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1 41 9 Vol41No JOURNALOFUNIVERSITY OFSCIENCEANDTECHNOLOGY OFCHINA Sep (2011) Logistic ( ) Logistic ; ; ;Logistic TP3097 A doi103969/jissn Aparalelcomputingmethodofchaoticrandomsequence basedonlogisticmapwithscalableprecision LIUJiahuiZHANG Hongli (Schoolof ComputerScienceand TechnologyHarbinInstituteof TechnologyHarbin150001China) AbstractThedynamicaldegradation ofchaos withfinitecomputing precision alwaystroublesdigital chaoticsystemsaparalelmethodforcomputingthelogisticmapwithscalableprecisionwasintroduced; inwhichchaotic map wasdividedintoseveralpartsforcomputingandtheresults wereplacedinthe dynamicalarraycomputingchaoticmapwithscalableprecisionbreakslimitofcomputerfiniteprecision alowingtheusertosetthecomputingprecisionfreelytheexperimentalresultsshowthatthehigherthe precisionisthegreaterthe mappingspace wilbewiththechaoticrandom sequences withscalable precisionapproachingtheidealstateofchaos Keywordsscalableprecision;randomsequence;paralelalgorithm;Logisticmap 0 DESAES ; (863) (2010AA012504) 2011 (1251H018) 1974 / bsuljh@163com / zhanghongli@hiteducn

2 Habutsu [4] Tentmap ; / Biham [5] 2 38 ; Habutsu 1995 Fridrich [6-7] 1996 Feldmann [8] (inversesystemapproach) 1997 Zhou [9-10] (piecewiselinearchaoticmap) 2003 Li [11] Zhou 1998 Baptista [12] Logistic 2001 Jakimoski [13] Baptista 2002 Wong [14] Logistic Baptista ; 2004 Li [15] Baptista Wong ; (dynamicaldegradation) 2004 Machado [16] Baker ; map 2006 ; ; Alvarez [17] IEEE 01 ; 1963 Lorenz [1] Blank [18-19] 1989 Mathews [2] Wheeler [3] Mathews Bakermap 2005 Li [20]

9 Logistic 839 [21-29] 2006 2009 AlvarezArroyo a ;x 0 1 Logistic 1 a<3 Logistic 03 ; a>357 1998 Fridrich [32] ;2005 Pareek [33] ;2007 Wei [34] Pareek 2003 Pareek [35] 128 1 Logistic Fig1

3 9 Logistic 839 [21-29] AlvarezArroyo a ;x 0 1 Logistic 1 a<3 Logistic 03 ; a> Fridrich [32] ;2005 Pareek [33] ;2007 Wei [34] Pareek 2003 Pareek [35] Logistic Fig1 Bifurcationdiagramsofthelogisticmap [36-37] Schuster [39] (1) a=4 Logistic ρ ( 烄 1 x)= π 槡 x(1-x ) 0<x <1; 烅烆 0 a=4 (Liapunov Logistic exponent) 1 Logistic ; 2 λ= 1 0dxρ ( x) f (x) =log2 Logistic ; 3 ; 珚 x 1 Logistic 1976 May [38] Logistic (Logistic [30-31] Logistic x n+1 =a x n (1-x n ) 0<a 40<x n <1n =01 (1) equation) ; 1 ac(m)= lim Logistic (Logisticmap)Logistic N ; 3<a<357 珚 x = lim N N-1 1 N i = i=0x 1 xρ ( x)dx = N-1 N i=0 (x i - 珚 x)(x i+m - 珚 x)=

4 xf m (x) 0 ρ (x)dx-x 2 = 0125 m =0; 0 m 0 f m (x)=f( f(x ) ) 烏烐烑 m 12 x 1 x 2 c 12 (m)= lim N 1 N-1 N i=0 (x i1 - 珚 x)(x(i+m)2 - 珚 x)= x 1f m (x 2 ) 0 ρ (x 1 ) ρ (x 2 )dx 1dx 2 -x 2 =0 64 IEEE ) 10 L ; mantissa exponent L 烇烉烋烇烉烋 (-1) b 63 (1b 51 b 0 ) 2 2 (b 62 b 52 ) b 63 b 0 b 51 b 52 b ; 15 x 0=01a= Logistic 2 Logistic 1 4 ; 2 ; 1985 IEEE ANSI 32 x L ( x 0=01a=39 x L x 2 Logistic Fig2 Iterationdiagramofthelogisticmapwithfiniteprecision

