Modified Logistic Maps for Cryptographic Application
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1 Applied Mathematics, 25, 6, Published Olie May 25 i SciRes. Modified Logistic Maps for Cryptographic Applicatio Shahram Etemadi Borujei, Mohammad Saeed Ehsai Faculty of Computer Egieerig, Uiversity of Isfaha, Isfaha, Ira etemadi@eg.ui.ac.ir, ehsai@eg.ui.ac.ir Received 3 October 24; accepted 7 May 25; published 2 May 25 Copyright 25 by authors ad Scietific Research Publishig Ic. This work is licesed uder the Creative Commos Attributio Iteratioal Licese (CC BY). Abstract I this paper, defiitio ad properties of logistic map alog with orbit ad bifurcatio diagrams, Lyapuov expoet, ad its histogram are cosidered. I order to expad chaotic regio of Logistic map ad make it suitable for cryptography, two modified versios of Logistic map are proposed. I the First Modificatio of Logistic map (FML), vertical symmetry ad trasformatio to the right are used. I the Secod Modificatio of Logistic (SML) map, vertical ad horizotal symmetry ad trasformatio to the right are used. Sesitivity of FML to iitial coditio is less ad sesitivity of SML map to iitial coditio is more tha the others. The total chaotic rage of SML is more tha others. Histograms of Logistic map ad SML map are idetical. Chaotic rage of SML map is fivefold of chaotic rage of Logistic map. This property gave more key space for cryptographic purposes. Keywords Chaotic Map, Modified Logistic Map, FML Map, SML Map. Itroductio I order to explai simple chaotic dyamical systems, oe-dimesioal map is used. Tet, Beroulli ad Logistic maps are commo examples of them. The retur map of Tet ad Beroulli are liear, while Logistic map is oliear []-[3]. Oe dimesioal map is simple as far as hardware implemetatio is cocered. I the other had, security of oliear maps is usually more tha liear fuctios. Logistic map is geerally used i most of cryptosystems ad pseudo radom geerators. It is used i chaos-based secure commuicatio system ad for geeratios of biary umbers. Li, Mou, ad Cai proposed statistical properties of digital piecewise liear chaotic maps ad their roles i cryptography ad pseudo-radom codig [4]. Addabbo proposed performace aalysis ad optimized desig of piecewise liear chaotic maps such as digital saw-tooth ad tet maps as a source of pseudo- How to cite this paper: Etemadi Borujei, S. ad Ehsai, M.S. (25) Modified Logistic Maps for Cryptographic Applicatio. Applied Mathematics, 6,
2 radom bits geerators [5] [6]. Pareek used chaotic maps for radom bit geerator [7]. Shastry, Nagaraj, ad Vaidya proposed a geeralizatio of Logistic map, ad its applicatios i geeratig pseudo-radom umbers [8]. Basios, Forti, ad Gilbert proposed statistical properties of time-reversible triagular maps of the square [9]. The key elemet of the cryptosystems is key space. The key space is correspodig to chaotic rage. Sice chaotic rage of logistic map is small, its key space is also small. I this paper, with the aim of expadig chaotic rage of Logistic map, two modified versios of Logistic map are proposed. Defiitio ad properties of Logistic map are reviewed i Sectio 2. The first ad secod modified versios of Logistic map are proposed i Sectio 3. Retur maps, orbit diagrams, bifurcatio diagrams ad Lyapuov expoets were also cocered. Compariso of the modified map with Logistic with respect to their sesitivity to iitial coditios, their chaotic rage ad histograms is cosidered i Sectio 4. Fially, coclusio ad refereces are itegrated. 