Applying the SVD to Image Steganography

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Applying the SVD to Image Steganography

• Jennifer Davidson
• Department of Electrical and Computer
  Engineering, Department of Mathematics
• Clifford Bergman
• Department of Mathematics
• Daniel Wengerhoff
• M.S. student, Department of Mathematics and
  Information Assurance Program
• IOWA STATE UNIVERSITY
• Ames, Iowa

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Background

• Use of images, video and audio data in covert
  channels and for content protection and control
  of distribution of data has exploded in last 5-10
  years
• Data in digital format has become more widely
  available and easy to use as computers are
  faster, more devices and software available
• Distribution of digital data is easier and faster
  as bandwidth is increasing
• Images large amounts of data, even compressed
  are easier to manipulate on computers and
  distribute over networks
• Large video files still not enough bandwidth to
  distribute as easily as still image data or audio
  data (music)
• Digital information hiding is the study of
  inserting digital information into data so that
  specific goals are met
• Steganalysis is the field of analyzing data to
  detect or extract hidden information
• Digital information hiding is maturing as a field
  but still in its infancy

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Background

• Steganography study of hiding information in
  data with intent to convey a secret message
• One main goal of steganography is keep the stego
  image as innocent-looking as possible so it does
  not appear to have a message hidden in it
• Digital watermarking study of information hiding
  with intent to protect content, to control
  distribution of content, and other intellectual
  property concerns
• One main goal for watermarking (and
  steganography) is that the watermark be embedded
  in a visually imperceptible way
• Cryptography study of scrambling a message so
  that if and when it is intercepted, it cannot be
  understood unless key information is known
• Steganalysis analyzing data in which information
  has been hidden
• Analysis techniques can detect, extract, change
  or ultimately destroy the hidden information
• Results of steganalysis can also be used to
  change or improve embedding techniques

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Background

• Watermarking applications include
• Authentication validating source of message
• Integrity assuring that data was not modified
  when transmitted, accidentally or deliberately,
  use fragile watermarks that find location of
  changed data and how it changed
• Authentication and integrity of medical image
  data very important
• content protection of intellectual property
  (photographs, music)
• copy control for DVDs and CDs
• Steganography applications include
• military covert communication
• detection of covert communication, for military
  purposes or illegal criminal operations

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Main idea of data hiding

• For images use visual redundancy to take out
  redundant data and insert your message
• Human eye sees logarithmically and locally, and
  cannot perceive all 256 different colors or
  grayscales across an image simultaneously
• 32 colors often suffices
• Data hiding makes very slight changes so that eye
  does not perceive change

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Problem Statement

• How can the singular value decomposition (SVD) be
  used for steganography?
• Change singular values slightly
• Recall that the visual energy contained in the
  singular values and related eigenvalues is
  directly related to the quality of the visual
  data we perceive in the image
• Doesnt affect what you see much as singular
  values are robust to slight perturbations
• Already been done (Liu, 2002) for watermarking
  purposes
• Change unitary matrices slightly
• Our approach perturbs the unitary matrix in the
  SVD

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Definitions and Notation

• Given a matrix A, its SVD is
• where
• U, V are unitary
• S is a diagonal matrix with the singular values
  of as its entries
• is the transpose of V

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Definitions

• For our purposes we put U in a standard form
  defined as follows
• A column is greater than 0 if the first entry in
  the column is positive
• A matrix is in standard form if all of its
  columns are greater than 0
• If a column in U is less than zero then multiply
  it (and the corresponding column of V) by -1
• If a column in U is greater than zero then leave
  it alone

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The Embedding Algorithm

• Message to be embedded
• Assume message is encrypted and is represented by
  a binary string
• The binary string thus has a uniform distribution
• Cover image (original image to hide message in)
• Any image that can be represented by a matrix of
  values
• Grayvalues must be integers in the range 0,255
• Output image (stego image) cannot be compressed
  (embedded data will be lost)

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The Embedding Algorithm

• Break cover image up into k ? k blocks (k 8)
  and perform the SVD embedding on these smaller
  blocks (not on entire image)
• Step 1 calculate the SVD of a block
• Step 2 decide how many columns to keep the same
  in the matrix U (2)
• Step 3 embed message bits into U by changing the
  signs on certain entries in U ( 1 ? , 0 ? - )
• Adjust UE (embedded matrix) to maintain
  orthogonality

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Example

• Keep first 2 columns unchanged
• Construct UE by changing uk2
• Last entries z72 and z82 calculated so that
  column 3 is orthogonal to columns 1 2
• Next column insert 4 bits last 3 entries
  calculated so that column 4 is orthogonal to
  columns 1, 2 3
• etc.

