Robust Digital Image Cryptosystem Based on Nonlinear Dynamics of Compound Sine and Cosine Chaotic Maps for Private Data Protection

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1 Robust Digital Image Cryptosystem Based o oliear Dyamics of Compoud Sie ad Cosie Chaotic Maps for Private Data Protectio S. Maksuapa, T. Veerawadtaapog, W. Sa-Um Itelliget Electroics Systems Research Laboratory Faculty of Egieerig, Thai-ichi Istitute of Techology Patthaakar Rd., Sualaug, Bagkok, THAILAD, 0250, Fax: , Tel: saru.maksuapa@gmail.com, taachad.pog@gmail.com, ad wimol@ti.ac.th Abstract this paper presets a digital image cryptosystem based o oliear dyamics of a compoud sie ad cosie chaotic map. The compoud sie ad cosie chaotic map is proposed for high-degree of chaos over most regios of parameter spaces i order to icrease high-etropy radom-bit sources. Image diffusio is performed through pixel shufflig ad bit-plae separatios prior to XOR operatios i order to achieve a fast ecryptio process. Security key coversios from ASCII code to floatig umber for use as iitial coditios ad cotrol parameters are also preseted i order to ehace key-space ad key-sesitivity performaces. Experimets have bee performed i MATLAB usig stadard colour images. oliear dyamics of the chaotic maps were iitially ivestigated i terms of Cobweb map, chaotic attractor, Lyapuov expoet spectrum, bifurcatio diagram, ad 2-dimesioal parameter spaces. Ecryptio qualitative performaces are evaluated through pixel desity histograms, 2-dimesioal power spectral desity, key space aalysis, key sesitivity, vertical, horizotal, ad diagoal correlatio plots. Ecryptio quatitative performaces are evaluated through correlatio coefficiets, PCR ad UACI. Demostratios of wrog-key decrypted image are also icluded. Keywords Digital Image Processig, Cryptosystem, Chaotic Map, Ecryptio, Decryptio, oliear Dyamics I. ITRODUCTIO Recet advaces i commuicatios have led to great demad for secured image trasmissios for a variety of applicatios such as medical, idustrial ad military imagig systems. The secured image trasmissios greatly require reliable, fast ad robust security systems, ad ca be achieved through cryptography, which is a techique of iformatio privacy protectio uder hostile coditios []. Image cryptography may be classified ito two categories, i.e. () pixel value substitutio which focuses o the chage i pixel values so that origial pixel iformatio caot be read, ad (2) pixel locatio scramblig which focuses o the chage i pixel positio. Covetioal cryptography such as Data Ecryptio Stadard (DES), Iteratioal Data Ecryptio Algorithm (IDEA), Advaced Ecryptio Stadard (AES), ad RSA algorithm may ot be applicable i real-time image ecryptio due to large computatioal time ad high computig power, especially for the images with large data capacity ad high correlatio amog pixels [2]. Recetly, the utilizatio of chaotic systems has extesively bee suggested as oe of a potetial alterative cryptography i secured image trasmissios. As compared to those of covetioal ecryptio algorithms, chaos-based ecryptios are sesitive to iitial coditios ad parameters whilst covetioal algorithms are sesitive to desigated keys. Furthermore, chaos-based ecryptios spread the iitial regio over the etire phase space, but cryptographic algorithms shuffle ad diffuse data by rouds of ecryptio [3]. Therefore, the security of chaos-based ecryptios is defied o real umbers through mathematical models of oliear dyamics while covetioal ecryptio operatios are defied o fiite sets. Such chaos-based ecryptio aspects cosequetly offer high flexibility i ecryptio desig processes ad acceptable privacy due to vast umbers of chaotic system variats ad umerous possible ecryptio keys. Chaos-based ecryptio algorithms are performed i two stages, i.e. the cofusio stage that permutes the image pixels ad the diffusio stage that spreads out pixels over the etire space. Most existig chaos-based ecryptios based o such two-stage operatios employ both