2. Logistic Map Logistic map is oe-dimesioal map which uses to model simple oliear discrete systems. Logistic map explai by a recursive fuctio as follows: ( ) ( ) x = Lrx, = r x + x () L, give by Equatio (), the pa-, 4. The retur map of Logistic fuctio is give i Figure for r = 4. Sesitivity of Logistic map to iitial coditio could be observed by plottig orbit diagrams with respect to two iitial coditios with small differece. The correspodig orbit diagrams with respect to two iitial coditios.35 ad.35 for fixed values of r = 4 is draw i Figure 2. There is suitable sesitivity to iitial coditio. I order to view chaotic properties of Logistic map, bifurcatio diagram ad Lyapuov expoet of it should be calculated ad plotted. Bifurcatio diagram of Logistic mapwith respect to r are calculated ad plotted i Figure 3. Lyapuov expoet of Logistic mapwith respect to r are also calculated ad plotted i Figure 4. Regardig Figure 3 ad Figure 4, Logistic map is chaotic whe parameter r lies i iterval [3.6, 4]. First, cofirm that you have the correct template for your paper size. where r is its parameter ad x [,]. Cosider Logistic map : [,] [,] rameter r lies i iterval [ ] 3. Modified Logistic Maps A discrete dyamic process is said to be two-segmetal if there exists a partitioig poit. The geeral equatio X(+) X() Figure. Retur map of logistic map with respect to r =
3 .9 x=.35 x= X() Figure 2. Orbit diagrams of logistic map with respect to two iitial coditios.35 ad.35 (r = 4) Figure 3. Bifurcatio diagram of logistic map with respect to r Figure 4. Lyapuov Expoet of Logistic map with respect to r. 775
4 of the process could be defied as depicted i Equatio (2), such that g( x ) ad ( ) had side fuctios, respectively [] []. ( ) ( ) g x, x < a; x+ = f ( x) = h x, x a. where x [,] ad a (,). The ecessary coditio for a two segmetal fuctio f { gh, } h x are the left ad right = to be a Lebesgue process is that the absolute value of slops must be greater tha uity all over the domai [2]. That is, the absolute of derivatives of two braches over the rage must be greater tha oe, Equatio (3). ( ) for : [,] f x > x (3) As far as Logistic map is cocered, its equatio could be separated as follows with respect to a =.5 : x + ( ) ( ) ( ) ( ) g x = r x x, x < a; = Lrx (, ) = h x = r x x, x a. h x. To h x. Actually, we modified secod part of Logistic map i order to improve chaotic rage of Logistic map i two maers. Cosiderig Equatio (4), the derivatives of g( x ) is exceedig uity, but this is ot true for ( ) solve this problem, we use symmetry ad trasform properties to modify ( ) 3.. First Modified Logistic (FML) Map g x aroud y = r 8, the trasform the result h x. The recursive equatio of First Modified Logistic (FML) map is defied i Equatio (5), where is a time idex, x is the iitial value, ad r is the cotrol parameter. We modified Logistic map, by obtaiig vertical symmetry of ( ) to right for x =.5, to geerate a ew ( ) x + where x [,], (, 4] ( ) ( ) ( ) ( ) ( ) g x = r x x, x <.5; = FML ( rx, ) = h x = r x.5 x.5 + r 4. x.5. r. The retur map of the result is draw i Figure 5 for r = 4. Orbit diagrams of FML map with respect to two iitial coditios.35 ad.35 for fixed values of r = 4 are draw i Figure 6. Sesitivity of FML map to iitial coditio is observed i the graph. There is ot suitable sesitivity to iitial coditio. Bifurcatio diagram of FML map with respect to r are calculated ad plotted i Figure 7. Lyapuov expoet of FML map with respect to r are calculated ad plotted i Figure 9. Accordig to Figure 7 ad Figure 8, FML map is chaotic whe parameter r lies i itervals [2.6, 2.9] or [3.2, 4]. (2) (4) (5) Figure 5. Retur map of FML map with respect to r =
5 .9 x=.35 x= X() Figure 6. Orbit diagrams of FML map with respect to iitial coditios.35 ad.35 (r = 4) Figure 7. Bifurcatio diagram of FML map with respect to r Figure 8. Lyapuov expoet of FML map with respect to r. 777
