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- 1. Maneckshaw. B, Krishna Kumar. V / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.811-815 Multiple FG Mapping Technique for Image Encryption Maneckshaw. B*, Krishna Kumar. V** *(Department of Mathematics, Karaikal Polytechnic College, PIPMATE, Karaikal, India) ** (Department of Computer Science Engineering, Karaikal Polytechnic College, PIPMATE, Karaikal, India)ABSTRACT The Fuzzy logic concept is implemented 𝜎: 𝑉 → 0,1 𝑎𝑛𝑑 𝜇: 𝑉 × 𝑉 → [0,1] such that all x,in several image processing operations such as y in V, 𝜇 𝑥, 𝑦 ≤ 𝜎 𝑥 ⋀𝜎 𝑦 , where ∧ refers toedge detection, segmentation, object recognition, minimum i𝑒. , 𝑎 ∧ 𝑏, 𝑚𝑒𝑎𝑛𝑠 min 𝑎, 𝑏 .etc. In this paper, a novel Image encryption Figure 2.1 shows an example of a fuzzy graphalgorithm based on multiple fuzzy graph (FG) where their vertices and edges are labeled by theirmapping technique is proposed. The Fuzzy membership grades.graphs are obtained from a matrix of size n andthen they are used to encrypt an image. Herefuzzy graphs with triangular and sigmoidmembership functions are discussed.Experimental results show that this proposedalgorithm is more efficient and robust.Keywords – Fuzzy graphs, fuzzy mappings,image encryption, image processing, . Fig. 2.1membership functions. A fuzzy graph can also be thought of having its fuzzy values on a continuous universe. Say, forI. INTRODUCTION instance, a fuzzy graph whose vertices are Relaxation of crisp boundaries of the represented by vectors and edges are representedclassical set redefines the inclusion and exclusion of by the set of vectors spanned by the linearmembers of the sets, this terminology, known as combination of the vectors represented on its pairFuzzy sets was studied by Zadeh. [1-2]. A. of incident vertices, then the edges have a set ofRosenfeld developed the theory of fuzzy graphs[3]. variable vectors traversing on them whoseIn this paper, we define fuzzy graphs based on [4] membership grades are derived from a desiredand relax the fuzziness of the edges of the graphs function, in such case the fuzzy graph will have afor the purpose of encryption. Various fuzzy related membership grades decided from a continuousalgorithms in the domain of image processing and universe represented by a membership function.pattern recognition have been discussed in [6-8]. Figure 2.2 shows a fuzzy graph whose membershipHere, we introduce some sketches for obtaining grades are decided by a membership function,fuzzy graphs such as fuzzy triangle, sigmoid fuzzy sigmoid function [5] sig(x: a, c), defined in (2.1)graphs from a square matrix of size n and discuss in and whose graph is given in Fig.2.3.detail how multiple numbers of fuzzy graphs are 1generated for a single given matrix and single given 𝑠𝑖𝑔 𝑥; 𝑎, 𝑐 = −𝑎 (𝑥 −𝑐) (2.1) 1+𝑒membership functions. We later employ this idea ofgenerating multiple fuzzy graphs [FG] as atechnique in image encryption. This paper isdivided into two parts; in the first part we define thefuzzy graph and generation of multiple FGs and thesecond part discusses image encryption techniquesin three phases-Pixel shuffling, FG mappings andImage encryption based on multiple FG mappingtechnique. We also propose some algorithms basedon the said techniques and have exhibited some 1 𝑤𝑒𝑟𝑒 𝑓 𝑥 = 𝑠𝑖𝑔 𝑥; 𝑎, 𝑐 =experimental results. 1 + 𝑒 −𝑎(𝑥−𝑐) Fig. 2.2II. FUZZY GRAPHS A Fuzzy Graph [1] 𝐺 = 𝑉, 𝜎, 𝜇 is anonempty set V together with a pair of functions 811 | P a g e
