Research on the Evaluation of Information Security Management under Intuitionisitc Fuzzy Environment

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1 Iteratoal Joural of Securty ad Its Applcatos, pp Research o the Evaluato of Iformato Securty Maagemet uder Itutostc Fuzzy Evromet LI Feg-Qua College of techology, X'a Iteratoal Uversty, , X'a Shaax Cha lfegqua@xau.edu.c Abstract Wth the rapd developmet of computer techology ad formato techology, formato has bee a ew asset of the eterprse ad played more ad more mportat role. How to protect formato securty s the problem that all compaes eed to solve together. I ths paper, we propose a ovel method to evaluate the eterprse s formato securty maagemet uder tutostc fuzzy evromet. The tutostc fuzzy set whch cosders ot oly membershp ad o-membershp, but also hestacy ca express the decso maker s prefereces more precse. The exteded TOPSIS approach wth correlato coeffcet stead of dstace measure s troduced the procedure of decso makg. Fally, the applcato ad comparso aalyss are demostrated to verfy valdty ad relablty of the method. Keywords: formato securty maagemet, Itutostc fuzzy set, TOPSIS, correlato coeffcet 1. Itroducto Wth the growg spread of commucato meda ad the expoetal crease of electroc storage ad trasmsso of formato, the eed for formato securty has rse to hgh levels. Systemzed ad stadardzed are ma tred the developmet of formato securty maagemet. The settg-up of formato securty maagemet system s the effectve way to guaratee formato securty. The evaluato of formato securty maagemet has receved more ad more atteto. L [1] troduced D-S evdece theory to evaluate the capacty of formato securty maagemet. The characterstcs of formato securty maagemet were aalyzed ad based o the PDRR, the ew formato securty maagemet model was proposed. Huag [2] cetered the study o the formato securty maagemet of the persoal data protecto to promote ther stadardzato. Belloe [3] proposed some specfc operatoal plas ad approaches to assess formato securty maagemet. To evaluate a set of alteratves, the decso maker should gve prefereces o each attrbute for each alteratve. However, t s dffcult to express the decso maker s prefereces accurately. Fuzzy logc ad fuzzy set are popular whe hadlg mperfect, vague or mprecse prefereces. Sce fuzzy set was proposed by Zadeh [4], t has bee successfully used for hadlg fuzzy decso makg problems. The ma characterstc of fuzzy sets s that: the membershp fucto assgs to each elemet x a uverse of dscourse X a membershp terval [0, 1] ad the o-membershp degree equals oe mus the membershp degree. Frutful research results of fuzzy sets have bee acheved may dfferet felds [5, 6]. I applcato, however, the formato combes ot oly evdece for x or evdece agast x, but also ucertaty. Because of the lack of kowledge, the decso maker does ot kow whether evdece for x or ot. I order to solve ths problem, Ataassov [7] troduced the cocept of tutostc ISSN: IJSIA Copyrght c 2015 SERSC

