Patent Application: US-200913125799-A

Abstract:
there is described a method of encrypting a set of 2d input data , preferably image data . the method comprises obtaining the hash value of a password and re - sizing the hash value to fir the size of the 2d input data . the re - sized data is transformed using an irreversible transform , and the output of the transform is then used to encode the 2d data .

Description:
an embodiment of the invention will now be described , by way of example only , with 20 reference to the accompanying drawings , in which : fig1 shows the different iterative results of applying eq . ( 1 ) with 1 = 2 on the image ‘ lena ’ of size ( 101 × 101 ) ( shih , 2008 ); fig2 shows colour sensitivity of the image “ mother nature ” to a number of cycles ( a = 3 . 9 and n = 75 ), where image ( a ) is encoded with j = 1 , and image ( b ) is encoded with j = 2 ( pisarchik et al ., 2006 ); fig3 shows the colour sensitivity of the image “ mother nature ” to a number of iterations ( a = 3 . 9 and j = 3 ), where ( a ) is the original image , ( b ) is the image encoded with n = 1 , ( c ) is with n = 30 , and ( d ) is with n = 75 ( pisarchik et al ., 2006 ); fig4 is an overview of the image encryption method of the invention ; fig5 shows the results of a sensitivity test of the method of the invention on a sample image ; fig6 shows a set of correlation analyses for the images of fig5 ; fig7 shows a histogram analysis performed on sample image ‘ lena ’, and the image when encrypted using the method of the invention ; fig8 shows the results of frequency tests performed on the encrypted version of the sample image ‘ lena ’; fig9 shows two greyscale images and the associated grey values of each image — 9 ( a ) shows a cropped plain . patch from a natural image , and 9 ( b ) shows the image of ( a ) encrypted using the method of the invention ; fig1 shows ( a ) an original image , and the encrypted version of the image when using ( b ) the method of the present invention , and ( c ) a baker map ; fig1 shows ( a ) an original image , and the encrypted version of the image when using ( b ) 128 - bit aes running in ecb mode , and ( c ) the method of the present invention ; fig1 is a diagram illustrating the trade - off between robustness and distortion in binary to integer conversion ; fig1 ( a ) shows a histogram analysis performed on a sample image of a patient &# 39 ; s ct scan , and the image when encrypted using the method of the invention , and fig1 ( b ) shows the procedure for embedding such encrypted data in a face image ; fig1 shows the a test image “ mother nature ”, and the difference in the encrypted versions of the test image when utilising relatively similar passwords ; fig1 illustrates the cryptographic diffusion produced through use of the method of the invention ; fig1 shows the result of encrypting the test image “ mother nature ” using the method of the invention , and the recovery of the original image from the encrypted data ; fig1 shows the result of encrypting a test image of an id card using the method of the invention , and the recovery of the original image from the encrypted data ; fig1 shows the results of two image processing attacks on a steganographic image encrypted with the method of the invention , illustrating the resistance of the method to the attacks ; fig2 is a further overview of the image encryption method of the invention , for a black - and - white image , further comprising a post - encryption for improving resistance to plain text attacks ; fig2 illustrates how the pixel substitution process is performed for a sample data set ; fig2 shows the result of a chosen - plaintext attack ( cpa ) on an image encrypted using the method of fig2 ; and fig2 shows the method of recovery of an encrypted image from the process shown in fig2 . it is intended to extend the 1 - d hashing algorithm sha - 1 to encrypt digital 2d data . the terminology and functions used as building blocks to form sha - 1 are described in the us secure hash algorithm 1 , see the reference . the introduction of fast fourier transform ( fft ) forms together with the output of sha - 1 a strong image encryption setting . it is shown that the sha - 1 algorithm , which is a one - time pad hash algorithm , can meet both requirements of confusion and diffusion with a hashed key . the encryption method of the invention is illustrated in fig4 . the method can be seen as being in the opposite direction of what fridrich and goljan ( 2000 ) proposed , where they use a key to a 64 × 64 image block to return a hash of length n = 50 bits . however , in the method of the present invention , the strength of a 1d encryption algorithm is exploited ( namely sha - 1 ), and it is extended to handle 2d data such as images . the fft is incorporated into the process to increase the disguise level , and thus generate a random - like output that does not leave any distinguishable patterns of the original image . the method of the invention starts with a password phrase p supplied by the user ( step 10 ). this password phrase p is then used to generate an sha - 1 based hash string h ( p ), by applying the hashing function to p ( step 12 ). h ( p ) is in “ char ”, or character format , which is then converted into the appropriate binary bit stream ( step 14 ). the bit stream vector of h ( p ) can then be transformed to a matrix of fixed dimension , e . g . 