Abstract:
The invention is for a method and system for encrypting and decrypting image/signal, based on new column and/or row operation of the image/signal, and a new digital watermark system, based on the new encryption/decryption system. The column and row operation are introduced for creating a chaotic image/signal so that the resulting image/signal is unreadable/inaudible with a fast computational speed. The new digital watermark technology can sustain cropping damage for verification.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
       [0001]    This application is related to U.S. Provisional Application No. 61/277,551, filed on Sep. 28, 2009. 
     
    
     BACKGROUND OF INVENTION 
       [0002]    1. Field of Invention 
         [0003]    A method of fast encrypting and decrypting image and data set, a method of encrypting and decrypting whole image using column and/or row operations, which are different to Fourier transform method, wavelet transform method, chaotic map method and other algebraic operations. 
         [0004]    2. Description of Related Art 
         [0005]    Image data security and authenticity have become more and more important thanks to the rapid development of internet and cloud computation. To protect the privacy or secrete for communicating via digital signal/image, encryption technology is needed. To provide authenticity and prevent piracy for digital image products, digital watermark technology is needed. 
         [0006]    A gray level digital image (or black-white image) of size m×n is given by a m×n matrix with each entry value given by the intensity of the image at the location of the entry. A digital color image usually refers to a RGB image, which is the combination of three color images (red, green and blue): each of them is a gray level image. The encryption and watermark methods in this description are referred to encrypting and watermarking gray images. 
         [0007]    A method and apparatus of image encryption is the combination of certain operations on the intensity and the location of each pixel so that the outcome image is not visible, or unreadable, or meaningless. 
         [0008]    Digital watermark is the process of embedding authentic information into a digital image which may be used to verify its authenticity or the identity of its owner, in the same manner as paper bearing a watermark for visible identification. The embedded invisible digital watermark usually is not noticeable: Either it is invisible, or it appears meaningless. 
         [0009]    An encryption system transfers a plaintext (the message to be encrypted) into a ciphertext (the encrypted message) via an encryption key. This is called the procedure of encryption. After the ciphertext is received through a public channel, the encryption system is used to transfer the ciphertext into a recovered plaintext via a decryption key. This is the procedure of decryption. An encryption system is also called a cipher, or a cryptosystem. There are two types of ciphers according to the relation between two cipher keys. If encryption key is the same as decryption key, the cipher is called symmetric cipher; if encryption key is different to decryption key, the cipher is called asymmetric cipher. There are also two different classes of ciphers according to the structure: block ciphers and stream ciphers. Block ciphers encrypt the plaintext block by block; stream ciphers encrypt the plaintex with a pseudo-random sequence (called keystream) controlled by the encryption key. If the plaintext is a digital image, the encryption system is called an image encryption system. 
         [0010]    Fast image encryption systems usually are block ciphers. There are various image encryption systems: (a) Fourier transform (FT) based encryption; (b) Wavelet transform based encryption; (c) Chaotic map based encryption; (d) Other algebraic operation based encryption, as discussed in the book by A. Uhl and A. Pommer: Image and Video Encryption, Springer 2005, pp 45-127. Usually, the FT-based encryption is very efficient in making image invisible (FT transfers the information on spatial domain to information on frequency domain). Other methods are easy to implement, but are not very efficient to make image invisible, thus the speed for encryption is not fast. Most existing methods for encryption suffer from cropping damage. That is: the original image cannot be recovered from a cropped encrypted image. These encryption methods cannot be used as digital watermark technologies for the image or video that easily suffers from cropping damage. 
         [0011]    Thus, there is a need for introducing a new encryption system that encrypts image fast and can recover the original image (or part of it) even the encrypted image has been cropped. The new system is also good for digital watermarking image that easily suffers from cropping damage. 
         [0012]    It is well known that the diffusion process via a heat equation may blur a given image u(x,y). See, for example, G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer 2002, pp 85-86, and pp 252-253. 
         [0000]    A heat equation with Dirichlet boundary condition is defined as 
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         [0000]    However, it is also well known that the backward heat equation is an ill-posed problem, thus cannot be used to decrypt the encrypted image. See, for example, W. Strauss, Partial Differential Equation, 2 nd  edition, Wiley 2007, pp 54-55. Serious modification is needed in order to apply heat equation for image encryption. 
         [0013]    Let the background of a given gray level image u(x,y) be a square: Ω=(0, a)×(0, b). Let a=lΔx, b=mΔy, and Δx=Δy. The discrete form of the image is given by 
         [0000]        u   i,j   0   =u ( iΔx,jΔy ), for  i= 1 , . . . ,l; j= 1 , . . . ,m.    
         [0000]    The heat equation (1) can be approximated by evolving the image along x-direction: 
         [0000]        u   i,j   n+1 =δ( u   i+1,j   n   +u   i−1,j   n )+(1−2δ) u   i,j   n   , u   1,j   n =u 1,j   0   , u   l,j   n   =u   l,j   0 ;  (2)
 
