Statistical models for Secure Steganography Systems

							           Statistical models for
           Secure Steganography
           Systems
           N.Muthiyalu Jothir
           Media Informatics




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Agenda
 Introduction
 Steganography
 Information theory
 Security Model
 Limitations
 Conclusion



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Introduction
   “Steganography is the art and science of
    writing hidden messages in such a way
    that no one apart from the intended
    recipient knows of the existence of the
    message”

   “Covered or Hidden Writing”

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    Steganography Vs Cryptography

   Steganography is the dark cousin of
    cryptography, the use of codes.

   Cryptography  Privacy

   Steganography  Secrecy


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Digital Still Images
   Larger the cover message – Easier to hide
    message

   For example: a 24-bit bitmap image will have 8
    bits representing each of the three color values
    (R,G,B)

   Watermarking, Fingerprinting etc.

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Information Theory

   “The fundamental problem of
    communication is that of reproducing at
    one point either exactly or approximately a
    message selected at another point.”
                                    -C.E. Shannon, 1948




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    Information Theory …cont
     According to Shannon, the entropy of a random variable X with
      probability distribution PX and alphabet ‫ א‬is defined as




     indicates the amount of information contained in x, i.e., the number of
      bits needed to code x.

     For example, in an image with uniform distribution of gray-level
      intensity, i.e. Px = 1/256, then the number of bits needed to code each
      gray level is 8 bits. The entropy of this image is 8.




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…cont
   Information  "only infrmatn esentil fo
    understandn mst b tranmitd."

   The amount of information, or uncertainty, output
    by an information source is a measure of its
    entropy.

   In turn, a source's entropy determines the
    amount of bits per symbol required to encode
    the source's information.
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Relative Entropy
   Let p and q be two probability distributions on a
    common alphabet X. Relative entropy / Kullback
    Leibler “distance” between p and q is defined as




   D(p || q) is a measure of the inefficiency of assuming
    that the distribution is q when the true distribution is p.


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Security Model : Proposed
   An information-theoretic model

   Presence of passive adversaries

   “Simmons' Prisoners„ Problem"

   “Hypothesis” testing problem
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Active adversaries

   Presence of hidden message is known
    Publicly

   E.g., Watermarking and Fingerprinting.




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Scenario with Passive Adversaries

   Players  Alice and Bob

   Passive Adversary  Eve

   “Cover Text, C”  Original, unaltered message

   “Stegotext, S”  Transformed message using
    Secret Key.

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Hypothesis testing
           Eve, the decision maker




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Model
   The security of a steganographic system is
    quantified in terms of the relative entropy
    D(PC | PS) (or discrimination) between PC
    and PS.

   D(PC | PS) = 0  stego system is perfectly
    secure

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Security System




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Observations
   H(S / CEKR) = 0  Certainty

   H(E) > 0  Uncertainty

   H(E / SK) = 0  Certainty



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…cont
   Alice is inactive  she sends cover text C

   Active  S is a concatenation of multiple
    messages from Alice

   The probability distributions of cover text
    (Pc) are assumed to be known to all
    parties
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Security Definition
   Definition 1 :
    A stegosystem as defined previously with cover
    text C and stegotext S is called Є – secure
    against passive adversaries if

                    D(PC|PS) ≤ Є

   If Є = 0, the stegosystem is called perfectly
    secure.


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Eve's decision process

 Binary partition (C0, C1) of the set C of
  possible cover texts
 Alice is active if and only if the observed
  message „c‟ is contained in C1.
 Type II error  Eve fails to detect
            Probability   β
   Type I error  Eve accuses Alice when
    she is inactive.
            Probabilty    ά. Assumed to be zero.
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Theorem
   The stegosystem that is Є-secure against
    passive adversaries, satisfy
                     d(ά, β) ≤ Є


   In particular, if ά = 0, then
                       β = 2-Є



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…cont
   In a perfectly secure system,

            D(PC|PS) = 0  PC = PS;

    Thus, Eve can obtain no information about
    whether Alice is active by observing the
    message.

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External Information Influence
   The modified stegosystem with external
    information Y , cover text C, and stegotext
    S is called Є - secure against passive
    adversaries if

            D(PC|Y |PS|Y ) ≤ Є


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One-time pad
   Security  The stegotext distribution is close to the
    cover text distribution without knowledge of the key.

   Cover text C is a uniformly distributed n-bit string

   The key generator chooses the n-bit key K

   S=e     K and Bob can decode by computing e = S         K



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Security of One Time Pad
   Uniformly distributed in the set of n-bit strings and
    therefore D(PC|PS) = 0.
   Perfect steganographic security
   One-time pad system is equivalent to the basic scheme
    of visual cryptography

   But,
    Wardens never allow random messages  Drawback



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Universal Data Compression
   Traditional data compression techniques
            Huffman coding
            require a priori knowledge about the distribution of the
             data


   Universal data compression algorithms
            Lempel and Ziv
            source statistics are unknown a priori or vary with time




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Willems' Repetition algorithm

 Parameters  block length L and delay D
 Binary source X producing {Xt} = X1, X2,…
  with values in {0,1}.
 Source output is divided into blocks Y1,
  Y2… of length L
 Encoding of a block Yt operates by
  considering
              Repetition time, the length of the interval since its last
               occurrence.
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…cont
   Repetition time is encoded using the following
    code




   where || denotes the concatenation of the bit
    strings.


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The Modification for Information
Hiding
   Information hiding takes place if the encoder
    or the decoder encounters a block y such that
    ty ≥ 1/ρ
   If this is the case, bit j of the message m is
    embedded in y‟ according to




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Limitations
   Embedding distortion DEmb can be large for
    the same secure statistics

   Future work by Joachim and Bernd,
    address the above issue.




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Conclusion
 A security model has been proposed
 Forms the basis for the hypothetical
  testing scenario
 Security of the Steganography system
  depends on the relative entropy between
  C and S.


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References
1.         Christian Cachin, “An Information - Theoretic Model for
           Steganography”, Cambridge, 1998.

2.         Joachim, Bernd, “A Communications Approach to
           Image Steganography”, Proceedings of SPIE, Jan
           2002.




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           Thank You…




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