8 × 35 . parallel to this , the original image a is provided in rgb colour space ( i . e . the image can be represented as three different channels of data representing the red , green and blue colour spaces of the image respectively ) ( step 18 ). it will be understood that any suitable colour space implementation may be used in place of rgb colour space . the three different channels are converted to a bit stream and reshaped to have the dimension of 8 ×( π ( m , n )) ( step 20 ), where m and n are the height and width respectively of the image a . ( the formula ( π ( a , b )) is used to refer to the product of terms a and b .) the dimension 8 × 35 is chosen for convenience sake , i . e . if the method is dealing with the encryption of 8 - bit grayscale images , or 24 - bit rgb colour image files , then 8 is from the maximum length of the binary representation of the maximum possible grayscale value ( 255 ). it will be appreciated that the algorithm also handles the encryption of binary data . in such a case , the above dimension would be changed to 1 ×(( π ( m , n ))= π ( m , n ). the binary key produced at step 14 , herein of dimensions 8 × 35 , is too short to accommodate the image bit stream . therefore , the key is resized towards the needed dimension , herein 8 ×( π ( m , n )) ( step 16 ). this step would normally result in repetitive patterns , that would turn the ciphered image prone to attacks , which was independently noticed by usman et al . ( 2007 ). to cope with this situation , a modified two - dimensional discrete cosine transform ( dct ) followed by a two - dimensional fast fourier transform ( fft ) is applied to provide the confusion and diffusion requirement and to tighten the security ( step 22 ). prior to the transform operation , a matrix permutation ( step 17 ) is performed on the resized key produced by step 16 . taking the hash string h ( p ) generated in step 12 , this is used as the seed for a pseudo - random number generator ( prng ) to produce a pseudo - random string . ( it will be understood that any suitable key may be used as the seed for the prng , e . g . the original password p .) this pseudo - random sequence is used to permute the 8 ×( π ( m , n )) matrix of the binary key from step 16 . the permuted matrix is then passed to the transform stage — step 22 . with regard to step 22 , let the resized and permuted key bit stream from step 17 be λ 8 , mn where the subscripts m and n denote the width and height dimensions of the image . in step 22 , the fft operates as shown in eq . ( 3 ) on the dct transform of λ 8 , mn , subject to eq . ( 4 ), below . where f ( x , y )= dct ( λ k , l ) , satisfying eq ( 4 ), and subject to : note that for the transformation at the fft and discrete cosine transform ( dct ) levels the whole coefficients are not utilised . rather , the following rule is used , which generates at the end a binary random - like map . given the output of eq . ( 3 ), the binary map can be derived straightforwardly by : where thr is an appropriately selected threshold value . for a balanced binary sequence and for robustness , thr should be chosen such that the probability p ( f ( u , v )& lt ; thr )= p ( f ( u , v )& gt ; thr ). as f ( u , v ) is a complex function , the thresholding of above map ( x , y ) function can be based on the imaginary part of the complex function . in general , the complex imaginary part of the signal f ( u , v ) is symmetrical around zero ( see fig8 for validity of this property ). therefore , thr = 0 can be an explicit solution . however , it will be understood that the threshold thr may be adjusted subject to the requirements of the system . since the coefficients using this calculation are converted to binary map , the reconstruction of the password phrase is impossible , hence the name irreversible fast fourier transform ( irfft ). in other words , it is a one - way hash function which accepts initially a user password . the map is then xored with the respective bit stream versions of the rgb channels of the image ( step 24 ). the separate xored channels are then converted back into decimal values using a binary to decimal conversion system ( step 26 ). these decimal values for the different channels ( which can be interpreted as greyscale values ) can then be combined and reshaped ( step 28 ) to form the output ciphered ( encrypted ) image . nested transforms are not scant in the literature , for example o &# 39 ; ruanaidh et al . ( 1997 ) use fast fourier transform followed by log - polar mapping and followed by fast fourier transform to embed a watermark . the coding phase of the invention uses the map ( eq . ( 4 )) to encrypt the bit stream of the image a and produce a new encrypted matrix a ′, in such a way that : where d ( a ′, map ) denotes the decoding of a ′ with the same key generated map . preferably , ε auth should be equal to { ø } ( i . e . the null set ), and starts to deviate from that when a ′ undergoes an image processing attack . another phenomenon that is noticed is the sensitivity of the spread of the fft coefficients to changes in the spatial domain . therefore , when coupled with the sensitivity of