On Optimal Detection for Matrix Multiplicative Data Hiding ∗ Babak Moussakhani Department of Electronics and Telecommunications NTNU, N-7491 Trondheim, Norway babak@iet.ntnu.no Mohammad Ali Sedaghat Department of Electronics and Telecommunications NTNU, N-7491 Trondheim, Norway mohammad.sedaghat@ iet.ntnu.no John T. Flåm Department of Electronics and Telecommunications NTNU, N-7491 Trondheim, Norway flam@iet.ntnu.no Tor Ramstad Department of Electronics and Telecommunications NTNU, N-7491 Trondheim, Norway tor.ramstad@iet.ntnu.no ABSTRACT This paper analyzes a multiplicative data hiding scheme, where the watermark bits are embedded within frames of a Gaussian host signal by two different, but arbitrary, em- bedding matrices. A closed form expression for the bit er- ror rate (BER) of the optimal detector is derived when the frame sizes tend to infinity. Furthermore, a structure is pro- posed for the optimal detector which divides the detection process into two main blocks: host signal estimation and decision making. The proposed structure preserves optimal- ity, and allows for a great deal of flexibility: The estimator can be selected according to the a priori knowledge about host signal. For example, if the host signal is an Auto- Regressive (AR) process, we argue that a Kalman filter may serve as the estimator. Compared to a direct implementa- tion of the Neyman-Pearson detector, this approach results in significantly reduced complexity while keeping optimal performance. Categories and Subject Descriptors D.4.6 [Security and Protection]: [Authentication]; H.2.0 [General]: [Security, integrity, and protection] Keywords data hiding, watermarking, optimal detection, Kalman filter ∗ This work was funded by the MELODY Project, which is sponsored by the Research Council of Norway Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. IH&MMSec’13, June 17–19, 2013, Montpellier, France. Copyright 2013 ACM 978-1-4503-2081-8/13/06 ...$15.00. 1. INTRODUCTION Watermarking refers to the process of embedding side in- formation in a digital medium in a seemingly innocuous way. Its use is widespread in broadcast monitoring, covert com- munication applications and for preventing illegal use of the copyrighted data [2, 6, 10]. It is desired that the watermark is robust against attacks, and at the same time is inaudible, i.e. the change introduced by the watermark should be per- ceptually undetectable. Typically, this trade off needs to be satisfied for a certain data rate requirement [6]. The main stream methods for data embedding include additive and multiplicative schemes [4],[1]. Due to their robustness and compatibility with human visual/auditory systems, multiplicative approaches attract more interest [4], [3]. The methods introduced in [3],[1] utilize the optimal detector based on the likelihood ratio test to retrieve the embedded information. Solachidis et al. [11] have designed the optimal detector for multiplicative watermarks in the Discrete Fourier Transform domain for a signal with a first order separable autocorrelation function. In [9], the authors propose an optimal detector for a highly correlated first or- der Auto-Regressive (AR) host signal. In [12] an improved multiplicative spread spectrum data hiding method has been proposed. It tries to minimize the interference effect of the host signal. However, the investigation in [12] is limited to i.i.d host signals. To the best of our knowledge the optimal detector for an arbitrarily correlated (e.g. AR process of order r) host signal has not been investigated so far. Most detectors in the literature assume that each sample within a frame of the host signal is identically and indepen- dently distributed (i.i.d.). This assumption imposes severe restrictions on the correlation structure, and for most real- world random signals it does not hold. Our work considers a host signal with an arbitrary correlation matrix. In addi- tion, in contrast to e.g. [4], [1], we do not limit the attention to diagonal embedding matrices. Here, the embedding ma- trices can have arbitrary structure. We derive the optimal Maximum Likelihood (ML) detec- tor, and propose a block structure for its implementation 203