Detecting Hidden Message Using Higher Order Statistical Models Hany Farid By Jingyu Ye Yiqi Hu

Outline Using lower order to detect hidden message How to counter it How does higher order statistical model work Experiment Result Conclusion

Hidden Message In the Image Guess which image hides information?

What message hides in the image? The right image is modified with first chapter of Lewis Carroll’s ‘The Hunting of the Snark”. After compression, it is about 14700 bits. Not possible for human eyes to find the difference.

What is DCT It is a first order statistical model DTC is discrete cosine transform. It is close to DFT, discrete Fourier transform, but only uses the real numbers. It is more efficient then sine transform.

Using DTC to find the difference This is the DTC (discrete cosine transform) coefficient.

How to counter the first order statistical model Digital images and voices have quantisation noise which provides space for data hiding. Hide information in the least significant bit will cause little or no visual change to human eyes.

How to Counter it By decrease the efficiency of the steganograph, we can use only fraction of the least significant bit Attackers can embed data using pseudo random algorithm When apply first order statistical model, it will assume just noise

Image Statistics The decomposition of images using basis functions that are localized in spatial position, orientation, and scale has proven extremely useful in a range of applications. One reason is that such decompositions exhibit statistical regularities that can be exploited. Quadrature mirror filters(QMFs) Applying separable lowpass and highpass filters along the image axes generating a vertical, horizontal, diagonal and lowpass subband. Subsequent scales are created by recursively filtering the lowpass subband. The vertical, horizontal, and diagonal subbands at scale i = 1,…,n are denoted as Vi(x, y), Hi(x, y), and Di(x, y), respectively.

Image Statistics Statistical Model: Mean, Variance, Skewness, Kurtosis Vertical, Horizontal, Diagonal Scales i=1,…,n-1 12(n-1) coefficient statistics

Image Statistics A linear predictor for the magnitude of these coefficients in a subset of all possible neighbors is given by: Q contain the neighboring coefficient magnitudes. 11

Image Statistics The coefficients are determined by minimizing the quadratic error function:

Image Statistics The log error in the linear predictor is then given by: 12(n-1) error statistics 12(n-1) coefficient statistics + 12(n-1) error statistics 24(n-1) statistics

Image Statistics

Classification Fisher Linear Discriminant Analysis The within-class means are defined as: The between-class mean is defined as:

Classification Fisher Linear Discriminant Analysis The within-class scatter matrix is defined as: Matrix contains the zero-meaned ith exemplar given by Matrix contains the zero-meaned jth exemplar given by The between-class scatter matrix is defined as:

Classification Fisher Linear Discriminant Analysis let be the maximal generalized eigenvalue-eigenvector of and Project the training exemplars and onto the one-dimensional linear subspace by and , we set a threshold to determine which class the test image belongs to.

Experiment Each 8-bit-per channel RGB image is cropped to a central 640 480 pixel area. Statistics from 1,800 such images are collected as follows. Each image is first converted from RGB to gray-scale. A four-level, three-orientation QMF pyramid is constructed for each image, from which a 72-length feature vector of coefficient and error statistics is collected. To reduce sensitivity to noise in the linear predictor, only coefficient magnitudes greater than 1.0 are considered. The training set of “no-steg” statistics comes from either 1,800 JPEG images 1,800 GIF images or 1,800 TIFF images.

Results Jsteg and OutGuess are transform-based systems embed messages by modulating the DCT coefficients. EzStego modulates the least significant bits of the sorted color palette index. LSB modulates the least-significant bit of a random subset of the pixel intensities.

Conclusion Messages can be embedded into digital images in ways that are imperceptible to the human eye, and yet, these manipulations can significantly alter the underlying statistics of an image. These higher order statistics used in the paper appear to capture certain properties of “natural” images, and more importantly, these statistics are significantly altered when a message is embedded within an image. This makes it possible to detect, with a reasonable degree of accuracy, the presence of hidden messages in digital images. To avoid detection, of course, one need only embed a small enough message that does not significantly disturb the image statistics.

Thank you!