the sha - 1 algorithm to changes of the initial condition , e . g . the password phrase , the algorithm can easily meet the shannon law requirements . for instance , a small change in the password phrase will , with overwhelming probability , result in a completely different hash . the following exemplifies such an assertion : input password : ‘ steganography ’ the corresponding hash function : ‘ 40662a5f1e7349123c4012d827be8688d9fe013b ’ input password : ‘ steganographie ’ the corresponding hash function : ‘ c703bbc5b91736d8daa72fd5d620536d0dfbfe01 ’ it is intended to transform these changes into the spatial domain where 2d - dct and 2d - fft can be applied that introduce the aforementioned sensitivity to the two dimensional space . as such , images can be relatively easily encoded securely with password protection . note that this scheme encrypts efficiently grayscale and binary images . however , for rgb images it is noticed that using the same password for the three colours ( r , g , and b ) will yield some traceable patterns inherited from the original image . this is easily overcome through use of one of two options : either the user supplies three passwords , each of which encrypts one colour channel or , which is more convenient , two unique keys are generated from the original supplied password . in fig4 , for instance , a single key is utilised to generate the following different hash functions h ( k ), h ( k ) and h ( h ( k )) to encrypt the r , g and b channels respectively . k denotes the supplied key , and the arrows indicate the string reading directions . regarding the security aspects of the invention , encryption algorithms are assumed to be robust to different statistical and visual attacks , and moreover key sensitivity and key space should be adequate . it is possible to analyse the security of the invention by considering key space analysis , key sensitivity , adjacent pixels analysis and statistical analysis and other security merits . the key space analysis of the algorithm of the invention comes down to analysing sha - 1 algorithm . the hashing algorithm sha - 1 is used , and implemented in php ( the popular web programming language ). sha - 1 accepts any key of any length less than 264 bits . the sha - 1 is called secure because it is computationally infeasible to find a message which corresponds to a given message digest , or to find two different messages which produce the same message digest 2 . sha - 1 is well adopted in several organisations and has received much scrutiny from the cryptography community . the algorithm of the invention is flexible enough in case of migrating to a newer version of sha &# 39 ; s family or other secure hash functions . a number of tests were carried out on image databases consisting of popular test images such as ‘ cameraman ’ or ‘ lena ’; images with different complexities and grayscale ; colour and binary images . the algorithm of the invention has been proven to be very sensitive to initial condition , as can be seen from fig5 , thanks to the plugged in hash algorithm and the irfft . fig5 shows the results of a key sensitivity test using a sample image . fig5 ( a ) is the encrypted image ; fig5 ( b ) is the encrypted image decrypted using the correct key ‘ steganography ’, having the hash ‘ 40662a5f1e7349123c4012d827be8688d9fe013b ’; fig5 ( c ) is the encrypted image decrypted using the wrong key ‘ steganographie ’, having the hash ‘ c703bbc5b91736d8daa72fd5d620536d0dfbfe01 ’; and fig5 ( d ) is the encrypted image decrypted using a slightly modified hash ‘ 40662a5f1e7349123c4012d827be8688d9fe013b ’. as can be seen from the images , even a minor change in the hash used does not result in a partially decrypted image . to test for statistical properties of the original image and the encrypted version , a test was carried out based on the linear relationship between two adjacent pixels horizontally , vertically and diagonally . it is observed that natural images with natural data have high correlation ratio between neighbouring pixels ( see fig6 ). to measure such a relationship the correlation coefficient is calculated , as appears in table 1 , of each pair pixels using the following system : in fig6 , a correlation analysis is shown of 5000 pairs of horizontal adjacent pixels chosen randomly from the following images : ( a ) is for the original unencrypted plain boat image ( fig5 ( b )); ( b ) is for the re - arrangement of the pixels of fig5 ( b ) using conventional permutation ; and ( c ) is for the encrypted image of fig5 ( b ) using the method of the invention . it can easily be seen that , while some patterning is still evident using the conventional permutation , the present invention results in a more thorough obfuscation of any data patterning . the comparison given in table 1 shows that the proposed algorithm outperforms other recent methods reported in the literature . to establish a fair evaluation , the same test image is used . in the horizontal , diagonal and vertical directions the encrypted version of the algorithm of the invention had the highest performance . unlike other methods , the algorithm of the invention implies no iterations , the encrypted image shown in fig7 is automatically generated once the program is invoked with a key . table 