2 Iteratoal Joural of Securty ad Its Applcatos fuzzy set (IFS), whch as a geeralzato of the cocept of fuzzy set cosders the membershp degree, the o-membershp degree ad the ucerta degree. Meawhle, Ataassov [8] defed some operatoal laws of tutostc fuzzy sets. The tutostc fuzzy set has receved more ad more atteto sce ts appearace [9-11]. Szmdt ad Kacprzyk [12] used tutostc fuzzy sets to solve group decso makg problems. Xu ad Yager [13] proposed some aggregato operators based o tutostc fuzzy sets, ad appled them to multple attrbute decso makg. Xu [14] developed some tutostc fuzzy aggregato operators, cludg the tutostc fuzzy weghted averagg operator, tutostc fuzzy ordered weghted averagg operator, ad tutostc fuzzy hybrd aggregato operator, whch ca be used to aggregate tutostc fuzzy formato. I [15], a automotve compay s desred to select the most approprate suppler for oe of the key elemets ts maufacturg process. It s dffcult how to select the best suppler from fve caddates. Meawhle, four attrbutes should be cosdered. TOPSIS(techque for order preferece by smlarty to a deal), proposed by Hwag ad Yoo [16], whose basc prcple s to choose the alteratve wth the shortest dstace from the postve deal soluto(pis) ad the farthest dstace from the egatve deal soluto(nis). It has gaed a lot of atteto. Mash Agarwal [17] compared Choquet tegral wth TOPSIS terms of solvg the suppler selecto problem uder fuzzy evromet. TOPSIS was appled to facal vestmet advaced maufacturg systems [18]. G. R. Jahashahloo [19] exteded the TOPSIS method to decso-makg problems wth fuzzy data, where the ratg of each alteratve ad the weght of each crtero are expressed tragular fuzzy umbers. I ths paper, motvated by the metoed method of tutostc fuzzy set, we propose a ew TOPSIS method wth correlato coeffcet. I tradtoal TOPSIS approach, dstace measure s used to measure devato of alteratves ad the postve deal soluto (PIS) or the egatve deal soluto (NIS). I the ew TOPSIS approach, we defe the weghted correlato coeffcet to measure the correlato of two tutostc sets stead of dstace measure. It ca provde more choce to make best decso. Correlato dcates how well two varable move together a lear fasho [20]. As a mportat cotet fuzzy mathematcs, correlato betwee fuzzy sets has gaed much atteto [21]. Przemyslaw Grzegorzewsk troduced Kedall s correlato coeffcet for fuzzy preferece [22]. Ye [23] used correlato coeffcet to tutostc, terval-valued tutostc fuzzy multple attrbute decso makg. The characterstcs of the weghted correlato coeffcet cludg commutatvty, boudary, ad reflexvty are demostrated ad prove. Smlar to tradtoal TOPSIS approach, the correlato coeffcet s used to compute the closeess coeffcet to rak all the alteratves. The greater the closeess coeffcet s, the better the alteratve wll be. The ma cotrbuto of ths paper cludes the followg: (1) the desg of ew evaluato model of formato securty maagemet uder tutostc fuzzy evromet; (2) the troducto of weghted correlato coeffcet of tutostc fuzzy set; (3) the creato of ew TOPSIS approach wth correlato coeffcet; ad(4) the practcal applcato of the ew method. The rest of ths paper s orgazed as follows. I Secto 2, we revew some basc cocepts related to tutostc fuzzy set. Secto 3 troduces the weghted correlato coeffcet uder tutostc fuzzy evromet. I Secto 4, a ew TOPSIS approach s proposed. I Secto 5, a evaluato problem of formato securty maagemet s vestgated to demostrate the proposed method, ad verfy ts valdty ad applcablty. Fally, Secto 6 cocludes ths paper. 2. Prelmares Itutostc fuzzy set whch geerates the cocept of Zadeh s fuzzy set s 44 Copyrght c 2015 SERSC

3 Iteratoal Joural of Securty ad Its Applcatos characterzed by a membershp degree, a o-membershp degree ad a hestacy degree to descrbe ucertaty ad vagueess more proper. I ths secto, we revew some cocepts of tutostc fuzzy set. Defto 1 [7] Let X be a ordary fte o-empty set. A tutostc fuzzy set X s a expresso A gve by A x, u ( x), v ( x) x X (1) A Where u : X 0,1 deotes the degree of membershp ad : 0,1 A A v X deotes the degree of o-membershp wth the codto: 0 ua va 1, for all x X. For each tutostc fuzzy set A X, f the amout ( x) 1 u ( x) v ( x), x X A A A The ( x ) s called the degree of determacy of x to A. Especally, A If A( x) 1 ua( x) va( x) 0, x X The the tutostc fuzzy set A s reduced to a fuzzy set. For computatoal coveece, ths paper, we call ua( x), va( x) a tutostc fuzzy value. Defto 2 [8] Let a% ( u, v ) ad b % ( u, v ) be two tutostc fuzzy values, the a % a % b % b % (1) a% b% ( ua %, u uau, vav ) b % % b % % b % (2) a% (1 (1 ua %),( va %) ), 0 Defto 3 [13] Gve two tutostc fuzzy values a ad b, the followg operatos are vald: (1) a% b% b% a% a% b% a% b% (2) (3) a% a% ( ) a% 3. The Correlato Coeffcet uder Itutostc Fuzzy Evromet Cosderg the operatoal characterstc of tutostc fuzzy set, we troduce the followg correlato coeffcets betwee tutostc fuzzy sets: the correlato coeffcet ad the weghted correlato coeffcet. Defto 4. (Gerstekor ad Mako, 1991)Gve two tutostc fuzzy sets M= x, um( x), vm( x) x X ad N= x, un( x), vn( x) x X, the correlato coeffcet betwee M ad N s gve as: C( M, N) IF M, N C( M, M ) C( N, N) 1 u ( x ) u ( x ) v ( x ) v ( x ) M N M N u ( ) ( ) 1 M x vm x u ( ) ( ) 1 N x vn x 1 1 Defto 5. Gve two tutostc fuzzy sets M= x, um( x), vm( x) x X ad N= x, un( x), vn( x) x X, the weghted correlato coeffcet betwee M ad N s gve as: IFW M, N C( M, N) C( M, M ) C( N, N) A (2) Copyrght c 2015 SERSC 45