         [0000]      Or along y-direction: 
         [0000]        u   i,j   n+1 =δ( u   i,j+1   n   +u   i,j+1   n )+(1−2δ) u   i,j   n   , u   i,1   n =u i,1   0   , u   i,m   n   =u   i,m   0 ,  (3)
 
         [0000]    where u i,j   n  is used to approximate ν(iΔx,jΔy,nΔt), δ=Δt/Δx 2  is the step length for iteration. See, for example, G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer 2002, pp 230-232. It is also well known that for δ&gt;½, the above iterations may not converge. See, for example, G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing, Springer 2002, pp 233-234. Therefore, it is possible to generate invisible image through the iteration with δ&gt;½. 
         [0014]    Iteration (2) can also be represented by a linear system: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0000]    And iteration (3) can also be represented by a linear system: 
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         [0015]    Using equation (4 x ) or (4 y ) with a chosen δ&gt;½, we can encrypt an image. However, there are two drawbacks. (a) Matrix A (l-2)×(l-2)  and B (m-2)×(m-2)  may not be invertible. Thus we cannot decrypt the encrypted image. (b) Even the matrix is invertible; the size of the matrix usually is too large so that the computation speed is slow. We will further modify the above iterations so that each iteration is invertible, and the matrix operation is avoided. 
       SUMMARY OF THE INVENTION 
       [0016]    We encrypt a given image u i,j   0  along x-direction through only column operations as follows. Assume that l is an even number, otherwise we can delete the first or the last column of the image. Define iteration for backward difference heat equation with a specially chosen δ=½: 
         [0000]        u   1,j   n   =u   1,j   0  for all  n; u   2k+1,j   n =2 u   2k−1,j   n  for  k= 1 , . . . ,l/ 2−1;  (5 o )
 
         [0000]      and 
         [0000]        u   l,j   n   =u   l,j   0  for all  n; u   l−2k,j   n =2 u   l−2k+1,j   n+1   −u   l−2k+2,j   n  for  k= 1 , . . . ,l/ 2−1.  (5 e )
 
         [0017]    Equation (5 o ) indicates that we can recover all odd columns from first column after one step of backward diffusion procedure (i.e. we can solve backward heat equation with Dirichlet boundary condition for all odd columns). Equation (5 e ) indicates that we can recover all even columns from the last column after one step of backward diffusion procedure. 
         [0018]    Since one of the coefficients in equations (5 o ) and (5 e ) is 2, which is bigger than 1, we know that above backward iteration shall create “blowup” sequences (the resulting sequence may not be bounded as the iteration goes on). We thus use equations (5 o ) and (5 e ) to encrypt an image along x-direction by using negative integer n: u i,j   0  is the original image, u i,j   −1  is the encrypted image after first iteration, and u i,j   −n  is the encrypted image after nth iteration. 
         [0019]    Assume that m is an even number, otherwise we can delete the first or the last row of the image. We encrypt an image u i,j   0  along y-direction by 
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         [0020]    An image can also be encrypted using combination of equations (5 o )-(5 e ) and (6 o )-(6 e ). The special combination (r,s) (number of iteration along x-direction is r, and number of iteration along y-direction is s) can be used as the encryption key. 
         [0021]    The encrypted image along x-direction can be decrypted via heat equation (2) with δ=½: 
         [0000]        u   i,j   n+1 =½( u   i+1,j   n   +u   i−1,j   n ),  u   1,j   n   =u   1,j   0   , u   l,j   n   =u   l,j   0 ;  (7)
 