1 shows a performance analysis of the method of the invention against known prior art methods , using the ‘ lena ’ test image . the correlation coefficients of pairs of adjacent pixels in different directions range from ‘ 1 ’ ( highly correlated ) to ‘− 1 ’ ( highly uncorrelated ). these coefficients ensure the two considered images are statistically independent but with different degrees . with regard to the conventional permutation , a permutation is a bijection function ( φ ) that maps each element x in a set s to a different index φ ( x )≠ x . it should be noted that this function , unlike the method of the invention , does not alter pixel values — it merely re - positions them . from this table , it can be seen that the method of the invention produces greater 20 performance than the prior art methods , as there is considerably less adjacent pixel correlation . fig7 ( a ) shows the sample image ‘ lena ’, and the image histogram analysis of the sample image . fig7 ( b ) shows the lena image encrypted using the method of the invention , and the image histogram of the encrypted image . the process does not retain any image statistics — this can be seen from comparing histograms of the plain and encrypted images , as the original histogram is flattened and has a uniform distribution for the encrypted version . given a randomly generated n - bit sequence , it is expected that approximately half the bits in the sequence to be ones and approximately half to be zeros . the frequency test checks that the number of ones in the sequence is not significantly different from n / 2 ( kanso and smaoui , 2007 ). in fig8 , an analysis of the frequency test is shown — fig8 ( a ) shows the complex imaginary part of f ( x , y ) in eq . ( 3 ), while fig8 ( b ) shows the corresponding binary map after applying eq . ( 4 ). the number of non - zero matrix elements is (≈ n / 2 )= 39998 , where n = 100 × 100 × 8 = 80000 . it is noticed that the complex imaginary part of the fast fourier transform exhibits conjugate symmetry in such a way that : fig8 shows such a property , where the magnitude of the transform is centred on the origin ( f ( u , v )= 0 ). in other words , eq . ( 4 ) yields a balanced binary sequence which passes this test . this assertion holds true for any 8 - bit image . however , for 1 - bit type , i . e ., binary image , eq . ( 3 ) employs no imaginary part , therefore , the real part is instead utilised which is also quasi - symmetrical . apart from the above performance of the method of the invention , two additional merits of the method are highlighted . the first feature is that the proposed scheme is capable of not just scrambling data like all chaos algorithms do , but also it changes the intensity of the pixels which contributes to the safety of the encryption . for convenience , fig9 illustrates a cropped grayscale matrix of size 4 × 5 from a natural image ( fig9 ( a )), along with its encrypted version ( fig9 ( b )), and the associated grey values of each image . notice that same gray values are producing different encryption values — this irregularity is very important to hamper any attempt to reverse attack the algorithm . as can be appreciated from the figure , the algorithm is operable to fuse the confusion and diffusion . the second feature of the proposed algorithm is the unbiased handling of both gray scale and binary images . chaos has a special case where they can be considered analogous to encryption , and that is when there is a binary plain image ( consisting of 0 and 1 values ). fig1 shows the encryption of an image consisting of a 10 × 10 black square on a white background — fig1 ( a ) shows the original binary image ; fig1 ( b ) shows the image when encrypted using the method of the invention ; and fig1 ( c ) shows the image encrypted after nine iterations using the baker map ( the dots were stretched using an erosion operation for better visualization ( fridrich , 1997 )). it is clear that the approach of the invention provides more confusion than its counterparts ( herein the baker map ). note that all images shown in the figure are of binary type . if an image contains homogenous areas , such as the one shown in fig1 ( a ), a large amount of redundant data will surf and thwart the efficiency of encryption algorithms , and laying the ground for a codebook attack . this is due to the consecutive identical pixels , which lead to the same repeated patterns when a block cipher is used in the electronic code book ( ecb ) mode ( shujun et al ., 2004 ). fig1 ( a ) shows an uncompressed plain - image containing many areas with fixed gray - levels ; fig1 ( b ) shows the corresponding encrypted image encrypted by 128 - bit aes running in ecb mode ( shujun et al ., 2004 ); and fig1 ( c ) shows the image encrypted using the algorithm of the invention . since the algorithm of the invention is not block based , the problem of homogenous areas does not impact on the efficacy of the encryption . after generating the encrypted payload , the colour transformation rgb → yc b c r is used on the cover image which will carry the encrypted data . the use of such a transformation is to segment homogeneous objects in the cover image , namely the human skin region . the yc b c r space can remove the strong correlation