4 Iteratoal Joural of Securty ad Its Applcatos 1 w u ( x ) u ( x ) v ( x ) v ( x ) M N M N w ( ) ( ) 1 um x vm x w ( ) ( ) 1 un x vn x 1 1 Theorem1 Let M ad N be two IFSs o a uverse of dscourse X x x x correlato coeffcet betwee M ad N should satsfy the followg propertes: (1) Commutatvty (2) Boudedess IFW ( M, N) ( N, M) IFW 0 ( M IFW, N ) 1 (3) 1, 2, L,, the (3) Reflexvty ( M IFW, N ) 1, f M=N Proof. C( M, N) (1) IFW M, N C( M, M ) C( N, N) 1 w u ( x ) u ( x ) v ( x ) v ( x ) M N M N w ( ) ( ) 1 um x vm x w ( ) ( ) 1 un x vn x w u ( x ) u ( x ) v ( x ) v ( x ) N M N M w ( ) ( ) 1 un x vn x w ( ) ( ) 1 um x vm x 1 1 ( NM IFW, ) (2) The equalty ( M IFW, N ) 0 s obvous. Below let us prove ( M IFW, N ) 1; 1 C( M, N) w u ( x ) u ( x ) v ( x ) v ( x ) N M N M wu ( x ) u ( x ) L w u ( x ) u ( x ) w v ( x ) v ( x ) L w v ( x ) v ( x ) 1 M 1 N 1 M N 1 M 1 N 1 M N w u ( x ) w u ( x ) L w u ( x ) w u ( x ) + 1 M 1 1 N 1 M N w v ( x ) w v ( x ) L w v ( x ) w v ( x ) Accordg to the Cauchy-Schwarz equalty: We ca obta: 1 M 1 1 N 1 M N 2 x1 y1 x2 y2 x y x1 x2 x x1 x2 x C( M, N) 2 w1 um ( x1 ) L w um ( x ) w1 vm ( x1 ) L w vm ( x ) ( ) L ( ) ( ) L ( ) 1 N 1 N 1 N 1 N w u x w u x w v x w v x w1 um x1 vm x w um x vm x w1 un ( x1) vn ( x ) L w un ( x ) vn ( x ) = w u ( x ) v ( x ) w u 2 ( x ) v 2 ( x ) 1 N N 1 M M ( ) ( ) L ( ) ( ) = C( M, M) C( N, N) 46 Copyrght c 2015 SERSC