5 9 Logistic 841 a L a x 0 (1)1-x 0; L x x=011-x= 09 1; (1) x(1-x) 2L x a x (1- x) Fig3 Chartofcomputingthelogisticmapstepbystep 2L x+l a 2 x=a x Logistic 1 x 1-x L xn =2 n L x0 + (2 n -1) L a (2) 2 x 0=01a=39 Logistic x (1-x) 1 x a x 3; ; 21 Logistic (Ⅰ) x 0 a 1 Logistic Tab1 Relationshipbetweeniterationnumber x 0 xx andmaximumprecisionofthelogisticmap n x = (s i D -i )D = X[1]=s 2 X X L x [40] (Ⅱ) x (1-x) y=1-x n=1-x n y = ; (r i D -i )r i 01 9n =12 i=1 r i = D i=n; -si (D -1)-s i i<n y x Y= r 1 r 2 r n Y 2 Logistic Logistic Logistic Y r 1 Y[0]=r 1 Y[1]=r x=x (1-x); 3 Logistic i=1 s i 01 9s n 0n =12 n x a n a = (a i D -i )D =10 i=0 a i 01 9a n 0n =01 X=s 1 s 2 s n X X s 1 X[0]=s 1 x (1-x)

6 熿 s 1r n s 2r n s n-1r n s nr n 燄 s 1r n-1 s 2r n-1 s n-1r n-1 s nr n s 1r 2 s 2r 2 s n-1r 2 s nr 燀 0 s 1r 1 s 2r 1 s n-1r 1 s nr 燅 x (1-x) M m 11 m 12 m 1n m 1(n+1) 熿 m 1(2n) 燄 m 21 m 22 m 2n m 2(n+1) m 2(2n) M = m n1 m n2 m nn m n(n+1) 燀 m n(2n ) 燅 n 2n Mi=r n-i+1 i X1 i n Mi Mi = mi 1 mi 2 mi n M Mi pi ; M m ik=01 i n1 k 2n Mi M x (1-x) Mi m ik = mi j k =j+n-i+11 ij n 2 2 Mi x (1-x) Mi αi = ( η i 烄 +d i )(modd)1 i 2n; pi = (i-1)(modn pros )+1 n 烅 η i = m ji ; j=1 烆 d i-1 =int[( η i +d i )/D]1 i 2n Mi int[] ; Mi d 2n=0 Mi Initialize MPI(intmyidintTotal processors); // MPI x (1-x) T if(myid==masterid)//iam masterprocessor T = α1α2 αnαn+1 α2n T Temp Temp 2n O(2 iteration (L x +L a )) iteration 22 N pros pi (Ⅰ) N pros L x X r i ( )pi (Ⅱ) N pros<l x i=12 pi =12 N pros i Mi Get parameter x (int*arr xintlength char*str par x);// x Get parameter a(int*arr aintlength a X char*str par a);// a Set precision (intlogistic precision);// 2n Temp Set iteration (intlogistic iteration);// X Temp (Ⅲ) x=a x if(myid==masterid) x=a x for (slaveid =0;slaveID < (Total processors-1); (Ⅳ) (2) slaveid++) ; (Ⅱ) MPI SEND (intslaveid );//

7 9 Logistic 843 else//iamslaveprocessor 2 MPI RECV (intslaveid );// MPI Barrier(MPI COMM WORLD);// ; 31 for(i=0;i<logistic iteration;i++) ParalelComputing step 1 (int*arr xint*arr yint Length x);// Logistic MPI Barier(MPI COMM WORLD);// ; t n-t 2 ParalelComputing step 2 (int *arr xintlength x 0=01a=39 Logistic int*arr aintlength a);// Logistic MPI Barier(MPI COMM WORLD);// ; 2 Logistic Tab2 Transientperiodandcycleperiod Logistic ofthelogisticmap ParalelComputing step 1 (int*arr xint*arr yint Length x); t if(myid==masterid) n-t for (slaveid=0;slaveid< (Total processors-1); slaveid++) 01 9 P(0) MPI SEND (intslaveid );// P(1) P(9) P(0)=P(1)= =P(9) MPI RECV (intslaveid );// =1/10 a=398765x 0= ChangePrecision(intLength x// else Length x intlogistic precision); CopyResult(int*arr xintlength x); // MPI RECV (intslaveid );// Computing( ); MPI SEND (intslaveid );// Logistic arr x 3 1 ; x 1 x 2 x t x t+1 x t+2 x n t Fig4 Probabilitystatisticschartofnumber 4