iitial coditios ad cotrol parameters of -D, 2-D, ad 3-D chaotic maps such as Baker map [4,5], Arold cat map [6,7], ad Stadard map [8, 9] for secret key geeratios. Furthermore, the combiatios of two or three differet maps have bee suggested [0, ] i order to achieve higher security levels. Despite the fact that such maps offer satisfactory security levels, iteratios of maps require specific coditios of chaotic behaviors through a arrow regio of parameters ad iitial coditios. Cosequetly, the use of iteratio maps has become typical for most of proposed ciphers ad complicated techiques i pixel cofusio ad diffusio are ultimately required. This paper therefore presets a alterative chaos-based digital image cryptosystem with three mai aspects. First, the compoud sie ad cosie chaotic maps, which potetially offers high-degree of chaos over most regios of parameter spaces, is proposed through oliear dyamics aalyses ad is cosequetly exploited as high-etropy radom-bit sources for ecryptio. Secod, image cofusio ad diffusio processes

2 [p0,p,p2,,p7] [p] [p0] TABLE. SUMMARY OF 6-CHARACTERS IPUT ASCII CODES FOR SETTIG IITIAL CODITIOS AD COTROL PARAMETERS Image to be ecrypted Dec2Bi Separate ASCII code Xm for Settig iitial coditios ASCII code Ym for settig cotrol parameters Grayscale Image 52x52 Pixels 8-Bit Biary umber per Pixel 3 Sets of RGB Image -Bit Biary umber per Pixel 24 Sets of Biary umber X : A A 4A 7A 0A 3A 6 Y : A A 3A 5A 7A 9A XOR X2 : A 2A 5A 8A A 4A Y2 : A 2A 4A 6A 8A 0A 2 6-Characters ASCII Code Key Geeratio System Iitial coditios Parameters a a2 a8 b b2 b8 Compoud Sie ad Cosie Maps X3 : A 3A 6A 9A 2A 5A 2 Y3 : A 3A 5A 7A 9A A 3 X4 : A 4A 7A 0A 3A 6A 3 Y4 : A 4A 6A 8A 0A 2A 4 X5 : A 5A 8A A 4A A 4 Y5 : A5A 7A 9A A 3A 5 [q0,q,q2,,q7] [q] [q0] [q7] X6 : A 6A 9A 2A 5A 2A 5 Y6 : A 6A 8A 0A 2A 4A 6 X7 : A 7A 0A 3A 6A 3A 6 Y7 : A 7A 9A A 3A 5A Ecrypted Image Bi2Dec 8-Bit Biary umber per Pixel Combie -Bit Biary umber per Pixel 3 Sets of RGB Image 24 Sets of Biary umber Figure. Proposed ecryptio algorithms usig the compoud sie ad cosie chaotic maps. are performed through ucomplicated pixel shufflig ad bitplae separatios prior to XOR operatios i order to achieve a fast ecryptio process. Last, security key coversios from ASCII code to floatig umber for use as iitial coditios ad cotrol parameters are also preseted i order to ehace keyspace ad key-sesitivity performaces. II. PROPOSED ECRYPTIO ALGORITHMS A. Compoud Sie ad Cosie Chaotic Maps A category of trigoometric fuctios, ivolvig sie ad cosie maps, have potetially offered rich dyamic behaviours as described i simple forms as [2] x +=si(ax ) ad x +=cos(bx ) where the costats a ad b ca be cosidered as parameters associated with the frequecies of sie ad cosie fuctios, respectively. Although such sie ad cosie maps offers relatively high complexity i terms of oliear dyamics, the chaotic regios i the bifurcatio diagram is still isufficiet due to periodic characteristics. This paper therefore cosiders the ehacemet of sie ad cosie maps through the combiatio betwee sie ad cosie maps, i.e. x cos( ax ) si( bx ) As will be see later, such a compoud sie ad cosie map i () offers high-degree of chaos over most regios of parameter spaces. As a ature of chaotic maps, the iitial coditios ad cotrol parameters ca be used as iteral security keys that etirely set the ecryptio characteristics. The proposed cryptography techique attempts to achieve simple-buthighly-secured image ecryptio ad decryptio algorithms i a category of chaos-based cryptosystems. Fig. shows the proposed ecryptio ad detectio algorithms usig compoud sie ad cosie maps. Three major processes are summarized as follows; X8 : A 8A A 4A A 4A 7 Y8 : A 8A 0A 2A 4A 6A 2 First, the origial image is prepared for diffusio. The origial color image with M image size is iitially coverted ito three sets of sub-images with RGB compoets cotaiig pixels i grey scale levels. Each sub-image will subsequetly be coverted ito biary matrix i which each