenever they exist.
We derive sublinear-time quantum algorithms for computing the Nash equilibrium of two-player zero-sum games, based on efficient Gibbs sampling methods. We are able to achieve speed-ups for both dense and sparse payoff matrices at the cost of a mildly increased dependence on the additive error compared to classical algorithms. In particular we can find ε\varepsilonε-approximate Nash equilibrium strategies in complexity O~(n+m/ε3)\tilde{O}(\sqrt{n+m}/\varepsilon^3)O~(n+m​/ε3) and O~(s/ε3.5)\tilde{O}(\sqrt{s}/\varepsilon^{3.5})O~(s​/ε3.5) respectively, where n×mn\times mn×m is the size of the matrix describing the game and sss is its sparsity. Our algorithms use the LP formulation of the problem and apply techniques developed in recent works on quantum SDP-solvers. We also show how to reduce general LP-solving to zero-sum games, resulting in quantum LP-solvers that have complexities O~(n+mγ3)\tilde{O}(\sqrt{n+m}γ^3)O~(n+m​γ3) and O~(sγ3.5)\tilde{O}(\sqrt{s}γ^{3.5})O~(s​γ3.5) for the dense and sparse access models respectively, where γγγ is the relevant "scale-invariant" precision parameter
Abstract : The additive infinite-dimensional Gaussian channel subject to jamming is modeled as a two-person zero-sum game with mutual information as the payoff function. The jammer's noise is added to the ambient Gaussian noise. The coder's signal energy is subject to a constraint is necessary in order that the capacity without feedback be finite. It is shown that use of this same RKHS constraint on the jammer's process is too strong; the jammer would then not be able to reduce capacity, regardless of the amount of jamming energy available. The constraint on the jammer is thus on the total jamming energy, without regard to its distribution relative to that of the ambient noise energy. The existence of a saddle value for the problem does not follow from the von Neuman minimax theorem in the original problem formulation. However, a solution is shown to exist. A saddle point, saddle value, and the jammer's minimax strategy are determined. The solution is a function of the problem parameters: the constraint on the coder, the constraint on the jammer, and the covariance of the ambient Gaussian noise.
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.
In this work, a four-terminal complex Gaussian network composed of a source, a destination, an eavesdropper, and a jammer relay is studied under two different set of assumptions: 1) The jammer relay does not hear the source transmission, and 2) The jammer relay is causally given the source message. In both cases, the jammer relay assists the eavesdropper and aims to decrease the achievable secrecy rates. The source, on the other hand, aims to increase it. To help the eavesdropper, the jammer relay can use pure relaying and/or send interference. Each of the problems is formulated as a two-player, noncooperative, zero-sum continuous game. Assuming Gaussian strategies at the source and the jammer relay in the first problem, the Nash equilibrium is found and shown to be achieved with mixed strategies in general. The optimal cumulative distribution functions (cdfs) for the source and the jammer relay that achieve the value of the game, which is the Nash equilibrium secrecy rate, are found. For the second problem, the Nash equilibrium solution is found and the results are compared to the case when the jammer relay is not informed about the source message.