         [0000]    The encrypted image along y-direction can be decrypted via heat equation (3) with δ=½: 
         [0000]        u   i,j   n+1 =½( u   i,j+1   n   +u   i,j−1   n ),  u   i,1   n   =u   i,1   0   , u   i,m   n   =u   i,m   0 .  (8)
 
         [0000]    For an encrypted image with key (r,s), we decrypt it along x-direction r times, and along y-direction times. Thus the new encryption system is a symmetric system. 
         [0022]    The new system of encryption and decryption uses row and/or column operations, thus has fast computational speed. 
         [0023]    The decryption procedure is based on neighborhood rows or columns, thus the partial information can be recovered from a cropped encrypted image, which is especially important in the application of digital watermark. For digital watermark, an add-on mark is encrypted first, and then encrypted mark with reduced intensity (invisible) is added on a given image. 
         [0024]    To recover the digital mark, one first computes the difference between the original image and the watermarked image, then decrypts the difference image, and increases the intensity so that the watermark is visible. 
     
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
         [0025]      FIG. 1  is the flow diagram illustrating how to encrypt a given image using key (r, s): iterate the procedure along x-direction r times, along y-direction s times. 
           [0026]      FIG. 2A  is a figure to show how to obtain the third column from the first column (of original image) and the second column of the image from previous iteration of encryption. 
           [0027]      FIG. 2B  is a figure to show how to obtain other odd column from the column that is two column before it (which is obtained in previous step) and the column before it from previous iteration of encrypting image. 
           [0028]      FIG. 2C  is a figure to show how to obtain the third column to the last from the last column (of original image) and the second column to the last of the image from previous iteration of encryption. 
           [0029]      FIG. 2D  is a figure to show how to obtain other even column from the column that is two column after it (which is obtained in previous step) and the column that is one column after it from previous iteration of encrypting image. 
           [0030]      FIG. 3A  is a figure to show how to obtain the third row from the first row (of original image) and the second row of the image from previous iteration of encryption. 
           [0031]      FIG. 3B  is a figure to show how to obtain other odd row from the row that is two row above it (which is obtained in previous step) and the row above it from previous iteration of encrypting image. 
           [0032]      FIG. 3C  is a figure to show how to obtain the third row to the last from the last row (of original image) and the second row to the last of the image from previous iteration of encryption. 
           [0033]      FIG. 3D  is a figure to show how to obtain other even row from the row that is two row below it (which is obtained in previous step) and the row that is one row below it from previous iteration of encrypting image. 
           [0034]      FIG. 4A  is the flow diagram illustrating how to decrypt an encrypted image using key (r, s): iterate the procedure along x-direction r times, along y-direction s times. 
           [0035]      FIG. 4B  is a figure to show how to obtain the middle column from two columns besides it. 
           [0036]      FIG. 4C  is a figure to show how to obtain the middle row from two rows above and below it. 
           [0037]      FIG. 5A  is the flow diagram illustrating how to obtain an invisible watermark image. 
           [0038]      FIG. 5B  is a figure to show how to obtain an invisible watermarked image. 
           [0039]      FIG. 5C  is the figure to show how to recover the watermark from an image with digital watermark. 
           [0040]      FIG. 6A  is a figure to show the result of the standard Lena picture being encrypted with key ( 2 ,  3 ). 
           [0041]      FIG. 6B  is a figure to show that part of original image can be recovered from a cropped encrypted image from  FIG. 6A  (part of the encrypted image). 
       