pixel is represeted by 8-bit biary umbers. For example, the pixel p(,) cotais the biary umber p 0-p 7. Each pixel will the be separated ito eight plaes correspodig to biary bits p 0 to p 7. As a result, there are 24 sets of bit plae images represeted i matrix forms with a sigle biary umber i each pixel, which is ready for further Excusive-OR (XOR) operatios. Secod, the iput security keys from users which is represeted i ASCII code with arbitrary 6 alphaumeric characters defied as A=A A 2A 3,...,A 6 will form two mai sets of ASCII codes, i.e X m ad Y m for settig the iitial coditios ad the cotrol parameters, respectively, where m =, 2, 3,...,8 as summarized i Table. Such two sets X m ad Y m will be coverted ito 48-bit biary represetatios deoted by B X to B X48 ad B Y to B Y48, respectively. The real umbers R Xm ad R Ym are subsequetly formed through the biary represetatio as follows; R Xm R Ym (B X 2 B 2... B 2 ) / 2 X X48 (B Y 2 B 2... B 2 ) / 2 Y Y48 As a result, the iitial coditios ad the cotrol parameters ca be achieved by a b m m ( RXm RYm) mod ( R ) mod Ym RYm It is see that the values of a m ad b m are i the regio of (0,) ad are ready for use as iteral security keys i the ecryptio algorithms. The desig algorithm realizes eight chaotic maps based o () as follows;

3 Parameter b = (0,0) Parameter a = (0,0) (a) Parameter b = (0,) Parameter a = (0,) (b) Figure 2. Plots of 2-D Lyapuov Expoet bifurcatio structure betwee parameters a ad b over the parameter space; (a) (0,0) ad (b) the zoomed i regio (0,). Lyapuov Expoet (LE) Chaotic Behaviors : LE>0 o-chaotic Behaviors : LE<0 umber of the iitial coditio a Values of the iitial coditio a Figure 4. Histrogram of umbers of iitial coditios a. =000 iteratios are adjusted to the biary umber through the zero thresholds for the subsequet XOR operatios. Last, the XOR operatios diffuse the geerated chaotic bit ad the 24 biary images i parallel process. The XOR operatio yields bit if the two iput bits are differet, but yields bits 0 if the two iputs are similar. The results obtaied from such XOR operatios are 24 matrices with sigle biary umber i each pixel. All the 24 matrices are combied ito three RGB matrices of a sigle 8-bit matrix i which each pixel is represeted by [b 0-b 7]. As a result, the ecrypted image ca be achieved. The decryptio process also follows the ecryptio process i a backward algorithms as log as the security keys are kow. Iterated Values X III. EXPERIMETAL RESULTS Experimetal results have bee performed i a computer aid desig tool MATLAB. oliear dyamics of a compoud sie ad cosie map was iitially simulated ad ecryptio ad decryptio security performaces were subsequetly evaluated. B. oliear Dyamics of Compoud Sie ad Cosie Map Sice chaotic behaviors of the compoud sie ad cosie maps determie overall performace of the cryptosystem, Lyapuov expoet (LE) has bee realized as a quatitative measure of chaoticity. The LE is defied as a quatity that characterizes the rate of separatio of ifiitesimally close trajectories ad is give by [3] Parameter a Figure 3. Plots of the Lyapuov expoet spectrum ad the bifurcatio diagram of parameters a over the parameter space (0,0) whe the parameter b is fixed at 0.5. xm, cos( bm0πx ) si( bm 0πx ) It is see i (6) that the costat 0π has bee icluded i order to sustai the parameters a ad b described i () i the regio of (0, 0) which is sufficiet to acquire chaos. The values of m are circularly shifted with to 8, i.e. if the operatio roud reaches m+=8 the the ext value is. As a result, a total umber of 6 keys are employed as security keys i the ecryptio process. Such keys are used to geerate chaotic sigals from the compoud sie ad cosie chaotic maps. The output sigals LE lim log 2 dx dx where is the umber of iteratios. Typically, the positive LE idicates chaotic behaviors. The larger value of LE results i higher degree of chaos. Fig.2 shows the plots of 2-imesioal Lyapuov Expoet bifurcatio structure betwee parameters a ad b over the parameter space (0, 0) ad the zoomed i regio (0, ) where the chaotic regio is represeted by the dark blue color while the o-chaotic regio is represeted i the white regio. It is show i Fig.2 that the chaotic behaviors of the compoud sie ad cosie map occupy most of parameter spaces, leadig to a very robust chaos for secret key geeratios. oetheless, the zoomed i regio at small values of parameters a ad b cotai some o-chaotic regios, which represet quasi-chaotic or periodic behaviors. The proposed

