Applied Poker Test for General Digital Sequences

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1 IOSR Joural of Mathematcs IOSR-JM e-issn: , -ISSN: X. Volume, Issue Ver. V Ja. - Feb. 6, PP Ale Poker Test for Geeral Dgtal Sequeces Sahar A. Mohamme Abstract: The Poker test s cosere oe of the mortat statstcal raomess tests. Ths test, for eeece, s base o the frequecy whch certa gts are reeate a seres of umbers. I ths aer we wll attemt to geeralze the bary oker test to be sutable to ale o ot oly bary sequeces, aother wor, the geeralze oker test coul be ale o gtal m- sequeces for m. The geeralze oker test calle Groue oker test. Keywors: Statstcal Raomess Tests, Poker Test, Hyothess Test, h-square Test, crytograhy. I. Itroucto The roblem of testg raomess s motvate by the ee to evaluate of the qualty of fferet raom umber geerators use by may ractcal alcatos clug comuter smulatos, crytograhy a commucatos ustry []. The oker test treats umbers groue together as a oker ha. The the has obtae are comare to what s execte usg the ch- square test []. Defto [3]: A Pseuo Raom Bt Geerator PRBG s a etermstc algorthm whch, gve a truly raom bary sequece of legth k, oututs a bary sequece of legth, k whch aears to be raom. The ut to the PRBG s calle the see, whle the outut of the PRBG s calle a seuoraom bt sequece. Remark [3]: The ch-square strbuto ca be use to comare the gooess-of-ft of the observe frequeces of evets to ther execte frequeces uer a hyothesze strbuto. The strbuto wth egrees of freeom arses ractce whe the squares of eeet raom varables havg staar ormal strbutos are summe. I 98, the fve statstcal tests for local raomess are beg fou to be ale o bary key stream sequeces [4]. I 994, a ew ackage of raomess trouce by RYPT-X [5]. Ths ackage s a mcrocomuter ackage that s tee to be use to test ether large bary strgs that are to be use as key stream stream chers or block cher algorthms. Oe of the tests of ths ackage s subblock test whch s a smlar to oker test, ts arttos the stream to F has of legth m bts. For a stream of sze, where F/ m 5, the total umber of has s /m, where eotes the teger value. The am of ths test s to show that there s a equal umber of each of m ossble has. If f eotes the frequecy of ha atter s, the the test statstc use s [6]: m m T= f F F Ths comare wth ch-square strbuto wth egree of freeom equal to m -. I 9, Al et al. [3] trouce a aer that clue the geeralzato of three basc a mortat tests of the staar statstcal basc raomess tests, these tests are frequecy, Ru a Autocorrelato tests. I 4, Ibraheem [7], extes the -tule to -tule 3 to aly bary seral test for bary sequeces. Seco, she geeralze the -tule bary seral test to -tule gtal seral test for gtal m- sequeces m3 geerate from gtal geerators. astly, she extes the -tule to -tule gtal seral test for gtal sequeces. The results of raomess tests of alyg the bary a gtal oker tests the bary a gtal m- sequeces were trouce tycal tables usg hyothess test usg h-square test. The bary a gtal oker test results are obtae by rogram usg verso. of MATAB aguage wth o tme metoe to obta the raomess results. II. Hyothess Test for Raomess Usg h-square Test Suose we have a exermet wth ossble outcomes, wth ukow robabltes,,...,, ow we attemt to ece whch of the two hyotheses or H A s true, where : = for all =,,..., a H A : for some =,,...,, where,,..., are kow. DOI:.979/ Page