    
    
     DETAILED DESCRIPTION 
       [0042]    Referring now to the drawings, and more particularly to  FIG. 1 , shown therein is the flow chat for the new image encryption method. To carry out the encryption, one first chooses two integer combination (r, s). r refers to encrypting a given image along x-direction r times; s refers to encrypting a given image along y-direction s times. Save the key (r, s) for decryption later. 
         [0043]      FIG. 2A-FIG .  2 D are figures to show one iteration of encrypting a given image along x-direction. 
         [0044]      FIG. 2A  is a diagram to show how to obtain the third column. In this embodiment, the third column (column  102 ) is the summation of negative one of the first column of the given image (column  104 ) and twice of the second column (column  106 ) from the original image or previous iteration. For example, for the first iteration of encryption, we use n=−1. So we choose the first column of the given image as column  104 , and the second column of the given image as column  106  to obtain new third column  102 . For the second iteration, we use n=−2. Thus we choose the first column of the given image as column  104 , and the second column from the image obtained from the first iteration as column  106  to obtain the new third column  102 . 
         [0045]      FIG. 2B  is a diagram to show how to obtain a general odd column. After we obtain the third column, the new fifth column is the summation of negative one of the new third column and twice of the fourth column from the original image or previous iteration. For example, for the first iteration of encryption, we use n=−1, k=2. So we choose the new third column as column  204 , and the fourth column of the given image as column  206  to obtain new fifth column  202 . For the second iteration, we use n=−2, k=2. Thus we choose the first column of the given image as column  204 , and the fourth column from the image obtained from the first iteration as column  206  to obtain the new fifth column  202 . The seventh, ninth columns, etc are obtained in the same manner. 
         [0046]      FIG. 2C  is a diagram to show how to obtain the third column to the last. In this embodiment, the third column to the last (column  302 ) is the summation of negative one of the last column of the given image (column  304 ) and twice of the second column to the last (column  306 ) from the original image or previous iteration. For example, for the first iteration of encryption, we use n=−1. So we choose the last column of the given image as column  304 , and the second column to the last of the given image as column  306  to obtain new third to the last column  302 . For the second iteration, we use n=−2. Thus we choose the last column of the given image as column  304 , and the second to the last column of the image obtained from the first iteration as column  306  to obtain new third to the last column  302 . 
         [0047]      FIG. 2D  is a diagram to show how to obtain a general even column. After we obtain the third column to the last, the new fifth column to the last is the summation of negative one of the new third column to the last and twice of the fourth column to the last from the original image or previous iteration. For example, for the first iteration of encryption, we use n=−1, k=2. So we choose the new third column to the last as column  404 , and the fourth column to the last of the given image as column  406  to obtain new fifth to the last column  402 . For the second iteration, we use n=−2, k=2. Thus we use the new third to the last column as column  404 , and the fourth to the last column of the image obtained from first iteration as column  406  to obtain new fifth to the last column  402 . The seventh to the last, ninth to the last columns, etc. are obtained in the same manner. 
         [0048]      FIG. 3A-FIG .  3 D are figures to show one iteration of encrypting a given image along y-direction. 
         [0049]      FIG. 3A  is a diagram to show how to obtain the third row. In this embodiment, the third row (row  502 ) is the summation of negative one of the first row of the given image (row  504 ) and twice of the second row (row  506 ) from the original image or previous iteration. For example, for the first iteration of encryption, we use n=−1. So we choose the first row of the given image as row  504 , and the second row of the given image as row  506  to obtain new third row  502 . For the second iteration, we use n=−2. Thus we choose the first row of the given image as row  504 , and the second row from the image obtained from the first iteration as row  506  to obtain the new third row  502 . 