• Message 0 0 1 1 0 1

Column 2 in U
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Example

• Reconstruct the block with the embedded bits
• where UE is the matrix constructed on previous
  slide, S, V are from original matrix (image) A
• round rounds to nearest integer clip values
  less than 0 get reset to 0 values larger than
  255 get reset to 255
• Do this for each block
• Capacity for 8 ? 8 block with two columns
  protected 15 bits per 64 pixels .234
  bits/pixel bit embedding rate
• .2 - .5 bits/pixel is typical for good
  stegoalgorithms
• May not be safe enough to avoid detection

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Extraction Algorithm

• Break image into blocks of size k ? k
• Compute the SVD of each block
• Put the U matrix in standard form
• Extract signs from values in appropriate
  locations
• Change signs into bit values

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The SVD for Steganography
Original image
Stegoimage
9x9 blocks, with the first column of each block
preserved and message embedded in the other
columns in the sign
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The SVD for Steganography
Original image
Stegoimage
9x9 blocks, with the first 4 columns of each
block preserved and message embedded in the other
5 columns in the sign
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Errors

• With this algorithm, the error between the
  extracted message and the embedded message is
  typically 8 - 10 of the total bits embedded
• Most of the error comes from the fact that the
  change from U to UE is large relative to the
  magnitudes of the entries in U
• We propose several solutions to reduce error

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Factors affecting error

• Iterative embedding
• Redundancy of message (error correcting)
• Number of columns protected
• Block size
• Relative spacing of singular values

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Iterative Embedding

• Embed message bits again in AE , that is, use AE
  as the original image and embed message again
• On the second pass of the algorithm, the entries
  in UE are much closer in value to previous
  unitary matrix

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Iterative Embedding

• Very effective due to smaller changes in unitary
  matrix after first pass
• 5 iterations are usually enough
• Greatly reduces error rates especially when
  combined with redundancy (reduces error from 8
  to less than 0.5)
• Computationally doesnt add much time

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Redundancy of Message Bits

• Greatly reduce error rate by using a simple error
  correcting code
• Embed same set of 15 bits into h consecutive
  blocks
• 5 times redundancy is good
• Combined with iterative embedding gives best
  results

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Block Size

• How big should blocks be?
• Considerations
• Data capacity
• k is the size of the blocks
• m is the number of columns protected
• Error rate
• Chart
• 8 x 8 is a good size

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Protecting Columns

• How many?
• Right, left, or both?
• Trade off between image fidelity and error rate
• Chart
• k2 is a good number for 8 x 8 blocks
• If the block size is larger (i.e. 16 x 16) m
  should be larger

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Adjusting Singular Values

• Does changing the singular values matter?
• Somewhat
• Chart
• Leave highest and lowest singular values alone
• Evenly space the rest

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Experiments Performed

• Basic embedding extraction of 100 random
  messages on ten images using different block
  sizes and cols protected
• Iterative embedding of 100 random messages on ten
  different images 8x8 blocks with 2 cols protected
• 5x redundancy on 100 random messages on ten
  different images using 8x8 blocks with 2 cols
  protected
• Iterative embedding using 5x redundancy while
  adjusting singular values on 100 random messages
  on 10 images with 8x8 blocks with 2 cols
  protected

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Calculation of Error Rates

• Hamming distance
• Subtract the extracted message from the embedded
  message and take the absolute value
• Add up all the 1s
• Divide the total by the message size

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Further Research

• Use of stream ciphers for encrypting messages
• Use of different encryption keys to avoid
  detection due to redundancy
• Do redundancy across images instead of within one
  image to reduce detection
• Use of other more sophisticated error correcting
  codes