- 2. Maneckshaw. B, Krishna Kumar. V / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.811-815 Fig 3.1 Then by finding membership function, Fig. 2.3 triangle(1.9,,1,2,3) we obtain the grade 0.9 which is given to vertex corresponding to first row.III. OBTAINING A FUZZY GRAPH FROM A Similarly, the other grades are obtained as shown inGIVEN SQUARE MATRIX Fig 3.2.In this section we explore some ins and outs ofobtaining a fuzzy graph from a given matrix. Wecan do it in number of ways, here we give twodifferent ideas of obtaining them, the first one isabout obtaining a fuzzy cycle and the second isabout obtaining a fuzzy graph. Fig.3.23.1 Fuzzy cycle from Mat M.Let M be any square matrix of size n; let the 3.2 Fuzzy graph from Mat Mvertices of the fuzzy graph represent the rows of We can also obtain a desired graph from a matrixthe matrices. We draw an edge between every pair by considering the vertices as the entries of theof vertices if they represent adjacent rows of the matrix and connecting edges between themmatrix. By assuming that the n th row is adjacent to whenever they are adjacent. Thus we will get a gridthe 1st row, we will have an edge between the n th like graph and for this graph if we applyvertex and the 1st vertex. Thus we obtain a cycle of membership grades we will obtain a fuzzy graph.n vertices. Now, by defining the MFs on the We can also consider omitting some edges withoutvertices/edges we get a fuzzy cycle. For a 3x3 connecting some adjacent vertices for a desiredmatrix we will obtain a fuzzy triangle as described purpose to obtain distinct graphs. In the following,in 3.1.1 we describe this idea of obtain a fuzzy graph with the help of sigmoid function defined earlier and we3.1.1 Fuzzy triangle from a 3x3 matrix call this graph as sigmoid fuzzy graph.Let M=[1 2 3;4 5 6;7 8 9], and the membershipfunction be the triangle(x;a,b,c) [5] as defined in 3.2.1 Sigmoid Fuzzy Graph(3.2) whose graph is shown in Fig 3.1. 0, 𝑥 ≤ 𝑎. Let A=(aij), be a 3x3 matrix, consider drawing a 𝑥−𝑎 graph as shown in Fig.3.3.We name the vertex a 22 , 𝑎≤ 𝑥≤ 𝑏 as the centre `c’ . Keeping c as centre we obtain the 𝑡𝑟𝑖𝑎𝑛𝑔𝑙𝑒 𝑥, 𝑎, 𝑏, 𝑐 = 𝑏−𝑎𝑐−𝑥 (3.2) , 𝑏≤ 𝑥≤ 𝑐 fuzzy values for each vertex of the graph by 𝑏−𝑎 applying sigmoid function sig(x; a, c) as defined in 0, 𝑥 ≥ 𝑐 (3.1). The value a can be randomly chosen from any other vertices, in this case we choose a =1. We notice that the fuzzy value of c is 0.5 and the corresponding values prior and later to centre c are lesser and greater than 0.5 respectively. 812 | P a g e
- 3. Maneckshaw. B, Krishna Kumar. V / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.811-815 𝑋 ′ = 𝑇𝑋, 𝑤𝑒𝑟𝑒 𝑋 ∈ 𝐵𝑗 𝑎𝑛𝑑 𝑋 ′ ∈ 𝐶𝑗 (4.1) The transformation is so chosen in such a way that there exists greater dissimilarity between the blocks Bj and Cj which is determined by using the correlation function ρ as the dissimilarity measure for a given threshold τ. The following algorithm describes the process involved pixel shuffling. Fig. 3.3 4.1.2 Pixel shuffling Algorithm Begin PixelNow, if we change the ‘a’ value to be the entry at Shufflinga23, i.e., a=6, then we will have the fuzzy graph as Step 1: Divide the image into blocks Bjshown in Fig. 3.4. In this case we have obtained a Step 2: Set up a transformation T=[Tm], where A isfuzzy graph with lesser vertices as the grades at the mx1 matrix and Tm is a 𝑗 × 𝑗 matrix where 𝑗 = 𝐶𝑗remaining vertices vanish for a precision of 3 Step 3: Randomly chose a Tm from Tdecimals. Here we can notice that still c’s value is Step 4: For each block Bj apply the transformation0.5. This means that the factor ‘a’ controls the given below to obtain a new position of the vectorsslope at the cross over vertex ‘c’, where the cross 𝑋′ = 𝑇 𝑚 𝑋over vertex is the one which has a fuzzy value 0. 