5 Iteratoal Joural of Securty ad Its Applcatos Therefore The (3) f M=N, IFW (, ) 2 C M N C( M, M) C( N, N) C( M, N) C( M, M) C( N, N) C( M, N) IFW ( M, N) 1 C( M, M ) C( N, N) M, M C( M, M ) =1 C( M, M ) C( M, M ) Whe w 1 ( 1,2, L, ), tutostc fuzzy weghted correlato coeffcet reduces to tutostc fuzzy correlato coeffcet. 4. Decso Makg Procedure based o TOPSIS Approach I tradtoal TOPSIS approach, the devato betwee the alteratve ad the postve deal soluto or the egatve deal soluto s measured by dstace. Its basc prcple s to choose the alteratve whch has the smallest dstace to the postve deal soluto ad the bggest dstace to the egatve deal soluto. I ths secto, we exted the TOPSIS wth correlato coeffcet stead of dstace measure the procedure of decso makg problem. The proposed TOPSIS approach s basc prcple s to choose the alteratve whch has the bggest value of correlato coeffcet to the postve deal soluto ad the smallest value of correlato coeffcet to the egatve deal soluto. The correlato s troduced to compare the alteratve wth the postve deal soluto or the egatve deal soluto. Uder tutostc fuzzy evromet, the tutostc fuzzy PIS, deoted by A +, ad the tutostc fuzzy NIS, deoted by A - ca be defed as follows: A x, max u( x ),m v( x ) 1,2, L, (4) A x,m u ( x ) max, v( x) L 1,2,, (5) I order to smplfy, we use the Eq. (6) ad Eq. (7) stead of the Eq. (4) ad Eq. (5). A x,1,0 1,2, L, (6) A x,0,1 1,2, L, (7) The relatve closeess coeffcet of a alteratve A wth respect to the hestat tutosc fuzzy PIS A + s expressed as follows: CC (8) Where 0 CC 1, 1,2,... m. Obvously, whe a alteratve s closer to the tutostc fuzzy PIS ad farther from the tutostc fuzzy NIS, CC wll be closer to 1. Hece, accordg to the closeess coeffcet CC, the rakg-order of all alteratves ca be determed ad the best alteratve ca be foud. Accordg to the above models, we ca preset a ovel approach to solve MADM problem, where attrbute values take the form of tutostc fuzzy formato. The approach volves the followg step: Step 1. For a MADM problem, we costruct the decso matrx D [% ] m, where all the argumets % ( 1,2,... m; 1,2,... ) are IFS, gve by the DM. As for every alteratve a ( 1,2,..., m), the decso maker s vted to express evaluato or preferece Copyrght c 2015 SERSC 47

6 Iteratoal Joural of Securty ad Its Applcatos accordg to each attrbute c ( 1, 2,..., ) by a tutostc fuzzy value a% ( u, v )( 1,2 L, m; 1,2, L, ), Where u dcates the hestat degree that the decso maker cosders what the alteratve should satsfy the crtera c, v dcates the hestat degree that expert e cosders what the alteratve should ot satsfy the crtera c.the we ca obta a decso makg matrx as follow: a% 11 a% 12 L a% 1 a21 a22 a2 D % % L % M M O M a% m1 L L a% m Step 2: Utlze Eq. (6) ad Eq. (7) to determe the correspodg tutostc fuzzy PISA + ad the tutostc fuzzy NISA -. PISA NISA Step 3: Utlze the tutostc fuzzy correlato coeffcet to calculate the correlato betwee the alteratve ad the PISA + ad the correlato betwee the alteratve ad the NISA -. IFW AA, * 1 1 C A A C A A * (, ) * * (, ) C( A, A ) w u ( x ) A w ua( x ) va( x ) w 1 va( x ) 1 (10) 2 w ( ) ( ) 1 ua x va x Step 4: Calculate the closeess coeffcet of each alteratve: ( a%%, a ) CC, 1,2,... m (11) ( a%, a% ) a% ( a%, ) Step 5: Rak all the alteratve a accordg to the closeess coeffcet CC, where the greater the value of the closeess coeffcet CC, the better the alteratve a. Step 6: Ed. 5. Numercal Example I ths secto, a evaluato problem of formato securty maagemet s aalyzed by the proposed method to demostrate ts applcablty ad valdty. 5.1 Descrpto of the Effcet Evaluato Problem The most ofte cted referece for whch protectve measures that comprse good formato securty s the ISO stadard. The scope of ISO s to establsh gudeles ad geeral prcples for tatg, mplemetg, matag ad mprovg formato securty maagemet a orgazato [25]. However, t does ot provde ay structos o how to acheve a evaluato problem of formato securty maagemet a specfc compay. The, we take some large state-owed eterprses as a example. A relevat expert from a famous uversty acts as the decso maker. He 1 2 (9) 48 Copyrght c 2015 SERSC