among r , g , and b matrices in a given image . in this approach , the concentration on skin tone is motivated by some applications of the final product . the algorithm starts first with segmentation of probable human skin regions : in eq . ( 8 ) c denotes the cover image , bck background regions and ( s 1 , s 2 , . . . , s n ) are connected subsets that correspond to skin regions . based on experiments carried out by the inventors , it has been found that embedding into these regions produces less distortion to the carrier image , compared to embedding in a sequential order or even in a noise - like fashion . in addition to this , the algorithm yields a robust output against reasonable noise attacks and translation . robustness against noise is due to the embedding in the 1st - level 2d haar dwt ( discrete wavelet transform ) with the symmetric - padding mode . dwt is a well known transformation that gained popularity among the image processing community , especially those who are dealing with image compression . its applications in different areas is growing however ( note that jpeg2000 uses dwt to compress images ). 2d dwt provides a decomposition of the approximation , and the details in three orientations ( horizontal , vertical , and diagonal ) by means of a convolution - based algorithm using high and low pass filters . in this case four filters associated with the orthogonal or bi - orthogonal of the haar wavelet are computed . a wavelet - based transformation is chosen over dct ( discrete cosine transform ) because : ( a ) the wavelet transform understands the human vision system ( hvs ) more closely than does dct ; ( b ) visual artefacts introduced by wavelet coded images are less evident compared to dct , because the wavelet transform does not decompose the image into blocks for processing ; and ( c ) dft ( discrete fourier transform ) and dct are full frame transforms — hence any change in the transform coefficients affects the entire image except if dct is implemented using a block - based approach . however , dwt has spatial frequency locality , which means if the signal is embedded , it will affect the image locally ( potdar et al ., 2005 ). hence a wavelet transform provides both frequency and spatial description for an image . more helpful to information hiding , the wavelet transform clearly separates high - frequency and low - frequency information on a pixel - by - pixel basis ( raja et al ., 2006 ). manipulating coefficients in the wavelet domain tends to be less sensitive , unlike other transformations such as dct and fft . for binary stream processing , there are two methods to convert decimal integer to a binary string : one is to use the conventional decimal to binary conversion , and the other is termed binary reflected gray code ( brgc ) 3 . this binary mapping is the key to the augmented embedding capacity introduced by the method named “ a block complexity data embedding ( abcde )” proposed in ( hioki , 2002 ). there is a trade - off , however , between robustness and distortion , which is summarized in fig1 . fig1 shows an 8 - bit ( 1 byte ) representation of the conventional integer to binary conversion . it is clear that choosing the right index for embedding is very crucial . this intricacy is less severe when using the brgc , since it produces seemingly disordered decimal - to - binary representation . the resistance to geometric distortion is feasible since , unlike s - tools and f5 , when skin tone blobs are selected , eye coordinates can be detected , which act as reference points to recover the initial position and orientation . thus , this makes the method of the invention invariant to both rotation and translation . the proposed encryption scheme is preferably applied to digital image steganography for two reasons , the first motivation is that embedding a random - like data into the least significant bits ( lsbs ) would perform better than embedding the natural continuous - tone data , and secondly for security and fidelity reasons the embedded data must undergo a strong encryption , so even if it is accidentally discovered ( which is unlikely to happen ), the actual embedded data would not be revealed . more specifically , identification cards ( id cards ), which are prone to forgery in aspects relating to biodata alteration or photo replacement , are an ideal implementation of the method . to evaluate the performance of the proposed system , a set of rgb images were used for this purpose . fig1 shows an example of the test data , with the associated psnr value ( discussed below ). fig1 ( a ) shows a set of image data to be encrypted — herein a ct scan of a young female with chronic breathlessness disease — and its encrypted version , each of which are shown along with their respective image histograms . fig1 ( b ) outlines the process for the concealment of the medical data of fig1 ( a ) in a face image . initially ( starting from the top right corner ), a face image is provided . the areas of skin tone present in the image are then detected , using any suitable skin tone detection method . the facial features are then extracted from the image , providing details such as the location of the eyes , nose , mouth , and the angle of rotation of the facial region . finally , the encrypted version of the secret data ( as shown in fig1 ( a )) is embedded in the facial region of the image . note that the use of biometric images facilitates having the embedding invariant to rotation and translation . this method is discussed in more detail in uk patent application no . 