         [0050]      FIG. 3B  is a diagram to show how to obtain a general odd row. After we obtain the third row, the new fifth row is the summation of negative one of the new third row and twice of the fourth row from the original image or previous iteration. For example, for the first iteration of encryption, we use n=−1, k=2. So we choose the new third row as row  604 , and the fourth row of the given image as row  606  to obtain new fifth row  602 . For the second iteration, we use n=−2, k=2. Thus we choose the first row of the given image as row  604 , and the fourth row from the image obtained from the first iteration as row  606  to obtain the new fifth row  602 . The seventh, ninth rows, etc are obtained in the same manner. 
         [0051]      FIG. 3C  is a diagram to show how to obtain the third row to the last. In this embodiment, the third row to the last (row  702 ) is the summation of negative one of the last row of the given image (row  704 ) and twice of the second row to the last (row  706 ) from the original image or previous iteration. For example, for the first iteration of encryption, we use n=−1. So we choose the last row of the given image as row  704 , and the second row to the last of the given image as row  706  to obtain new third to the last row  702 . For the second iteration, we use n=−2. Thus we choose the last row of the given image as row  704 , and the second to the last row of the image obtained from the first iteration as row  706  to obtain new third to the last row  702 . 
         [0052]      FIG. 3D  is a diagram to show how to obtain a general even row. After we obtain the third row to the last, the new fifth row to the last is the summation of negative one of the new third row to the last and twice of the fourth row to the last from the original image or previous iteration. For example, for the first iteration of encryption, we use n=−1, k=2. So we choose the new third row to the last as row  804 , and the fourth row to the last of the given image as row  806  to obtain new fifth to the last row  802 . For the second iteration, we use n=−2, k=2. Thus we use the new third to the last row as row  804 , and the fourth to the last row of the image obtained from the first iteration as row  806  to obtain new fifth to the last row  802 . The seventh to the last, ninth to the last rows, etc. are obtained in the same manner. 
         [0053]    Referring now to the drawings, and more particularly to  FIGS. 4A-4C , shown therein is the method of the new image decryption. To carry out the decryption, we use decryption key (r, s): decrypting the encrypted image along x-direction r times, and decrypting the encrypted image along y-direction s times.  FIG. 4A  is the flow chart. 
         [0054]      FIG. 4B  is a diagram to show one iteration of decrypting an encrypted image along x-direction. The first and the last column will be kept the same. Other new column is obtained as the average of the column before it and the column after it from the encrypted image or previous decrypting iteration. 
         [0055]      FIG. 4C  is a diagram to show one iteration of decrypting an encrypted image along y-direction. The first and the last row will be kept the same. Other new row is obtained as the average of the row above it and the row below it from the encrypted image or previous decrypting iteration. 
         [0056]      FIG. 5A  is the flow diagram illustrating how to obtain an invisible watermarked image. 
         [0057]      FIG. 5B  is a diagram to show how to obtain an invisible watermarked image. The mark  902  first is encrypted into mark  904 , and the intensity of the encrypted mark is reduced α 100% so that the resulting image  906  is invisible. The invisible encrypted mark  906  is then added to the carrier image  908  to obtain a watermarked digital image  910 . 
         [0058]      FIG. 5C  is a diagram to show how to recover the watermark from an image with digital watermark. The different image  1006  is obtained from the marked image  1002  and the original image  1004 . The intensity of the difference is increased to (1/α)100% to obtain the encrypted image  1008 , and then image  1008  is decrypted to recover the mark image  1010 . 
         [0059]      FIG. 6A  is a diagram to show the result after encrypting the standard 256×256 Lena image with key ( 2 , 3 ). 
         [0060]      FIG. 6B  is a diagram to show that the part of Lena image can be recovered from a part of encrypted image after it is decrypted using key ( 2 , 3 ).