4 Wrog Key Decryptio Correct Key Decryptio Ecryptio Origial Image Images Histograms of RED Compoet of Images Histograms of GREE Compoet of Images Histograms of BLUE Compoet of Images 2D Power Spectrum of Images Figure 5. Histograms ad 2D power spectrums of origial image, ecrypted image, decrypted image, ad decrypted image with wrog keys. Pixel values o (x+,y) Pixel values o (x,y+) Pixel values o (x+,y+) Pixel values o (x+,y) Pixel values o (x,y+) Pixel values o (x+,y+) Pixel values o (x,y) Pixel values o (x,y) Pixel values o (x,y) Figure 6. Image correlatio tests i origial ad ecrypted images, icludig horizotally, vertically, ad diagoally adjacet pixels. key geeratio system has bee desiged to potetially geerates secret keys potetially sice the ochaotic sigals will ultimately be achcived. Fig.3 shows LE spectrum ad the bifurcatio diagram of parameters a over the parameter space (0, 0) whe the parameter b is fixed at 0.5. It is apparet i Fig.4 that the LE spectrum is greater tha zero ad growig to ifiity. I additio, the bifurcatio diagram shows dese area of the maximum values of X over the etire rage. As for a particular example, Fig.5 shows the histograms of the umbers of the secret key a for,000 iteratios. It ca be see from Fig.5 that the oliear dyamics of the compoud sie ad cosie maps provide the radom secret keys that distribute over the regio (0, ) radomly. Such characteristics have also foud i other secret keys. The simulatios have bee esured that the proposed compoud the oliear dyamics of the compoud sie ad cosie maps ad the key geeratio

5 systems ca potetially provide truly radom values for diffusio process i the proposed cryptosystem. C. Key Space Aalysis The ecryptio ad decryptio realizes the 6-character ASCII code ABCDEFG as a iput key ad the wrog key chages the last character to 5. The resultig eight iitial coditios ad eight parameters, i.e. a total of 6 keys, are represeted by 8-digit floatig-poit umbers. Cosiderig each key i the form S 2 E where S is a sigificad ad E is a expoet, the keys that represeted by 8 digits of a floatigpoit umber (~ ) results i 28 ucertai digits, which is greater tha the miimum requiremet of the 56-bit data (~ ) ecryptio stadard (DES) algorithm [4]. D. Histograms ad 2D Power Spectral Aalysis The image histogram is a graph that illustrates the umber of pixels i a image at differet itesity values. I particular, the histogram of a color image ca be separated ito three subimages with Red (R), Gree (G), ad Blue (B) compoets. Each sub-image has 256 differet grey itesity levels, graphically displayig 256 umbers with distributio of pixels amogst these grayscale values. I additio, the 2D power spectrum that shows the power of image itesity ca be obtaied through a Discrete Fourier Trasform (DFT) aalysis ad the algorithm is give by [5] F( u, v) M x0 y0 f ( x, y)exp( j(2 / M ) ux)exp( j(2 / ) vy) where x ad y are a coordiates pair of a image, M ad are the size of image, f(x, y) is the image value at the pixel (x, y). Fig.5 shows the histograms of three R, G, B compoets ad 2D power spectrums of origial image, ecrypted image, decrypted image, ad decrypted image with wrog keys. As for a particular demostratio, the origial image is Lea image with image size. It ca be see from Fig.5 that the itesities of all origial images i the histogram are cotributed with differet values i a particular shape ad the power spectrum is ot flat havig a peak of itesity i the middle. The ecrypted image has a flat histogram ad