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Zero Divergence Secure Steganography

People

Kaushal Solanki, Ken Sullivan, Anindya Sarkar, U. Madhow, B.S.Manjunath, S. Chandrasekaran

Objective

Steganography is the art and science of communicating in such a way that the very existence of communication is not revealed to a third party.  In order to communicate without being detected, the data-hider must obey following two conditions.

  1. Perceptual constraint. The perceptual distortion between the original and stego image should not be more than a certain maximum amount, D1 , for some perceptual distance measure.
  2. Statistical constraint. The embedding process should not modify the statistics of the host signal more than a very small number, epsilon, for some statistical distance measure.

The objective in this project is to investigate steganographic schemes that can provide provable security by achieving zero Kullback-Leibler divergence between the cover and the stego signal distributions, while communicating at high rates.

Approach: Statistical Restoration

We have used the principle of statistical restoration, where a certain fraction of the available coefficients are used for hiding while the rest is used to compensate for the changes in the host statistics due to hiding. By avoiding hiding in the low probability regions of the host distribution, we are able to achieve zero Kullback-Liebler divergence between the cover and stego distributions, even while embedding at high rates. The framework is applied to design practical schemes for image steganography, which are evaluated using supervised learning on a set of about 1000 natural images. For the presented JPEG steganography scheme, it is seen that the detector is indeed reduced to random guessing.

Code

MATLAB code for JPEG-based steganography: zeroDivJPEGstego.m

Supporting files:
  • getjqm.m to compute the JPEG quality factor matrix for a given quality factor
  • genused.m to compute the sequence of the 8x8 block based AC DCT coefficients as they occur in the zigzag scan occur.

  • Here, are some sample images in which you can embed data securely using JPEG-based steganography:
    airplane, baboon, catcrowd.

    Examples

    Presented below are some sample results where we have applied the secure steganography algorithm on the baboon image. We have embedded 23300 bits in the 512 x 512 image, and we have used 19 AC DCT coefficients per 8 x 8 block for hiding. We use a hiding fraction of 30%; we hide in coefficients whose magnitude ≤ 30.

    Original baboon image
    Composite baboon image with 23300 bits embedded
    Original baboon image, before hiding
    Composite baboon image, with 23300 bits embedded in it


    Histogram of AC DCT coefficients available for hiding
    Histogram of AC DCT coefficients after hiding but before compensation
    Histogram of AC DCT coefficients available for hiding
    Histogram of AC DCT coefficients after hiding but before compensation


    Desired histogram for the compensation coefficients, to ensure zero KL divergence
    Difference ebtween the histograms of the original and composite images, after compensation
    Desired histogram for the compensation coefficients, to ensure zero KL divergence
    Final difference between the original and composite images, after compensation

    Acknowledgements

    This research is supported by a grant from ONR #N00014-01-1-0380, ONR #N00014-05-1-0816.

    Publications

    1. K. Solanki, K. Sullivan, U. Madhow, B. S. Manjunath and S. Chandrasekaran,
      "Statistical Restoration for Robust and Secure Steganography"
      Proc. IEEE International Conference on Image Processing, Genova, Italy, Sep. 2005.
      VRL ID 149: [abstract] [PDF] [BibTex]

      Abstract preview: "We investigate data hiding techniques that attempt to defeat steganalysis by restoring the statistics of the composite image to resemble that of the cover. The approach is to reserve a number of host ..." [more]

    2. K. Sullivan, K. Solanki, B.S. Manjunath, U. Madhow and S. Chandrasekaran,
      "Determining achievable rates for secure zero divergence steganography"
      Proc. IEEE International Conference on Image Processing 2006 (ICIP06), Atlanta, GA USA, Oct. 2006.
      [abstract] [PDF] [BibTex]

      Abstract preview: "In steganography (the hiding of data into innocuous covers for secret communication) it is difficult to estimate how much data can be hidden while still remaining undetectable. To measure the inherent..." [more]

    3. K. Solanki, K. Sullivan, U. Madhow, B. S. Manjunath and S. Chandrasekaran,
      "Provably secure steganography: Achieving zero K-L divergence using statistical restoration"
      Proc. IEEE International Conference on Image Processing 2006 (ICIP06), Atlanta, GA USA, Oct. 2006.
      [abstract] [PDF] [BibTex]

      Abstract preview: "In this paper, we present a framework for the design of steganographic schemes that can provide provable security by achieving zero Kullback-Leibler divergence between the cover and the stego signal d..." [more]

    4. A. Sarkar, K. Solanki, U. Madhow, S. Chandrasekaran and B. S. Manjunath,
      "Secure Steganography: Statistical Restoration of the Second Order Dependencies for Improved Security"
      Proc. 32nd IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Honolulu, Hawaii, Apr. 2007.
      [abstract] [PDF] [BibTex]

      Abstract preview: "We present practical approaches for steganography that can provide improved security by closely matching the second-order statistics of the host rather than just the marginal distribution. The methods..." [more]

    5. A. Sarkar and B. S. Manjunath,
      "Estimating Steganographic Capacity for Odd-Even Based Embedding and its Use in Individual Compensation"
      Proc. IEEE International Conference on Image Processing (ICIP), San Antonio, TX, Sep. 2007.
      [abstract] [PDF] [BibTex]

      Abstract preview: "We present a method to compute the steganographic capacity for images, with odd-even based hiding in the quantized discrete cosine transform domain. The method has been generalized for varying orders ..." [more]

    6. A. Sarkar, K. Sullivan and B. S. Manjunath,
      "Steganographic Capacity Estimation for the Statistical Restoration Framework"
      Proc. SPIE - Security, Steganography, and Watermarking of Multimedia Contents (X), San Jose, California, Jan. 2008.
      [abstract] [PDF] [BibTex]

      Abstract preview: "In this paper we attempt to quantify the "active" steganographic capacity - the maximum rate at which data can be hidden, and correctly decoded, in a multimedia cover subject to noise/attack (hence - ..." [more]