844 41 32 5 Logistic 7 14a=39 x 0=01 x 0=0100001 A = α1α2 αnb = β 1 β 2 β n αi i 5 Fig5 Sensitivitytestforinitialvaluesofchaos withdiferentgivenprecision 50 0000001 0001 (0000001) (Ⅰ) ( 5 30 5000) 6

8 Logistic 7 14a=39 x 0=01 x 0= A = α1α2 αnb = β 1 β 2 β n αi i 5 Fig5 Sensitivitytestforinitialvaluesofchaos withdiferentgivenprecision ( ) (Ⅰ) ( ) x 0 = 01 S 1 ; x 0 = (Ⅱ) S [1]LorenzE NDeterministicnon-periodicflow [J]J AtmosSci Fig6 Diferentialchartofcorrespondingposition withabsolutevalueinchaoticrandomsequences (References)

9 9 Logistic 845 [2]MathewsR A JOnthederivation ofa chaotic encryptionalgorithm [J]Cryptologia198913(1) [3]Wheeler D DProblems with chaoticcryptosystems [J]Cryptologia (3) [4]HabutsuTNishio YSasaseIetalA secretkey cryptosystembyiteratingachaotic map [J]Lecture NotesinComputerScience [5]Biham E Cryptanalysis of the chaotic-map cryptosystemsuggestedateurocrypt 91 [J]Lecture NotesinComputerScience [6]FridrichJGeerJReconstruction ofblurred orbits underfiniteresolution [J]JApplMathComp [7]FridrichJGeerJReconstruction ofchaoticorbits underfiniteresolution [J]JApplMathComp [8]Feldmann UHasler MSchwarz WCommunication bychaoticsignalstheinversesystemapproach [J] International Journal of Circuit Theory and Applications [9]Zhou HLingX TProblemswiththechaoticinverse systemencryptionapproach [J]IEEE TransCircuits SystemsI (3) [10]Zhou HLingX TYuJSecurecommunicationvia one-dimensional chaotic inverse systems [C ]// Proceedingsof1997IEEEInternationalSymposiumon CircuitsandSystemsVol2PiscatawayIEEEPress [11]LiSMouXCaiYetalOnthesecurityofachaotic encryptionschemeproblemswithcomputerizedchaos infinitecomputingprecision [J]ComputerPhysics Communications (1)52-58 [12]Baptista M SCryptographywithchaos[J]PhysLet A [13]JakimoskiGKocarev LAnalysisofsomerecently proposedchaos-basedencryptionalgorithms[j]phys LetA (6) [14]Wong KA fastchaoticcryptographicscheme with dynamiclook-up table [J] Phys Let A (4) [15]LiSChen GWong KetalBaptista-typechaotic cryptosystemsproblems and countermeasures [J] PhysLetA (5/6) [16]MachadoRFBaptista M SGrebogiCCryptography withchaosatthephysicallevel[j]chaossolitons & Fractals200421(5) [17]AlvarezGLiSBreakinganencryptionschemebased on chaotic Baker map [J] Phys Let A (1/2)78-82 [18]Blank M Pathologies generated by round-of in dynamicalsystems [J]Physica D199478(1/2) [19]Blank MDiscretenessandContinuityinProblemsof Chaotic Dynamics (Translations of Mathematical Monographs ) [ M ] Providence American MathematicalSociety [20]LiSChenGMouXOnthedynamicaldegradation of digital piecewise linear chaotic maps [J] InternationalJournalofBifurcationandChaos (10) [21]LiCLiSLoKetalAdiferentialcryptanalysisof Yen-Chen-Wu multimediacryptographysystem [J] JournalofSystemsandSoftware201083(8) [22]Orue A B Alvarez GPastor Get al A new parameterdeterminationmethodforsomedouble-scrol chaotic systems and its applications to chaotic cryptanalysis [J] Communications in Nonlinear Scienceand NumericalSimulations201015(11) [23]ArroyoDAlvarezGLiSetalCryptanalysisofa new chaoticcryptosystem based on ergodicity [J] InternationalJournal of Modern Physics B (5) [24]ArroyoDLiCLiSetalCryptanalysisofanimage encryption scheme based on a new total shufling algorithm [J]ChaosSolitons & Fractals (5) [25]AlvarezGLiSCryptanalyzinganonlinearchaotic algorithm (NCA ) for image encryption [J] Communicationsin Nonlinear Scienceand Numerical Simulations200914(11) [26]LiCLiSChenGetalCryptanalysisofanimage encryption scheme based on a compound chaotic sequence [J]Imageand Vision Computing (8) [27]Arroyo DRhouma R Alvarez Getal On the securityofanewimageencryptionschemebasedon chaotic map latices [J ] Chaos (3) [28]LiSLiCChen GetalCryptanalysisofRCES/ RSES image encryption scheme [J] Journal of SystemsandSoftware200881(7) [29]Alvarez G Montoya F Romera M et al Cryptanalysisofachaoticencryptionsystem [J]Phys LetA [30]Alvarez G Li S Some basic cryptographic requirements for chaos-based cryptosystems [J] InternationalJournalofBifurcationandChaos2006