5 Step 5: Calculate the correlation ρ using theand the corresponding values prior and later to correlation function given in (4.2) for the blocks Bjcentre c are lesser and greater than 0.5 respectively. and Cj where Bj is the original block and Cj is the newly obtained block 𝑖,𝑗 𝐵 𝑖𝑗 −𝐵 𝑖𝑗 𝐶 𝑖𝑗 −𝐶 𝑖𝑗 𝜌= 2 2 (4.2) 𝑖𝑗 𝐵 𝑖𝑗 −𝐵 𝑖𝑗 𝐶 𝑖𝑗 −𝐶 𝑖𝑗 Step 6: If 𝜌 < 𝜏 then place the pixels of Bj in Cj Otherwise repeat the process by randomly choosing 𝑇 𝑘 𝑓𝑜𝑟 𝑘 ≠ 𝑚. In case all such choices are exhausted retain the lastly chosen transformation End Pixel Shuffling Fig. 3.4From this approach we understand that for a single 4.2 FG Mapping onto an imagematrix M of size n, and for a single membership In this part, we describe the process of mapping thefunction, we will get a multiple number of fuzzy fuzzy graph onto an image. We have seen in thegraphs as we change our ‘a’ factor. Again when we previous section how to generate a multiplechange the centre c, we will still get some more numbers of fuzzy graphs such as fuzzy triangle,new fuzzy graphs depending upon the matrix fuzzy sigmoid graphs etc., from a given matrix ofchoice. Hence, we are comprehended with the size n. We can utilize these multiple FGs toscale of generating a multiple number of fuzzy construct a fuzzy graph covering over the imagegraphs. We utilize this technique to generate that is considered for encryption. By constructingmultiple numbers of fuzzy graphs which will aid us FGs randomly over the image so that the union ofto have a fuzzy graph covering over an image to such constructed FGs covers the entire image to aencrypt it in an efficient manner. Section that larger extend. The number of fuzzy graphsfollows is about implementing such a technique. required for construction is proportional to the size of the target image. The security of the encryptionIV. IMAGE ENCRYPTION increases as the number of fuzzy graphs mapped This section is devoted to image onto the image increases. This patents that it isencryption using the technique of multiple FG highly infeasible to decrypt the encrypted imagemapping. We undergo the process of image without knowing the patterns and the orders of theencryption by dividing it into three stages – fuzzy graphs so mapped onto it. The mapping isshuffling, mapping and encryption in order to have done by undergoing the following steps.a greater security levels. 4.2.1 FG Mapping Algorithm4.1 Pixel Shuflling BeginForemostly, we consider grouping the images Step 1: Choose a set of matrices M1, M2,…, Mkpixels into image blocks Bj of size n-each such Step 2: Choose a set of Membership functions MF1,that B=UBj, where B is the image. For all pixels in MF2, …, MFs. (s need not be equal to k as we know 𝐵𝑗 , a unique block Cj is found through a that for one MF we will get a multiple FGs, sotransformation represented in (4.1). s<=k). 813 | P a g e
- 4. Maneckshaw. B, Krishna Kumar. V / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.811-815Step 3: Apply the MFs to obtain the Fuzzy Graphs 4.4 Experimental Results:FG1, FG2 , …, FGt. (t will be greater than t for a The proposed algorithm is implemented insimilar reason ,i.e., t>=k>=s). MATLAB software and tested with various images.Step 4: Divide the shuffled image into blocks B1, The results are presented in this section. TheB2 , … Bk of sizes n1, n2 ,… nk where each n j will chosen matrix is [2 0 1; 4 9 3; 1 10 7] and the blockrepresent the number of vertices of FG1, FG2 , …, size of the image is 3x3. We select the triangularFGt and and the sigmoid fuzzy graphs mentioned in the section 3.1.1 and in 3.2.1 and apply them 𝑘 𝑡 alternatively onto the image. The range for the 𝑁= 𝑖=1 𝑛𝑖 = 𝑗 =1 𝑉(𝐹𝐺𝑗 (4.3) membership function is 0 to 10 with increment ofStep 5: Plot the fuzzy graphs FG1, FG2 , …, FGt 0.1 (i.e) a total of 100 fuzzy floating values areonto the image blocks B1, B2 , … Bk . obtained and a non-zero floating value is selectedEnd for XOR operation. Fig 4.1 shows the lena image and its multiple