0819407 . 8 , filed oct . 23 , 2008 , which is incorporated by reference herein . it is believed that there are numerous different applications for such an extended 2d - sha - 1 algorithm , one of which is in the field of steganography . this technology can overcome the difficulties mentioned previously . it is shown that the results of the algorithm of the present invention is superior to the work of ( pisarchik et al ., 2006 ) in terms of algorithm complexity and parameter requirements . moreover , the algorithm is securely backed up by a strong id hash function . in ( pisarchik et al ., 2006 ) the desired outcome converges after some iteration , which needs to be visually controlled to flag the termination of the program . however , the algorithm of the invention is run only once for each colour component ( r , g and b ). the algorithm of the invention needs only one input from the user ( the password ) and it will handle the rest of the process , while in ( pisarchik et al ., 2006 ) three parameters — namely the reported a , j , and n — are required . the method of the invention can be applied to gray scale images as well as binary images . these extensions are not feasible in ( pisarchik et al ., 2006 ) as they incorporate into their process relationships between the three primary colours ( r , g and b ). finally , time complexity which is a problem admittedly stated in ( pisarchik et al ., 2006 ) would be reduced greatly by adopting the method of the invention . the algorithm was tested on the same test image described in ( pisarchik et al ., 2006 ) to establish a fair judgement , namely “ mother nature in the new millennium ”, as shown in fig1 ( a ). to demonstrate visually the diffusion requirement being met , fig1 ( b ) illustrates the encrypted output of the test image with ‘ steganography ’ as the password , with fig1 ( c ) showing the output with ‘ steganographie ’ as the password . even though only a small change has occurred in the password used , the final two chaotic maps differ dramatically as can be seen from fig1 ( d ), which shows the difference between ( b ) and ( c ). fig1 shows the sensitivity of the algorithm to alterations on the 2d spatial data ( i . e . the image ). fig1 ( a ) shows the altered test image ( a black box is added to the lower right - hand corner of the image ), with fig1 ( b ) showing the altered image when encrypted using ‘ steganography ’ as the password . to illustrate the sensitivity of the algorithm , fig1 ( c ) shows the difference between fig1 ( b ) and fig1 ( b )— i . e . the difference in the output of the method of the invention when the original test image is altered slightly . this sensitivity , combined with the sensitivity shown in fig1 , forms an excellent property of the algorithm of the invention . from fig1 and fig1 it is clearly seen that the 2d encryption of the invention meets the diffusion requirement for steganography . pisarchik et al . ( 2006 ) altered the test image by adding a black box at the lower right corner of the image and tried to visualise the difference by means of image histograms . even though an image histogram is a useful tool , unfortunately it does not tell much about the structure of the image and in this case about the displacement of colour values . histograms accumulate similar colours in distinguished bins regardless of their spatial arrangements . a better alternative would be to use similarity measurement metrics , such as the popular peak signal to noise ratio ( psnr ). psnr values will run into infinity if the two examined sets are identical . psnr is defined by the following system : where mse denotes the mean square error , which is given by : and max c holds the maximum value in the examined image , for example : wherein c max ≦ 1 in double precision intensity images , and c max ≦ 255 in 8 - bit unsigned integer intensity images ; x and y are the image coordinates , m and n are the dimensions of the image , s xy is the original data and c xy is the modified data . psnr is often expressed on a logarithmic scale in decibels ( db ). psnr values falling below 30 db indicate a fairly low quality ( i . e ., distortion caused by embedding can be obvious ); however , a high quality steganographic application should strive for 40 db and above . beneath are some key points that should be kept in mind when calculating psnr . note 1 : many authors take the above values ( 1 , 255 ) as the default values for c max , in binary and 8 - bit images respectively , regardless of the range of the examined intensity values . however , it can be the case for example that an 8 - bit original image has its values range only from 3 to 240 , and thus its c max would be then 240 . hence , c max is defined as the maximum fluctuation in the observed input image data . this makes c max an image dependent value . note 2 : the psnr is a universal formula , which can be straightforwardly applied when dealing with grayscale images . however , one can face a problem when