10 (8) [31]ArroyoDAlvarezGLiSSomehintsforthedesign ofdigitalchaos-basedcryptosystemslessonslearned fromcryptanalysis [C]//Proceedingsofthe2ndIFAC Conference on Analysis and Control of Chaotic SystemsLondonQueen MaryUniversityofLondon 2009TD1 2 [32]Fridrich J Symmetric ciphers based on two- dimensionalchaoticmaps[j]intjbifurcationchaos [33]Pareek NPatidar VSud KCryptography using multiple one-dimensional chaotic maps [J ] Communicationsin Nonlinear Scienceand Numerical Simulation (7) [34]WeiJLiao X Wong KetalCryptanalysisofa cryptosystem using multiple one-dimensionalchaotic maps [J]Communicationsin NonlinearScienceand NumericalSimulation [35]Pareek N Patidar V Sud K Discrete chaotic cryptographyusingexternalkey [J]Phys Let A (1/2)75-82 [36]LiCLiSAlvarezGetalCryptanalysisofachaotic blockcipherwithexternalkeyanditsimprovedversion [J]ChaosSolitons & Fractals (1) [37]SolakEOkalCYildizO TetalCryptanalysisof Fridrich schaoticimageencryption [J]International JournalofBifurcationandChaos201020(5)1-9 [38]May R MSimple mathematical models with very complicated dynamics [J] Nature [39]Schuster H GJust W Deterministic chaos An introduction[m ]4th ed Weinheim Wiley-VCH VerlagGmbH & CoKGaA [40]Dito WMunakataTPrinciplesandapplicationsof chaoticsystems [J]Communicationsofthe ACM (11) ( 836 ) [10]PiresJrW RdePaulaFigueiredoT HWongH Cet al Malicious node detection in wireless sensor networks[c]//proceedingsofthe18thinternational Paralel and Distributed Processing Symposium PiscatawayUSAIEEEPress [11]MiQStankovicJAStoleruRSecurewalkingGPS Asecurelocalizationandkeydistributionschemefor wirelesssensornetworks[c]//proceedingsofthethird ACM conferenceon WirelessnetworksecurityNew YorkUSAACM [12]KhalilIBagchiSRotaru C Netal UnMask Utilizingneighbor monitoringforatack mitigationin multihop wireless sensor networks [J] Ad Hoc Networks20108(2) [13]KhalilIBagchi SShrof N BLITEWORP A lightweightcountermeasureforthewormholeatackin multihopwirelessnetworks [C]// Proceedingsofthe 2005InternationalConferenceon DependableSystems andnetworkspiscatawayusaieeepress [14]BianchiGPerformanceanalysisoftheIEEE80211 distributedcoordinationfunction [J]IEEEJournalon Selected Areas in Communications (3)

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