MARKOV MODEL: Analyzing its behavior for Uncertainty conditions

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1 MARKOV MODEL: Anlyzing its behvior for Unertinty onditions Llith.R.V.S Sri Si Adity Institute of Siene nd Tehnology,Surmplem e- mil: Sri Divy.R Sri Si Adity Institute of Siene nd Tehnology,Surmplem e-mil: Abstrt: Mrkov model is used to nlyze the dynmi behvior of the system in prediting the next stte with the previous stte. The proess of ttempting to guess the next hrter revels informtion bout the pssword strtegy. In this pper, we give fuzzy inferenes bout the guessing psswords, by exmining with the previous stte nd omputing the possible outomes of probbility of eh hrter. For some problems there nnot be omplete solutions. For suh problems Fuzzy inferenes llow us to evlute sub expressions. In the present pper, we disuss how to tre out some unertinty onditions nd nlyze their behvior using fuzzy inferene system nd finlly test the system for finding stedy stte behvior in guessing the hrters in the pssword. Keywords: Mrkov model, Fuzzy sets, Trnsition mtrix, Membership funtions, Fuzzy Logi 1. Introdution: To understnd the nture of the thret to pssword-bsed systems, we hve to hve informtion bout the guessble psswords. Mrkov model [1] gives representtion of guessble psswords by reting trnsition mtrix using prefixes. If the users re ssigned psswords tht ontin eight rndomly seleted printble hrters, then remembering pssword is diffiult, so pssword rking is very diffiult. The tehniques used to eliminte guessble psswords re 1. User Edution-users re given the guidelines for seleting hrd psswords. 2. Computer generted psswords- If the psswords re rndom, diffiult to remember.3--retive pssword heking system periodilly runs rker to find guessble psswords for nelling psswords guessed by the user.4. Protive pssword heking user is llowed to selet his or her own pssword. While guessing psswords guessing entire string is diffiult. But when subset of string is guessed we n predit the next ourrene by storing its previous stte. The next hrter to be rrived depends only on the previous stte but not independent. This onept is useful in plying rds, rolling die et. In this pper, we give mthemtil representtion for few mthemtil models for the omputtion of next possible stte bsed on the urrent stte. In pssword heking, guessble pssword is ompred with the ditionry of psswords stored. When the probbility of the ertin word is rehed, we n forget the ourrene of the prtiulr hrter in the next subsequent sttes. For tht, we explin some of the mthemtil representtions for omputing vrious ombintions of subsets of the string. The ide of fuzzy sets ws born in July The onept of grde of membership is the bkbone of fuzzy set theory. Fuzzy set is set with smooth boundry. Fuzzy sets re used where to generlize query or to del with prtil membership. Prtil membership ours when either onsequent or nteedent hve prtil stisftion. Fuzzy rules re helpful to give nlysis for sub expressions. By tking onlusion, by ombining ll the outomes of the fuzzy sub expressions we will obtin solution whih overs lmost ll the possible entities. In setion 2 we disussed the bsi priniple of Mrkov model. In setion 3 we disussed the onepts of unertinty onditions by tking n exmple. In setion 4, we disussed the behviorl study of Fuzzy inferenes for some of the sttes n be dedued using inferene logi of Fuzzy theory. The inferene logi helps us to retrieve informtion either by omprtive nlysis or by probbilisti pproh. We onsider the probbilisti pproh for evluting subsets. In setion 5, the signifine of tringulr membership funtion, operting between lower nd upper bounds is disussed. Finlly, in setion 6, we hve given mthemtil representtion, for some of the strings guessed, nd 1

2 omputed their probbilities. In setion [7] we gve few smples for prourement of probbilities nd to hve some history of previous sttes. 2. Priniple of Mrkov model:- The problems ssoited with the bove pprohes re 1. Spe nd 2. Time. Mrkov model helps in prepring list of guessble psswords, whih re to be rejeted by the system Fig : 1 M={ m, A, T, k} M= {3, {, b, }, T, 1} T= A Mrkov model is qudruple [m, A, T, k], where m is the number of sttes in the model, A is the stte spe, T is the mtrix of trnsition probbilities, nd k is the order of the model. Mrkov model is pplible if eh row of trnsition mtrix sums to 1. The sum of row of elements of row 1 = =1.0 The sum of elements of row 2 = =1.0 The sum of elements of row 3 = =1.0 The Mrkov model is not pplible, if the elements of ny row is less thn or greter thn Anlyzing the behvior for some unertinty onditions:- In the Fig 2, entire informtion is sinked into the node b, s there is no outgoing stte. No outgoing stte identifies tht there will be no ourrene of next hrter. If we nlyze the behvior of stte b we n derive the previous stte nlysis tht is rehing b. 2

3 b Fig:2 The trnsition mtrix for Fig.2 is T= By this, we n onlude tht the guessble psswords for this trnsition mtrix n be b, or b, or b, or b or b or b or et. Whih shows the position of b is lwys lst for most of the substrings nd the position of n be either t the beginning or t the lst. The probble ourrenes of the next hrter for n be, b or.the probble ourrenes of the next hrter for is or b. And the probble ourrenes of the next hrter for b is nothing. 4.Fuzzy Inferenes:- We n pply fuzzy inferenes logil resoning, for the sttes tht output to the sme stte. To know how to onvert the risp set into fuzzy one nd fuzzy into risp one, observe the following rules for the exmple sttes tken s below: Let x is pssword submitted by the user. The fuzzy rule for the fig.1 n be written s Rule 1: if x is then y is b with 0.7 probbility Rule 2: if x is then y is b with 0.3 probbility Obviously, multiple inputs produe the sme output upto some extent. Fuzzy resoning is used to generlize logil resoning lled modus ponens. This n be referred generliztion, in the sense whtever the inputs submitted we re obtining the stte b for some probbilities. Fuzzy rule bsed inferene: As the probbilities of getting b for nd re different, we n ssure tht we n predit b fter or for 0.3 probbility lso. This type of inferenes n by represented using membership funtions. 3