confronting true rgb colour images . some authors treat each colour channel ( r , g and b ) separately when calculating the mse , prior to calculating the average mse ( amse ) ( saenz et al ., 200 0 ; yuan - hui et al ., 2007 ). matlab , on the other hand , advises that the rgb model be completely converted into ycbcr colour space , where the image primary colours ( rgb ) are represented by a weighted average in the luma channel ( mathworks , see the reference ). consequently , the latter component ( y ) is recommended to calculate the psnr . the distortion that needs to be measured might have affected only colours ; therefore , de - correlating such colours would not stipulate accurate results , at least from steganography point of view , mathworks &# 39 ; hint is not appropriate . note 3 : the psnr can easily be drawn based on incorrect attempts to calculate the mse ( the denominator in the psnr eq . ( 9 )). hence , image subtraction should be applied on double precision values , since deriving image differences based on 8 - bit unsigned integers would yield different results since intensity values truncation would have taken place . table 2 shows the psnr values of the different generated chaotic maps ( the unit measurement of psnr is decibel ( db )), which provides further detail regarding the diffusion aspect . pisarchik &# 39 ; s algorithm ( pisarchik et al ., 2006 ) involves a rounding operator applied each time the program is invoked by the different iterations . the present invention does not adopt this feature , as it is believed that there will be a loss of information when the embedded data is reconstructed . in the present invention , the algorithm works in one direction , and the recovery would be initiated by the same password and goes in parallel , i . e . not in the reverse order . fig1 ( a ) shows an input image (“ mother nature ”), 17 ( b ) the encrypted image , and 17 ( c ) the recovered image . in fig1 , the psnr equals infinity , which means the two images are identical . fig1 shows the output of the algorithm when applied to a binary image , 17 ( a ) being the original image , 17 ( b ) the encrypted image , and 17 ( c ) the recovered image with psnr = infinity . three types of attacks were carried out on the algorithm , namely noise impulses , rotation , and cropping attacks , as demonstrated in fig1 . fig1 shows the resistance of the algorithm to image processing attacks , when carried out on the steganographic image of fig1 . fig1 ( a ) shows the carrier steganographic image top attacked with a joint attack of cropping and rotation of − 12 degrees , with fig1 ( b ) showing the extracted secret data . in fig1 ( c ), the carrier steganographic image is attacked with salt and pepper noise , with the recovered secret data shown in fig1 ( d ). fig1 clearly demonstrates the resistance of the algorithm of the invention to image processing attacks . the algorithm is capable to survive jpeg compression attack up to 75 %— below that the hidden data will be totally destroyed . it is believed that that surmounting jpeg compression was enhanced by the encryption of the payload data , since encryption often significantly changes the statistical characteristics of the original multimedia source , resulting in much reduced compressibility ( mao and wu , 2006 ). this resilience to attacks is deemed to be essential in image steganography or watermarking . in this case , the algorithm of the invention performs better than peng &# 39 ; s algorithm ( peng and liu , 2008 ). the retrieved encrypted data was hit severely because the embedding strategy , for perceptibility reasons , took place in the least significant bits of the carrier image ( lsbs ). if robustness of the encryption is considered alone without the embedding phase , much better performance can be seen , as depicted in fig1 . in fig1 ( a ), a visual example of an encrypted image is attacked with salt and pepper noise of 20 % density . in fig1 ( b ), the decrypted image is shown − psnr = 14 . 7057 db −. fig1 ( c ) shows the psnr versus noise density shown as a function . a further enhanced version of the method of the invention is shown in fig2 . this method includes a post - encryption step which improves the resistance of the algorithm to a chosen - plaintext attack ( cpa ). cpa is an attack model in which an attacker is presumed to have the ability to encrypt a plain image to obtain its corresponding cipher . the purpose of this attack is to exploit weaknesses in the encryption algorithm in the hope to reveal the scheme &# 39 ; s secret key , as shown in equation ( 11 ). where a is the decrypted image , a ′ is the encrypted image , b ′ is the attacker &# 39 ; s encrypted neutral image , { circle around ( x )} is the xor operation , and map is the key ( see equation 4 above ). with reference to fig2 , the method is largely similar to the method illustrated in fig4 . here , a password key k (“ securemeplease ”) is passed through an appropriate hash function , e . g . sha , to provide hash string h ( k ). as with above , the hash string is then converted to a binary bit stream , and is re - shaped to fit the size of the image in question , and permuted using a prng seeded with h ( k ). the modified dct followed by fft is then performed using eqns . ( 