7 Iteratoal Joural of Securty ad Its Applcatos chooses fve large state-owed eterprses deoted as a 1, a 2, a 3, a 4, a 5. Meawhle, he detfes four attrbutes based o the former studes deoted as c 1, c 2, c 3, c 4 Table 1 ad specfes w = (0.35, 0.25, 0.3, 0.1) Attrbutes c 1 c 2 c 3 c 4 Table 1. The Meag of the Seve Attrbutes Meag Physcal securty maagemet (It cludes equpmet securty maagemet, Evromet securty maagemet etc.,) Operato securty maagemet (It cludes operatg system mateace securty maagemet, etwork mateace securty maagemet, database mateace securty maagemet etc.,); Rsk maagemet (It cludes evromet rsk, operato rsk, database rsk, cotrol rsk etc.,,) Huma resources ad polcy maagemet (It cludes huma resources maagemet, strategy maagemet, eterprse s relevat polcy maagemet, etc.,) 5.2 Soluto to the Evaluato Problem wth Novel TOPSIS Method As metoed above, the proposed method whch exteds the TOPSIS approach wth correlato coeffcet s troduced to evaluate ths fve eterprses formato securty maagemet. The evaluato model s demostrated the Fgure 1. Copyrght c 2015 SERSC 49

8 Iteratoal Joural of Securty ad Its Applcatos Fgure 1. The Evaluato Model of Iformato Securty Maagemet Based o the evaluato model, the procedure of the decso makg problem s as follows: Step 1: Wth respect to attrbute c ( 1, 2, L, ) the evaluato of the alteratve a ( 1,2, L, m) s made by a tutostc fuzzy value a% ( u, v ) 1,2, L, m; 1,2, L,. The we ca obta a decso makg matrx as follow: Table 2. The Traspose of Itutostc Fuzzy Decso Matrx Attrbutes a 1 a 2 a 3 a 4 a 5 c 1 {0.7}{0.2} {0.5}{0.3 } {0.4}{0.4} {0.4}{0.6} {0.7}{0.1 } c 2 {0.6}{0.2} {0.5}{0.3} {0.3}{0.6} {0.5}{0.4} {0.6}{0.3} c 3 {0.7}{0.1} {0.2}{0.7} {0.2}{0.6} {0.6}{0.2 } {0.7}{0.3} c 4 {0.3}{0.6} {0.5}{0.4} {0.6}{0.2} {0.6}{0.2} {0.3}{0.6} Step 2: The correspodg tutostc fuzzy PISA + ad the tutostc fuzzy NISA - ca be acqured: PISA Copyrght c 2015 SERSC

9 Iteratoal Joural of Securty ad Its Applcatos NISA 01 Step 3: The correlato coeffcet betwee the each alteratve betwee the each alteratve ad the NISA - ca be calculated: Table 3. The Correlato Coeffcet of Each Alteratve a a a a a ad the PISA + ad Step 4: Based o the correlato coeffcet, the closeess coeffcet of each alteratve ca be obtaed: Table 4. The Closeess Coeffcet of Each Alteratve CC a a a a a Step 5: Accordg to the closeess coeffcetcc, the rak-order of all the alteratve a ca be demostrated: a 1 f a 5 f a 4 f a 2 f a 3 Therefore, we ca acqure that the a 1 s the best eterprse of formato securty maagemet. I the ext step, we ca aalyze the advatage of a 1 to mprove all the eterprses formato securty maagemet. 5.3 Comparso Aalyss wth Tradtoal TOPSIS Approach I order to further realze the ew approach, we compare ths proposed TOPSIS method wth the tradtoal TOPSIS approach. I the tradtoal TOPSIS approach, step 1 ad step 2 s as same as them the ovel method. Step 3: The Eucldea dstace measure betwee the each alteratve ad the PISA + ad betwee the each alteratve ad the NISA - ca be calculated: d a a 1 fw 2 w u u v v %%, 1 d a a 1 fw 2 w u u v v %%, 1 Copyrght c 2015 SERSC 51

