Abstract
We introduce a zero-sum game problem of soft watermarking: The hidden information (watermark) comes from a continuum and has a perceptual value; the receiver generates an estimate of the embedded watermark to minimize the expected estimation error (unlike the conventional watermarking schemes where both the hidden information and the receiver output are from a discrete finite set). Applications include embedding a multimedia content into another. We study here the scalar Gaussian case and use expected mean-squared distortion. We formulate the problem as a zero-sum game between the encoder & receiver pair and the attacker. We show that for linear encoder, the optimal attacker is Gaussian-affine, derive the optimal system parameters in that case, and discuss the corresponding system behavior. We also provide numerical results to gain further insight and understanding of the system behavior at optimality.
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Notes
- 1.
This distinction is reminiscent of the differentiation between hard and soft decoding methods in classical communication theory.
- 2.
One alternative approach for this problem may involve using a separated setup, where we first apply lossy compression to the information to be hidden that possesses perceptual value, and subsequently embed the compression output into the unmarked host using a conventional capacity-achieving data hiding code. It is not immediately clear which approach is superior; a comparative assessment constitutes part of our future research.
- 3.
In classical (information-theoretic) watermarking literature, a pseudo-random key sequence is shared between the encoder and the decoder, mainly to render the attacker strategies memoryless. In this paper, we do not consider the key sequence in the problem formulation since our formulation is based on single-letter strategies.
- 4.
The investigation of a potential relationship between (2) and imposing an upper bound on \(\mathbb {E} \left( Y^2 \right) \) for the data hiding setup constitutes part of our future research.
- 5.
One way to get around this problem is to introduce soft constraints into the objective of the attacker. Then, the problem is no longer a zero-sum game. Another way is to define the attacker constraint for each realization, in almost sure sense, in which case the attacker can satisfy its constraint for any encoding strategy. These are beyond the scope of this paper.
- 6.
This is a zero-sum game where the objective is linear (hence, concave) in the attacker mapping for a fixed decoder map, and the optimal decoder mapping is unique (conditional mean) for a given attacker mapping. \(\mathcal{M}\) is weak*-compact, and the minimizing h can be restricted to a compact subset of \(\varGamma _R\) (with (3) bounded away from zero); hence a saddle point exists due to the standard min-max theorem of game theory in infinite-dimensional spaces [7].
- 7.
- 8.
In the proof of Theorem 1, all expressions that involve \(\alpha \) are, in fact, functions of \(\alpha ^2\); therefore if \(\alpha ^*\) is optimal, so is \(-\alpha ^*\). To account for such multiple trivial solutions, we use the term “essential uniqueness”.
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Acknowledgement
Research of Akyol and Başar was supported in part by the Office of Naval Research (ONR) MURI grant N00014-16-1-2710. The work of C. Langbort was supported in part by NSF grant #1619339 and AFOSR MURI Grant FA9550-10-1-0573.
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Mıhçak, M.K., Akyol, E., Başar, T., Langbort, C. (2016). Scalar Quadratic-Gaussian Soft Watermarking Games. In: Zhu, Q., Alpcan, T., Panaousis, E., Tambe, M., Casey, W. (eds) Decision and Game Theory for Security. GameSec 2016. Lecture Notes in Computer Science(), vol 9996. Springer, Cham. https://doi.org/10.1007/978-3-319-47413-7_13
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DOI: https://doi.org/10.1007/978-3-319-47413-7_13
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