2 Ale Poker Test for Geeral Dgtal Sequeces We coser eeet reettos of the exermet wth the raom varables N eotg the umber of tmes the th outcome occurs, for =,,, where N. We use the test statstc [7]: N we obta the P-value of the test aroxmately for large, usg the ch-square table for P=P, where s the observe value of. I artcular, a small value of, where was obtae from h-square table at freeom egree = - a sgfcat value =. or =.5 that leas to a large P- value s strog evece favor of rove that the ata are really geerate from a raom samle. I orer to test the hyothess, we ale the followg stes [8]: State the ull a alteratve hyotheses. Select the strbuto to use. Determe the rejecto a o-rejecto regos. alculate the value of the test statstc. Make a ecso. III. Bary Poker Test et be a ostve teger such that 3, a let k=. Dve the sequeces to k o-overlag arts each of legth, a let be the observe umber of occurreces of the th tye of sequece of legth,. The oker test etermes whether the sequeces of legth each aear aroxmately the same umber of tmes S, as woul be execte for a raom sequece. The execte value of the strg whch cossts of s wth o coserato to arragemet of s s [9]: E The statstc use s: whch aroxmately follows a strbuto wth = egrees of freeom. Note that the oker test s a geeralzato of the frequecy test: settg = the oker test yels the frequecy test. Examle : lets have the followg sequece wth legth =3, m=. et S=. For =3, =,3,3,, the E =.5,3.75,3.75,.5 resectvely. From the sequece we have the followg frequeces usg relato 3: Samles Freq.,, 4,, =.333, ote =.333 < =7.8, ths mea we accet a reject H A, ths sequece ass the bary oker test. Examle : We test two sequeces, S whch s geerate from raom geerator raom fucto of the comuter whle the sequece S geerate from o-raom geerator two shft regsters wth AND boolea fucto usg bary or classcal oker test PT for m= a =3,4,5. Table clues the formato of the legth of the two sequeces wth the execte E a the frequeces values. Whle table shows the DOI:.979/ Page

3 Ale Poker Test for Geeral Dgtal Sequeces values a the accete or rejecto ecso for the two sequece S a S, the formato of alyg the relato 3 obtae from table. =7.85, a.7for ==3,4 a 5 resectvely. Table : Iformato of the two sequeces S a S. = E S S = E S S = D E S S Seq. S S Table : classcal Poker test PT results for two sequeces. PT Values Decso Accet Accet Accet Accet Accet Accet Accet Accet Accet Decso Remark : From examle, f we re-calculate the frequecy a execte the values for the samles of bary sequece: jk Samles Freq. These samles have the same execte value s.t. E =E=/ 3 */ =/ 3 *3/3=.5. So relato 3 ca be extee to followg relato for =3: E jk j k E , Notes the close results for examle relatos 3 a 4. 4 I the ext roosto we wll rove that f the bary sequece S asses the oker test, relato 3, the t wll satsfes oker test for relato 4 a vce versa., Proosto : The sequece S asses bary oker test for relato 3 f a oly f t asses bary oker test for relato 4. Proof: : ths mea E =, where =//. For smlcty let's choose =, the: =, = +, = a = =, = + =+=, = =. DOI:.979/ Page

4 Ale Poker Test for Geeral Dgtal Sequeces DOI:.979/ Page otce a a b Now c Sce E =, the Substtute c, obta: e From a, b a e we obta: j k jk E E I the same way we ca rove the other se. Proosto ca be rove for ay, a ow ts o matter f we ale relato 3 or 4 to ece whether that S wll asses bary oker test. IV. Geeralzg the Bary -Tule Dgtal Poker Test I ths secto we wll geeralze the bary -tule sequeces to calculate the gtal oker test for gtal m- sequeces. et S be the gtal m-sequece geerate from gtal geerator wth legth, where s S, a s m-, =,,,,-. Now we wll roose a ew oker test, we calle t Groue oker test GPT, s.t. we wll ve the samles of -tule subsequece to m sjot sets of samles. et A={,,,m -} be the set of all fferet teger sum of the all robable -tule samles of the teste gtal sequece S. we arttoe the set A to equal orer subsets A ={x =,,,x -}, A ={x,x +,,x -},, A m- ={x m-,x m- +,,x m =m -}, s.t. x =m - =h, where =,,,m-. Notce that:

5 Ale Poker Test for Geeral Dgtal Sequeces m A A. A k A j, where k j, k, j m- ay two fferet subsets of A are sjot. A =m a A =h=m -, m-. Examle 3: For m=3 a =3, we have the followg samles:,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, Ths meas we have m =3 3 =7 samles. We have A={,,,6}, A ={,,,8}, A ={9,,,7} a A ={8,9,,6}. A =3 3 =7, A = 3 =9=h, x =,9,8, =,,. Now let N, =,,,m-, be the total observe frequecy of all elemets of the set A occurre gtal sequece S. The robablty P for each gt,..., m - the set A s: Ps k = / m, k=,,,m -. 5 Sce the orer of the set A, A =h=m -, the the robablty for each gt set A s: Ps t =h/ m =/m, t=,+,,h. 6 The the execte value E s: E=/m./=/m. 7 whch s equal for all gts s t. Whe alyg ch-square test, wll comare wth table value wth freeom egree =m. Its obvous that f the frequecy N of each stct gt s t s aroxmate to other frequeces, the the gtal sequece s satsfes the oker ostulate, so must be: N N N m- The the GPT wll be: m N m. m. Examle 4: et's call examle, =3, m=, =3. x =, 4, A ={,,,3} a A ={4,5,6,7}, E=.5*3/3=5, N Samles Freq. N,,, 5 N,,, 5 8 m N m. m. = <7.85, the S asses GPT Examle 5: let's call examle, usg GPT for m= a =3,4,5. Table 3 clues the ew formato of the two sequeces S a S wth the ew execte E a ew frequeces N values. Whle table4 shows the values a the accete or rejecto ecso for the sequece S a S, the formato of alyg the relato 8 obtae from table 3. =5.99for =m=. DOI:.979/ Page

6 Ale Poker Test for Geeral Dgtal Sequeces Table 3: Iformato of the two sequeces S a S., E, E =3, E= =4, E=5 =5, E= S N S , E =3, E= =4, E=65 =5, E=5 5 S N S , E =3, E=666.7 =4, E=5 =5, E= S N S Seq. S S Table 4: GPT results for two sequeces for m=. GPT Values Decso Accet Accet Accet Accet Accet Accet Accet Accet Accet Decso Examle 6: Now lets choose m=3,4 a 5, a =3,4,5 for the two sequeces S a S wth legth =5 gts. Table 5 llustrates the GPT results usg ch-square test for the two chose sequeces a.7for =m=3,4 a 5 resectvely. =7.85, Seq. S S Table 5: GPT results for two sequeces for m=3,4,5 a =3,4,5. GPT Values =5 m Decso Accet Accet Accet Accet Accet Accet Accet Accet Accet Decso V. oclusos. As kow, the roof of raomess for ay sequece s robablstc, so t's referable to choose as log as ossble a choose may examles of sequeces to be teste by oker test to obta accurate ecso.. I orer to obta accurate raomess ecso for ay gtal sequece, we have to choose fferet values for. 3. From Al et al. [3] Ibraheem [7] wth ths work we wll obta fve gtal raomess test for gtal sequeces so we suggest a gtal statstcal raomess ackage usg fve gtal tests; frequecy, ru, autocorrelato, seral a oker tests. Refereces []. Abel-Rehm W. M. F., Ismal I. A. a Morsy E., "Testg Raomess: Poker Test wth Has of Three Numbers", Joural of omuter Scece 8 8: , ISSN ,. []. Stewart, W. J., "Probablty, Markov has, Queues, a Smulato: The Mathematcal Bass of Performace Moelg", st E., Prceto Uversty Press, Prceto, ISBN-: 69466, : 758, 9. [3]. Al F. H, Mohamme, S. A. a Shammra, M. A., Geeralze the Raomess Tests to Test the Dgtal Sequeces Prouce from Dgtal Stream her Systems, Iraq Joural for Scece, Bagha Uversty, ollege of Scece, 9. [4]. Beker, H. a Per, F., her Systems: The Protecto of ommucatos, Joh Wley & Sos, New York, 98. [5]. Gustafso, H., Dawso, E., Nelse,. a aell, W., A omuter Package for Measurg the Stregth of Ecryto Algorthms, omuters & Securty, 3, , 994. DOI:.979/ Page

7 Ale Poker Test for Geeral Dgtal Sequeces [6]. Brassar, G., Moer rytology: A Tutoral, NS 35, Srger-Verlag, New York, 988. [7]. Ibraheem S. K., "Seral Test Exteso a Geeralzato to Test the Dgtal Sequeces", Iraq Acaemc Scetfc Jourals, ISSN: 84635X, Volume: 5 Issue:4 Pages: 83-96, 4. [8]. htt:// [9]. Al-Shammar, A. G., Mathematcal Moelg a Aalyss Techque of Stream her rytosystems, Ph. D. Thess, Uversty of Techology, Ale Sceces, 9. DOI:.979/ Page