10 Iteratoal Joural of Securty ad Its Applcatos Table 5. The Eucldea Dstace Measure of Each Alteratve d d a a a a a Step 4: Based o the Eucldea dstace measure, the closeess coeffcet of each alteratve ca be obtaed: d d ( a%%, a ) CC, 1,2,... m d d d( a%, a% ) d( a%, a% ) Table 6. The Closeess Coeffcet of Each Alteratve CC a a a a a Step 5: Accordg to the closeess coeffcet CC, the rak-order of all the alteratve a ca be demostrated: a 1 f a 5 f a 4 f a 2 f a 3 Therefore, we ca acqure that the a 1 s the best eterprse of formato securty maagemet. The, t s obvously that the fal result the tradtoal approach s as same as the proposed method ths evaluato problem. But the extet of chages betwee two curves s ot qute as same Fgure2. The black le ad the red le deote the chages of closeess coeffcet wth correlato coeffcet ad closeess coeffcet wth dstace measure, respectvely. The extet of chages of the closeess coeffcet wth correlato coeffcet s chage s bgger tha the extet of chages of closeess coeffcet wth Eucldea dstace measure. So t s easy to compare the alteratves the TOPSIS approach proposed ths paper, especally there are may complex ad smlar alteratves. 52 Copyrght c 2015 SERSC

11 Iteratoal Joural of Securty ad Its Applcatos Fgure 2. The Comparso betwee the Propose Method ad the Tradtoal Method 6. Coclusos Because of the heret vagueess of huma prefereces, the attrbutes volved decso makg problems are ot always expressed real umbers, ad some are better suted to be deoted by fuzzy values, such as tutostc fuzzy values. I ths paper, cosderg membershp, o-membershp ad the ucertaty of the decso maker s preferece, we exted the TOPSIS approach wth correlato coeffcet uder tutostc fuzzy evromet. The cotrbuto of ths paper clude the followg: (1) the desg of ew evaluato model of formato securty maagemet uder tutostc fuzzy evromet; (2) the troducto of weghted correlato coeffcet of tutostc fuzzy set; (3) the creato of ew TOPSIS approach wth correlato coeffcet; ad (4) the developmet of the evaluato problem of formato securty maagemet; (5) comparso aalyss betwee the ew TOPSIS approach ad tradtoal TOPSIS approach ths evaluato model. Although the proposed method ths paper s a good soluto to the evaluato problem of formato securty maagemet, t caot aalyze more complex problems where may dfferet decso makers take part ths evaluato problem. I the future, we wll exted the method to aalyze the group decso makg problem of formato securty maagemet. Refereces [1] J. N. L, Harb: Research o formato securty maagemet metrcs, Harb Uversty of Scece ad Techology, (2007). [2] C. C. Huag, K. J. Far ad F. Y. L, A Study o Iformato Securty Maagemet wth Persoal Data Protecto, Proceedgs of the Iteratoal Coferece o Parallel ad Dstrbuted Systems, Los Alamtos, CA, Uted States IEEE Computer Socety, (2011), pp [3] J. Belloe, S. Basquat ad J. Rodrguez, Reachg Escape Velocty: a Practced Approach to Iformato Securty Maagemet System Implemetato, Iformato Maagemet ad Computer Securty, vol. 16, 1, (2008), pp [4] L. A. Zadeh, Fuzzy sets, Iformato ad Cotrol, vol. 8, (1965), pp [5] R. R. Yager ad D. P. Fler, Essetals of Fuzzy Modelg ad Cotrol, Wley, New York, (1994). [6] D. Dubos ad H. Prade, A revew of fuzzy sets aggregato coectves, Iformato Scece, vol. 36, (1985), pp [7] K. Ataassov, Itutostc fuzzy sets, Fuzzy Sets ad Systems, vol. 20, (1986), pp [8] K. Ataassov, New operatos defed over the tutostc fuzzy sets, Fuzzy Sets ad Systems, vol. 61, (1994), pp [9] K. Ataassov ad G. Gargov, Itutostc fuzzy logc, C. R. Acard, Bulgare, Sc., vol. 43, o. 3, (1990), pp [10] K. Ataassov, G. Pas ad R. Yager, Itutostc fuzzy terpretatos of mult-crtera mult-perso ad mult-measuremet tool decso, Iteratoal Joural of systems Scece, vol. 36, (2005), pp [11] C. Q. Ta ad X. H. Che, Itutostc fuzzy Choquet tegral operator for mult-crtera decso makg, Expert Systems wth Applcato, vol. 37, (2010), pp Copyrght c 2015 SERSC 53

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