FG4.3 Image Encryption based on Multiple FG mapped encrypted image and Fig 4.2 shows theirMapping technique. corresponding histograms.The membership functions of the fuzzy graphs playan important role on the outcome of the imageencryption process. The fuzzy value lies in theinterval [0,1], we can always have a one-onemapping for any interval [a,b] of the real line, sowe choose an interval [a,b] of length b-a, first. Forevery 𝑥𝜖[𝑎, 𝑏], we obtain a set of fuzzy floatingpoint (FFP) values for each fuzzy graphs FGjcontained in the block Bj. For a set of p- pixelsbelonging to block Bj we will choose that numberof pixels using the FFP values. The selected FFPvalues are then XOR’ ed with each pixel of theimage block Bj to obtain the encrypted imageblock Ej as represented by the following equation. 𝐸𝑗 = 𝑚𝑜𝑑(𝐹𝐹𝑃𝑗 ∗ 103 , 256)⨁𝑃𝑗 (4.4)The following algorithm represents the processdescribed above. Experimental results show that this proposed algorithm has large key space analysis and the4.3.1 Multiple FG mapping Technique Image histogram of the encrypted image is uniformlyProcessing Algorithm distributed.BeginStep 1: Shuffle the image to be encrypted as givenshuffling algorithm of sectionStep 2: Map the fuzzy graphs as given in FGmapping algorithmStep 3: Select a interval range [a,b]Step 4: For each 𝑥𝜖[𝑎, 𝑏], obtain a set of fuzzyfloating point (FFP) values for each fuzzy graphsFGj contained in the block BStep 5: For a set of p- pixels belonging to block Bjchoose that number of pixels by using the FFPvalues.Step 6: Apply XOR with each pixel of the imageblock Bj to obtain the encrypted image block Ej asrepresented by the following equation (4.4). 𝐸𝑗 = 𝑚𝑜𝑑(𝐹𝐹𝑃𝑗 ∗ 103 , 256)⨁𝑃𝑗Step 7 : obtain the encrypted image E by taking theunion of Ej’s Fig4.1 𝐸= 𝐸𝑗 𝑗End 814 | P a g e
- 5. Maneckshaw. B, Krishna Kumar. V / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 3, Issue 1, January -February 2013, pp.811-815 [5] J.S.R. Jang,C.-T. Sun and E. Mizutani, Neuro-Fuzzy and Soft Computing, Prentice- Hall of India, 2002. [6] Pal, S.K, “Fuzzy sets in image processing and recognition”, IEEE International Conference on Fuzzy Systems, pp. 119 – 126, Mar 1992. [7] Chi, Z., Yan, H., Pahm, T., Fuzzy Algorithms: With Applications to Image Processing and Pattern Recognition((Advances in Fuzzy Systems: Application and Theory), World Scientific Pub Co Inc, 1996. [8] Tizhoosh, Hamid R. Fuzzy Image Processing: Introduction in Theory and Applications, Springer-Verlag, 1997. Fig. 4.2Thus for the proper decryption of the encryptedimage, we need to know the initial matrix, thechosen fuzzy graphs and their membershipfunctions, and the order in which they are appliedto the image. The chosen platform was a PC withan Intel i3 processor with 2 GB RAM runningMicrosoft Windows XP Professional with SP2.V. CONCLUSION The proposed method have a larger keyspace, i.e., the randomly obtained pixel valueswhich are combined with the original pixels of theimage are influenced by the factors such as thechosen matrix, the size of the matrix, the types ofgraphs obtained, the types of membership functionschosen, the patterns of fuzzy graphs obtained, theblocks they are plotted. Hence this techniqueprovides a multiple levels of securities for theimage encryption.REFERENCES:[1] L.A. Zadeh. Fuzzy sets Inform. Control 8(1965)[2] Zadeh L.A. Similarity relations and fuzzy ordering Information Sciences 971, 3(2):177- 200.[3] A. Rosenfeld, Fuzzy Graphs: In fuzzy sets and their applications to cognitive and Decision processes. Zadeh. L.A, Fu.K.S.Shimura M. Eds. Academic Press. New York 1975 (77-93)[4] Mordeson. J. Nair. P.S, Fuzzy Graphs and Fuzzy Hyper Graphs, Physica-Verlag,(2000) 815 | P a g e

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