power spectrum, idicatig that the itesity values are equally cotributed over all the itesity rage ad the origial images are completely diffused ad ivisible. The decrypted images with right keys provide similar characteristics of the origial images while the decrypted images with wrog keys are still diffused ad the origial images caot be see. These results qualitatively guaratee that the image is secured. E. Correlatio Coefficiet Aalysis I order to quatify the ecryptio performace ad key sesitivity aalysis, correlatio betwee image pairs, which is a measure of relatioships betwee two pixels itesities of two images, of the three realized images have bee aalyzed. The covariace C v ad the correlatio coefficiet γ xy ca be obtaied as follows [6]; C ( x, y) v i ( xi E( x))( yi E( y)) xy cov( x, y) D( x) D( y) where the fuctios E(x) ad D(x) are expressed as E( x) i x i ad 2 D( x) ( E( )) () x i i x ad the variables x ad y are grey-scale values of pixels i correspodig pixels i differet images or two adjacet pixels i the same image. Typically, the value of γ xy is i the regio [-, ]. I other words, the values of γ xy i the regio (-,0) ad (0,) respectively idicate positive ad egative relatioships, while the larger umber close to or - have stroger relatioships. Usig a radom selectio of 2,048 pairs of pixels, Fig.6 shows image correlatio tests i origial ad ecrypted images, icludig horizotally, vertically, ad diagoally adjacet pixels. It ca qualitatively be cosidered from Figs.6 that the adjacet pixels of all ecrypted images are highly ucorrelated as depicted by scatters plots of correlatios. For the quatitative measures, the correlatios betwee pairs of origial images ad correspodig ecrypted images through the computatio of correlatio coefficiet betwee RGB compoets of the origial images ad correspodig ecrypted images have bee aalyzed. Table 2 summarizes correlatio coefficiets of 2,048 pixels of each image pair. It ca be see i Table 3 that the correlatio coefficiets are very small closig to zero, idicatig that each pair of images are completely idepedet of each other. As for ivestigatios of other images with differet characteristics, comparisos of correlatio coefficiets of four stadard images i MATLAB are also studied. Table 3 summarizes correlatio coefficiets of 2,048 pixels of each pair of images Apparetly, the correlatio coefficiets are also very small. These results quatitatively guaratee that the image is secured. F. Origial Image Sesitivity Aalysis Oe mior chage i the plai image causes sigificat chages i the ecrypted image the such differetial aalysis may become iefficiet, ad therefore much differece betwee ecrypted forms is expected i order to maitai high security level. PCR (et Pixel Chage Rate) ad UACI (Uified Average Chagig Itesity) are two most commo measures. PCR cocetrates o the absolute umber of pixels which chages value i differetial attacks while the UACI focuses o the averaged differece betwee two paired ecrypted images [7]. For the two ecrypted images i which the correspodig origial images have oly oe pixel differece are deoted by C ad C 2. Label the greyscale values of the pixels at pixel (i,j) i C ad C 2 by C (i,j) ad C 2 (i,j), respectively. Defie a bipolar array D, with the same size as images C ad C 2. Cosequetly, D(i,j) is determied by C (i,j) ad C 2 (i,j), if C (i,j) =C 2 (i,j) the D(i,j)=, otherwise, D(i,j)=0. The PCR [2] is defied as

