New Results on Robustness of Secure Steganography Mark T. Hogan, F´elix Balado, Neil J. Hurley and Gu´enol´e C.M. Silvestre School of Computer Science and Informatics, University College Dublin, Belfield, Dublin 4, Ireland. ABSTRACT Steganographic embedding is generally guided by two performance constraints at the encoder. Firstly, as is typical in the field of watermarking, all the transmission codewords must conform to an average power constraint. Secondly, for the embedding to be statistically undetectable (secure), it is required that the density of the watermarked signal must be equal to the density of the host signal. Assuming that this is not the case, statistical steganalysis will have a probability of detection error less than 1/2 and the communication may be terminated. Recent work has shown that some common watermarking algorithms can be modified such that both constraints are met. In particular, spread spectrum (SS) communication can be secured by a specific scaling of the host before embedding. Also, a side informed scheme called stochastic quantization index modulation (SQIM), maintains security with the use of an additive stochastic element during the embedding. In this work the performance of both techniques is analysed under the AWGN channel assumption. It will be seen that the robustness of both schemes is lessened by the steganographic constraints, when compared to the standard algorithms on which they are based. Specifically, the probability of decoding error in the SS technique increases when security is required, and the achievable rate of SQIM is shown to be lower than that of dither modulation (on which the scheme is based) for a finite alphabet size. 1. INTRODUCTION The term steganography refers to the family of techniques used to hide data within a host multimedia signal. Ide- ally, the corresponding modified signal, referred to as a stegotext, is perceptually and statistically indistinguish- able from the host. The classical representation of steganographic communication is given by the prisoners’ problem. 1 Alice produces a stegotext using the message that she wants to communicate and a given host, and sends it to Bob through an insecure communications channel. Usually, Alice and Bob make use of secret keys for their covert communication. The warden Wendy monitors the channel between Alice and Bob, and performs a detection test to decide if the signal being sent includes hidden information by exploiting potential imperfections of the steganographic method used. This detection procedure is known as steganalysis. Now, consider the nature of Wendy’s actions. Typically she can be either passive or active. If passive then a detection test is all that is performed on the received document. If she is active, then the document is attacked regardless of the outcome of any detection test. In this work we consider that Wendy is basically passive, but we also assume that the transmitted document undergoes a channel distortion before it is decoded. This operation may be interpreted as an active warden or attacker pre-emptively jamming the transmitted signal irrespective of the detection test outcome; for the sake of comparison with prior research, we assume that this distortion is additive white Gaussian noise (AWGN) independent of the host. The success of detection tests lies in the location of statistical differences between the host signal and the stegotext signal. This idea has been formalised by Cachin, 2 where the security of steganography has been defined in terms of the Kullback Leibler distance (D KL ) between the densities of the host and stegotext signals. The D KL is equal to zero iff the two distributions are equal. The implication is that a non-negligible value for D KL for any embedding scheme leads to detectable statistical differences. A major goal of embedding is, therefore, to keep D KL as low as possible, such that the communication passes unhindered. We now specify two cases of steganographic communication, namely perfect and non-perfect steganography. If the embedding is such that D KL = 0 between the host and stegotext densities then we have perfect steganography. Further author information: (Send correspondence to M.H.) - M.H.: E-mail: markhogan@ihl.ucd.ie, Telephone: +353 (0)1 716 2454.