6 3.2. Secod Modified Logistic (SML) Map h x by symmetries ad trasformatio. Here, we fid vertical symmetry of = ad horizotal symmetry of the result with axis of symmetry x =.25. The trasformatio of the result to right with x =.25 is performed. Therefore, recursive equatio of Secod h x, ad is defied i Equatio (6): We make aother modificatio to ( ) g( x ) with axis of symmetry y r 8 Modified Logistic (SML) map is formig by modifyig ( ) g( x) = r x ( x), x <.5; x+ = SML ( rx, ) = h( x) = r x ( x ) + r 4, x.5. x is the iitial value, x [,] ad the cotrol parameters [,] where is a time idex, r. I order to explai the performace of Equatio (6), the retur map of the result is draw i Figure 9. The orbit diagrams of SML map with respect to two iitial coditios.35 ad.35 for fixed value of r = 4 are draw i Figure. Sesitivity of SML map to iitial coditio is observed i the figure. There is superior sesitivity to iitial coditio. Bifurcatio diagram of SML map with respect to r are calculated ad plotted i Figure. Lyapuov expoet of SML map with respect to parameter r are calculated ad plotted i Figure 2. Accordig to Figure ad Figure 2, SML map is chaotic whe parameter r lies i itervals [2, 4]. (6) Figure 9. Retur map of SML with respect to r = 4..9 x=.35 x= X() Figure. Orbit Diagrams of SML map with respect to iitial coditios.35 ad.35 (r = 4). 778
7 Figure. Bifurcatio diagram of SML map with respect to r Compariso Figure 2. Lyapuov expoet of SML map with respect to r. With the purpose of expadig chaotic rage, two modified versio of Logistic map are proposed. I order to compare the performace of the proposed maps with respect to applicatios, orbit diagrams, bifurcatio diagram, Lyapuov expoet ad histogram of outputs are cosidered. Orbit diagram shows the sesitivity of the map to iitial coditios. Bifurcatio diagram ad Lyapuov expoet is used to evaluate chaotic behavior of the maps. Histogram of outputs which could be plotted by observig outputs of large umber of iteratios, simulate probability desity fuctio of the maps. 4.. Sesitivity to Iitial Coditio (Orbit Diagram) I order to compare the sesitivity of the proposed maps, FML ad SML, with Logistic map to iitial coditio, their orbit diagrams with respect to two iitial coditios with small differece are cosidered. They are show i Figure 2, Figure 6 ad Figure, respectively. As it explaied earlier, sesitivity of FML to iitial coditio is less ad sesitivity of SML map to iitial coditio is more tha the others Chaotic Rage (Bifurcatio Diagram, Lyapuov Expoet) Bifurcatio diagram ad Lyapuov expoet of Logistic mapwith respect to r are plotted i plotted Figure 3 ad Figure 4, respectively. Meawhile, bifurcatio diagram ad Lyapuov expoet of FML map are also plot- 779
8 ted i Figure 7 ad Figure 8. Bifurcatio diagram ad Lyapuov expoet of SML map are also plotted i Figure ad Figure 2. Regardig the related figures, Logistic map is chaotic for r [ 3.6,4], FML map is chaotic for itervals [2.6, 2.9] ad [3.2, 4]. I additio, SML map is chaotic for rage of r [ 2, 4]. Therefore, the total chaotic rage of SML is more tha the others. The compariso of these values is depicted i Table. Chaotic rage of SML map is fivefold of chaotic rage of Logistic map Statistical Characteristics (Histogram) Simulatio of probability desity fuctio could be performed to show the statistical characteristics of the map. This simulatio is ru for, iteratios o the map ad draws its histogram. Figure 3 shows the histogram result of Logistic map for fixed parameter value of r = 4. The probability desity fuctio of the SML map is also simulated with, iteratios o SML map. Figure 4 shows the histogram of SML map for fixed value of r = 4. It is appearig that histograms of Logistic map ad SML map are idetical, while FML map could ot perform acceptable result. 