Soft-SCS: Improving the Security and Robustness of the Scalar-Costa-Scheme by Optimal Distribution Matching

Abstract

In this paper we propose an extension of the Scalar-Costa-Scheme (SCS), called Soft-SCS, which offers better or equal achievable rates than SCS for the AWGN channel. After recalling the principle of SCS we highlight its secure implementations regarding the Watermarked contents Only Attack, and we also describe the relations between the alphabet size and the secure embedding parameters. Since the gap between the achievable rates of secure-SCS and SCS is important for low Watermark to Noise Ratios (WNR) regimes, we introduce Soft-SCS, a scheme which enables to achieve security by matching a given distribution of watermarked content while minimizing the embedding distortion. The embedding is given by the optimal transport and the distortion is computed using the transportation theory. Contrary to SCS, the distribution of watermarked contents is not piecewise uniform of width $\text{(1-\ensuremath{\alpha})\ensuremath{\Delta}}$, but contains affine portions parametrized by a new embedding parameter β used to maximize the robusness of Soft-SCS. As a consequence, the achievable rates of Soft-SCS for low WNR regimes for both its secure and robust implementations are higher than for SCS. Our conclusions are that (1) the loss of performance between the secure and robust implementations of Soft-SCS for WNR regimes smaller than 0 dB is negligible and (2) the robust implementation of Soft-SCS is equal to SCS for WNR regimes over 0 dB.