6 TABLE 2: COMPARISOS OF CORRELATIO COEFFICIETS OF LEA IMAGE AT DIFFERET SIZES. Image Sizes C RR C RG C RB C GR C GG C GB C BR C BG C BB TABLE 3: COMPARISOS OF CORRELATIO COEFFICIETS OF DIFFERET IMAGE WITH IMAGE SIZE. Images C RR C RG C RB C GR C GG C GB C BR C BG C BB Brai Madril Peppers Jet Plae TABLE 4: SUMMARY OF PCR AD UACI OF DIFFERET IMAGE WITH IMAGE SIZE. Images PCR R PCR G PCR B UACI R UACI G UACI B Lea Brai Madril Peppers Jet Plae D( i, j) i, j PCR 00% (2) T (, ) D i j i, j UACI 00% (3) T where T deotes the total umber pixels i the ecrypted image, F deotes the largest supported pixel value compatible with the cipher image format, ad. deotes the absolute value fuctio. Table 4 summarizes the values of PCR ad UACI for differet image with the sizes of It ca be see that the PCR are relatively close to 00% ad the UACI are also i the acceptable regio of approximately 33%. COCLUSIO A robust digital image cryptosystem based o oliear dyamics of a compoud sie ad cosie chaotic map has bee preseted. The compoud sie ad cosie chaotic map has bee proposed for high-degree of chaos over most regios of parameter spaces i order to icrease high-etropy radombit sources. Image diffusio has bee performed through pixel shufflig ad bit-plae separatios prior to XOR operatios i order to achieve a fast ecryptio process. Security key coversios from ASCII code to floatig umber for use as iitial coditios ad cotrol parameters were also preseted to ehace key-space ad key-sesitivity performaces. oliear dyamics of the chaotic maps have bee ivestigated i terms of chaotic attractor, Lyapuov expoet spectrum, bifurcatio diagram, ad 2-dimesioal parameter spaces. Ecryptio qualitative performaces were evaluated through pixel desity histograms, 2-dimesioal power spectral desity, key space aalysis, key sesitivity, vertical, horizotal, ad diagoal correlatio plots. Ecryptio quatitative performaces were evaluated through correlatio coefficiets, PCR ad UACI. Demostratios of wrog-key decrypted image are also icluded. The proposed cryptosystem offers a potetial alterative to private data protectio systems. REFERECES [] M. Philip, A Ehaced Chaotic Image Ecryptio Iteratioal Joural of Computer Sciece, Vol., o. 5, 20. [2] G.H. Karimia, B. Rashidi, ad A.farmai, A High Speed ad Low Power Image Ecryptio with 28- bit AES Algorithm, Iteratioal Joural of Computer ad Electrical Egieerig, Vol. 4, o. 3, 202. [3] G. Che, Y. Mao, C.K. Chui, A symmetric image ecryptio scheme based o 3D chaotic cat maps, Chaos, Solitos ad Fractals, Vol. 2, pp , [4] X. Tog, M. Cui, Image ecryptio scheme based o 3D baker with dyamical compoud chaotic sequece cipher geerator, Sigal Processig, Vol. 89, pp , [5] J.W. Yoo, H. Kim, A image ecryptio scheme with a pseudoradom permutatio based o chaotic maps, Commu oliear Sci umber Simulatio, Vol. 5, pp , 200. [6] X. Ma, C. Fu, W. Lei, S. Li, A ovel Chaos-based Image Ecryptio Scheme with a Improved Permutatio Process, Iteratioal Joural of Advacemets i Computig Techology, Vol. 3, o. 5, 20. [7] K. Wag, W. Pei, L. Zou, A. Sog, Z. He, O the security of 3D Cat map based symmetric image ecryptio scheme, Physics Letters A, Vol. 343, pp , [8] K. Wog, B. Kwok, ad W. Law, A Fast Image Ecryptio Scheme based o Chaotic Stadard Map, Physics Letters A, Vol. 372, pp , [9] S. Lia, J. Su, Z. Wag, A block cipher based o a suitable use of the chaotic stadard map, Chaos, Solitos ad Fractals, Vol. 26, pp. 7 29, [0] K. Gupta, S. Silakari, ew Approach for Fast Color Image Ecryptio Usig Chaotic Map, Jour. of Iformatio Security, pp , 20 [] F. Huag, Y. Feg, Security aalysis of image ecryptio based o two-dimesioal chaotic maps ad improved algorithm, Frot. Electr. Electro. Eg. Chia, Vol. 4, o., pp. 5-9, [2] G.Lee ad. H. Farhat, Parametrically Coupled Sie Map etworks,electrical Egieerig Departmet, Uiversity of Pesylvaia, Philadelphia, PA, USA,5 December 2000 [3] F.Su, S. Liu, Z. Li, Z. Lu, A ovel image ecryptio scheme based o spatial chaos map,college of Cotrol Sciece ad Egieerig, Shadog Uiversity, Jia, PR Chia, pp , 2008 [4] Q. Gog-bi, J. Qig-feg,Q. Shui-sheg, A ew image ecryptio scheme based o DES algorithm ad Chua's circuit, Imagig Systems ad Techiques, pp , [5] Z. Peg, T.B. Kirk, Two-dimesioal fast Fourier trasform ad power spectrum for wear particle aalysis, Tribology Iteratioal, Vol. 30, Issue. 8, pp , 997. [6] A. Yahya ad A. Abdalla, A Shuffle Image-Ecryptio Algorithm, J. Comput. Sci,pp [7] Y.Wu, Joseph P. ooa, S.Agaia, PCR ad UACI Radomess Tests for Image Ecryptio, Departmet of Electrical ad Computer Egieerig Tufts UiversityMedford, MA,USA