3 ), (*), & amp ; ( 4 ), as above , to obtain a binary random - like map ( output at m in the diagram ). in parallel , the original image b is converted to a bit stream and reshaped to have the required dimensions . ( in contrast to the method of fig4 , the method shown in fig2 is performed for a black - and - white image , removing the requirement to split the image up into the three separate rgb channels .) the binary map is xored with the bit stream version of the image b . the result is then converted into grayscale values , then reshaped to form an encrypted image ( indicated at b , at the output of the binary to decimal conversion ). in order to reduce the threat of an attack using cpa , a new map k2 for pixel substitution is formed by hashing the hash of the original key , i . e . k2 = h ( h ( k )). the purpose of this random map is to exchange the encrypted values falling on the on pixels in the map with those falling on the off pixels and vice - versa . a new encrypted matrix b ′ can then be created , using equation ( 5 ) and the new pixel substitution map created from k2 = h ( h ( k )). with reference to fig2 , the pixel substitution process is illustrated in both matrix form ( fig2 ( a )) and vector form ( fig2 ( b )). a sample set of 2d data of dimensions 4 × 4 is indicated at 100 ( representative of the encrypted 2d image matrix b resulting from the binary to decimal conversion in fig2 ) . the pixel substitution map ( e . g . k2 as described above ) is indicated at 102 . the appropriate swap configuration is determined based on the index of 1 &# 39 ; s and 0 &# 39 ; s in the pixel substitution map ( the index shown at 104 in fig2 ( b )), with the output of the pixel swap being the new encrypted matrix b ′ ( indicated at 106 ). with reference to the pixel substitution map 104 , this means that the value in position ( 2 ) of the b array is swapped with the value in position ( 1 ) of the array , the value in position ( 3 ) is swapped with position ( 4 ), position ( 5 ) swapped with ( 6 ), position ( 9 ) swapped with ( 7 ), position ( 11 ) swapped with ( 8 ), position ( 14 ) swapped with ( 10 ), position ( 15 ) swapped with ( 12 ), and position ( 16 ) swapped with ( 13 ). the resultant 2d matrix 106 is equivalent to the encrypted matrix b ′ of fig2 . with reference to fig2 , the results of a sample cpa cryptanalysis attack are shown , according to equation ( 11 ) above . the original image is shown in fig2 ( a ), with the encrypted version shown in fig2 ( b ). the attacker &# 39 ; s neutral image is shown in fig2 ( c ), which is encrypted with the same encryption key to produce to produce the attacker &# 39 ; s version of b ′. ( the same encryption key is used in this sample to simulate a worst - case attack .) the attacker &# 39 ; s b ′{ circle around ( x )} map is shown in fig2 ( d ), which is used to decrypt fig2 ( b ) to produce fig2 ( e ), which in this case bears no resemblance to the original , unencrypted image . as can be seen from the figures , the use of the post - encryption step results in a resistance of the algorithm to worst - case attacks using cpa . with reference to fig2 , the decryption process for the method shown in fig2 is illustrated . essentially , the decryption process is the encryption process in reverse . knowing the encrypted image b ′, and the original key k “ securemeplease ”, the key k is hashed to produce h ( k ), converted to binary and reshaped . eqns . ( 3 ), (*), & amp ; ( 4 ) are performed to produce the binary random - like map ( output indicated at m ). taking the encrypted image b ′, the new pixel - substitution map k2 = h ( h ( k )) is applied to provide a binary stream , which is then xored with the binary random - like map and the output converted to decimal and reshaped to form the original , unencrypted image b . while the above describes a pixel substitution map for an image , it will be understood that the map may equally be applied to substitute elements in any 2d array . a new encryption algorithm for two - dimensional data such as images has been shown . the algorithm is initiated by a password supplied by the user . then an extension of the sha - 1 algorithm is provided to handle 2d data . an irreversible fast fourier transform ( irfft ) is applied to generate a more scattered data . it has been shown that the method of the invention outperforms that of ( pisarchik et al ., 2006 ) in many ways . a security analysis for the proposed system is also presented . a comparison to other current systems is also highlighted , which shows the superiority of the algorithm of the invention . finally , a useful application of the proposed cryptographic scheme in steganography has been described . the invention is not limited to the embodiments described herein but can be amended or modified without departing from the scope of the present invention . 1 dutch public transit card broken , [ online ]. available from : & lt ; http :// www . cs . vu . nl /˜ ast / ov - chip - card /& gt ;, accessed on feb . 3 , 2008 at 15 : 37 . 2 rfc3174 — us secure hash algorithm 1 ( sha1 ), [ online ]. available from : http :// www . faqs . org / rfcs / rfc3174 , accessed on 8 jul . 2008 at 19 : 06 . 3 weisstein , eric w . “ gray code ”. 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