5. Coclusios I order to evaluate the performace of Logistic map, after cosiderig defiitio ad properties of it, orbit diagrams, Lyapuov expoet ad histogram of Logistic map were cosidered. Orbit diagram showed that the sesitivity of Logistic map to iitial coditio was medium. Bifurcatio diagram ad Lyapuov expoet were used to evaluate chaotic properties of the map ad recogize the rage of parameters. The total chaotic rage of Logistic map was small. With the purpose of expadig chaotic rage, two modified versios of Logistic map are proposed. They are oe-dimesioal ad two-segmetal oliear maps. We called them First ad Secod Modified Logistic (FML & SML). We foud vertical symmetry of first segmet, trasformed the result to right for the secod segmet, ad called it FML map. Defiitio ad properties of FML map were also cosidered. Sesitivity of FML map to Table. Compariso of chaotic rages. Map Chaotic rage (out of 4) Chaotic rage ratio (%) Logistic.4 % FML.2 36% SML 2 5% Figure 3. Histogram of logistic map iteratios for r = 4. 78
9 Figure 4. Histogram of SML map iteratios for r = 4. iitial coditio is ot suitable. However, FML map is chaotic whe parameter r lies i itervals [2.6, 2.9] or [3.2, 4] accordig to the graphs of bifurcatio diagram ad Lyapuov expoet. To defie a secod versiomodified Logistic map, we foud vertical ad horizotal symmetry of its first segmet ad trasformed the result to right. Recursive equatio of Secod Modified Logistic (SML) map is formig. Sesitivity of SML map to iitial coditio is observed i the figure. There is superior sesitivity to iitial coditio. Accordig to Figure ad Figure 2, SML map is chaotic whe parameter r lies i itervals [2, 4]. It was cocluded that sesitivity of FML to iitial coditio was less ad sesitivity of SML map to iitial coditio was more tha the others. Histograms of Logistic map ad SML map are idetical, while FML map caot perform acceptable results. The total chaotic rage of SML is more tha the others. Chaotic rage of SML map is fivefold of chaotic rage of Logistic map. This property expads key space for cryptographic purposes. Refereces [] Alligood, K., Sauer, T. ad Yorke, J. (996) Chaos: A Itroductio to Dyamical Systems. Spriger-Verlag, New York. [2] Strogatz, S. (994) Noliear Dyamics ad Chaos. Perseus Books, Cambridge. [3] Schuster, H.G. ad Just, W. (25) Determiistic Chaos: A Itroductio. 4th Editio, WILEY-VCH Verlag GmbH, Weiheim. [4] Li, S., Li, Q., Li, W., Mou, X. ad Cai, Y. (2) Statistical Properties of Digital Piecewise Liear Chaotic Maps ad Their Roles i Cryptography ad Pseudo-Radom Codig. Cryptography ad Codig, 226, [5] Addabbo, T., Alioto, M., Berardi, S., Fort, A., Rocchi, S. ad Vigoli, V. (24) The Digital Tet Map: Performace Aalysis ad Optimized Desig as a Source of Pseudo-Radom Bits. Proceedigs of the 2st IEEE Istrumetatio ad Measuremet Techology Coferece, IMTC 4, 2, [6] Addabbo, T., Alioto, M., Berardi, S., Fort, A., Rocchi, S. ad Vigoli, V. (24) Hardware-Efficiet PRBGs Based o -D Piecewise Liear Chaotic Maps. Proceedigs of the th IEEE Iteratioal Coferece o Electroics, Circuits ad Systems, ICECS 24, 3-5 December 24, [7] Pareek, N., Patidar, V. ad Sud, K. (2) A Radom Bit Geerator Usig Chaotic Maps. Iteratioal Joural of Network Security,, [8] Shastry, M., Nagaraj, N. ad Vaidya, P. (26) The B-Expoetial Map: A Geeralizatio of the Logistic Map, ad Its Applicatios i Geeratig Pseudo-Radom Numbers. eprit arxiv.org:cs/6769. [9] Basios, V., Forti, G.L. ad Gilbert, T. (29) Statistical Properties of Time-Reversible Triagular Maps of the Square. Joural of Physics A: Mathematical ad Theoretical, 42, [] Huag, W. (25) Characterizig Chaotic Processes That Geerate Uiform Ivariat Desity. Chaos, Solitos & Fractals, 25,
10 [] Aiki, V., Arkadaksky, S., Kuptsov, S., Remizov, A. ad Vasileko, L. (28) Lyapuov Expoet for Chaotic D Maps with Uiform Ivariat Distributio. Bulleti of the Russia Academy of Scieces: Physics, 72, [2] Huag, W. (25) Costructig a Opposite Map to a Specified Chaotic Map. Noliearity, 8,
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