4 Input Crisp Set Fuzzy Fuzzy Fuzzifier b Inferene Logi z Defuzzifier Crisp Output Fig:3 5. Representtion of Fig 3, using Tringulr membership funtion: This funtion is designed, to distinguish the lower nd upper bounds of two inputs to be onsidered for produing the sme output. A fuzzy is defined by funtion tht mps objets in domin of onern to their membership vlue in the set. Suh funtion is lled membership funtion, denoted by Greek symbol µ for reognition nd onsisteny. The membership [4] of fuzzy set A is defined s µ nd the membership vlue of x in A is µ (x). Let A= {, b, } µ µ (b) 1.0 b x Fig:4 In the bove fig, with 0.7 probbility nd with 0.3 probbility outputs to. The next ourrene of hrter b ppers only with the probbilities of 0.7 nd b 0.3. Beyond tht heking for the ourrene of hrter b is not required. If there is proess with 4 elements, we n go for trpezoidl membership funtion. These membership funtions llow us to give informtion bout the rnge of vlues to be llowed for omputtion. To obtin the region of informtion tht is ommon to both the sttes, we dopt sling nd lipping methods, to dedue inferene logi. The following two digrms, gives informtion bout the inferred onlusion with mthing degree

5 µ 1.0 b 0.3 x 0.0 Fig. 5 inferred onlusion using lipping method 1.0 µ 0.3 b x 0.0 Fig. 6 inferred onlusion using sling method 6. Computtion of no. possible ourrenes of eh hrter [2]:- To evlute the behvior of the stedy stte vetor, Consider Mrkov hin with N sttes. A stte vetor for Mrkov hin is row vetor X=[x 1,x 2..x n ] Our exmple Mrkov proess is with 3 sttes (,b,). Row vetor 1= 1, 1b, 1 Row vetor 2 =b 2, b 2b, b 2 Row vetor 3= 3, 3b, 3 1 is the probbility tht the probbility of in first stte so tht it gets the next hrter s in the next stte nd so on. Let X= [ b ] Let T is n x n mtrix, k is the k th X=lim T power of mtrix, then[8], 5

6 Let X 0 denote the initil stte of the Mrkov hin. In generl, Xn = X n 1 T Let Identity mtrix I= And Trnsition mtrix T= is the mtrix obtined initilly by multiplying identity mtrix with the trnsition mtrix for finding the initil stte[3]. Let us find probbilities for some exmple strings. Probbility of obtining only b s in the string. x 0 x 1 = x 0 T Probbility of obtining only s in the string Probbility of obtining only s in the string[6]: Finding probbility of two hrter word (bi-grm) in the pssword- Probbility of getting b in the string =[.6 1 0] Finding probbility of three hrter word(trigrm)[5] in the pssword- Probbility of getting b in the string =[.6 1.4] nd so on. The strings re heked for probbilities tking initil stte s. If we hnge the initil stte, we go for ny ombintion of strings, like b, b et. 6

7 7. Unertinity onditions: Set s is the set of eptble strings. And w1, w2, w3... re the probble strings/substrings in the set. S={ w 1,w 2 w n } Let N ij be the no. of trnsitions mde by string from stte i to stte j. Totl no. of trnsitions(n ij ) from stte i to stte j depends on the probbility of the prtiulr hrter tht n hve trnsition from stte i to stte j. N i tul no. of trnsitions mde. Then trnsition probbility(tp) from stte i to stte j is N ij / N i Then emission probbility n be omputed s Probbility of stte I - N ij / N i For eg. Emission probbility (ep) for stte is 0.7 N b /N As the emission probbility inreses, the probbility of guessing pssword hrter dereses. 8. Conlusion:- The probbilities omputed for eh hrter from trnsition mtrix nd the probbilities derived from the vetor, both beome sme. With the bove omputtion nlysis, we n restrit our guess in prediting the next hrter up to some extent. 9. Future sope: This nlysis n be extended to seond order Mrkov model. By giving nlysis of more nd more sub expressions of fuzzy nlysis, leds to omprehensive study of the prediting psswords. Referenes: 1. Network Seurity Essentils Applitions nd Stndrds by Willim Stllings, Person Edution Fuzzy mrkov modeling in utomti ontrol of omplex dynmi systems V.Arkov, Institute of Mehnis, Russin Ademy of Siene, Uf, Russi G.G.Kukikov, T.V. Breikin, Uf Stte Avition, Tehnil University, Uf, Russi 4. FUZZY LOGIC - INTELLEGENCE,CONTROL, AND INFORMATION by JOHN YEN, REGA LANGARI 5. Introdution to Informtion Retrievl - Serh Engines Mihel Hwthornthwite works t Aid Computer Servies (Mnhester) who speilize in web design, web development nd bespoke softwre development Regulr Mrkov Chins Tom Lewis Winter Term

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