Generalized logistic map and its application in chaos based cryptography

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1 Journal of Physics: Conference Series PAPER OPEN ACCESS Generalized logistic ma and its alication in chaos based crytograhy To cite this article: M Lawnik 207 J. Phys.: Conf. Ser View the article online for udates and enhancements. This content was downloaded from IP address on 02//208 at 09:57

2 Generalized logistic ma and its alication in chaos based crytograhy M Lawnik Faculty of Alied Mathematics, Silesian University of Technology, ul. Kaszubska 23, Gliwice, Poland marcin.lawnik@olsl.l Abstract. The logistic ma is commonly used in, for examle, chaos based crytograhy. However, its roerties do not render a safe construction of encrytion algorithms. Thus, the scoe of the aer is a roosal of generalization of the logistic ma by means of a wellrecognized family of chaotic mas. In the next ste, an analysis of Lyaunov exonent and the distribution of the iterative variable are studied. The obtained results confirm that the analyzed model can safely and effectively relace a classic logistic ma for alications involving chaotic crytograhy.. Introduction Chaotic mas are widely used in many science branches, e.g. [-4]. One of the most commonly used chaotic dynamical system in ractice is the logistic ma (LM) given by the following exression: xk+ = 4qxk ( - xk ), () where q Î[ 0, ] and x Î [ 0,]. Its universal nature is certified by many alications, among others, in crytograhy based on chaos theory, e.g. [5,6]. Chaos based crytograhy regards chaotic mas as generators of seudo-random numbers, and the values of their initial conditions and arameters as the secret keys. From such oint of view, recurrence () has many disadvantages, which were indicated in rofessional ublications [7]. They include, among others, a small sace of allowable values of arameter q, not uniform distribution of the iterative variable, or unstable value of Lyaunov exonent. The Lyaunov exonent denotes the sensitivity of dynamical system x k+ = f ( x k ) to changes in the initial conditions and is calculated in accordance with the deendence: n å - k= 0 l = lim ln f '( xk ). (2) n n In the above-mentioned alications, it is definitely recommended that the values of arameter q of () are close to. This results from its distribution, which for such set of values is flat in the middle, and determined by the density function: r ( x) = (3) x ( - x) Furthermore, for q = the Lyaunov exonent for () reaches the maximum value equal to l = ln 2 among all allowable values of arameter q. The above results show that the admissible values of arameter q are considerably limited. This has a negative imact on the safety of algorithms and may Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by Ltd

3 lead to successful brutal attack. In view of this, the designation of a chaotic ma that could combine the simlicity of () and, at the same time, render much better roerties, seems relevant. In [8] the following family of chaotic functions was resented: ì æ r ö ï ç - xk q -, 0 < x < ï ç x è ø k + = í, (4) ï æ r ö ç - xk ïq -, x < ç ï - î è ø where, q, r reresents arameters. If = 0.5, r = 2 and q Î[ 0, ], ma (4) may be reduced to LM (). Thus, LM is a secial case of (4). In [8] only for selected values of arameters, q and r an casual analysis of Lyaunov exonent of (4) was resented, indicating the domains for which it has a ositive value. From the oint of view of above mentioned alications, such analysis is insufficient. 2. The generalized logistic ma The generalized logistic ma (GLM) can be given by the formula: ì - q 2 ( ) ï - x +, 0 2 k q xk x k+ = í, (5) - q 2 ï ( - x ) +, < 2 k q xk ïî ( - ) where Î( 0, ) and q Î[ 0, ]. GLM (5) is a secific case of (4) for the arameter value r = Analysis In figure, the grah of Lyaunov exonent for the values of arameters and q is resented. As shown in the grah, a large range of the domain [ 0.89,] [ 0,] has a ositive value of Lyaunov exonent. Furthermore, it may be reasoned that for the value of arameter q close to and the entire variation range of arameter, the GLM (5) has a ositive Lyaunov exonent, as shown in detail in figure 2. Accordingly, the analysis of Lyaunov exonent for this ma renders a considerably wider range of the allowable values in comarison with a LM (). 2

4 Figure. Grah of the Lyaunov exonent of GLM (5). The non-ositive value of the exonent is marked in black. As indicated in figure 2 the value of Lyaunov exonent is stable for arameter q close to. This means that Lyaunov time, which determines the number of iterations that the system requires to forget the initial condition, is also stable. In alications, Lyaunov time determines how many initial iterations of the system must be rejected, in order to obtain a chaotic solution, which has also an imact on the calculations effectiveness of the algorithms involving crytograhy. Figure 2. The grah of the Lyaunov exonent of the GLM (5) for values of arameter q close to. The distribution of the iterative variable for the value of arameter q close to is flat in the middle (figure 3), just as in a LM (). However, in the discussed examle, the distribution is the same for the entire variation of arameter, and not only for one value. This significantly extends the range of the allowable arameters, and, at the same time, enhances the safety of the algorithm - see figure 4. Figure 3. Distributions of the GLM (5) obtained in a numerical way for values of arameter q close to and =

5 4. Conclusions Figure 4. Distribution of GLM (5) obtained in a numerical way for set values of arameter and q = (continuous line). The broken line marks density function (3). The analysis of the generalized logistic ma with the use of function family (4) was conducted. The resented calculations indicated that the analyzed model has much better roerties than a classic logistic ma, such as wider range of the allowable arameters with a ositive value of Lyaunov exonent and flat distribution of the iterative variable in the middle art. From the oint of view of crytograhy based on the chaos theory, the roerties of the analyzed function rovide its successful alication. On the other hand, the analyzed model combines the simlicity of its classic corresondent. This should lead to the relacement of a logistic ma by its generalized form in crytograhy alications. References [] Berezowski M and Lawnik M 204 Identification of fast-changing signals by means of adative chaotic transformations Nonlinear Analysis: Modelling and Control 9(2) [2] Lawnik M and Berezowski M 204 Identification of the oscillation eriod of chemical reactors by chaotic samling of the conversion degree Chem. Process Eng. 35(3) [3] Lawnik M 204 Generation of numbers with the distribution close to uniform with the use of chaotic mas Proceedings of the 4th International Conference on Simulation and Modeling Methodologies, Technologies and Alications [4] Lawnik M 204 The aroximation of the normal distribution by means of chaotic exression Journal of Physics: Conference Series [5] Batista M S 998 Crytograhy with chaos Physics Letters A [6] Pareek N K, Vinod Patidar and Sud K K 2006 Cover image Image encrytion using chaotic logistic ma Image and Vision Comuting 24(9) [7] Arroyo D, Alvarez G and Fernandez V 2008 On the inadequacy of the logistic ma for crytograhic alications eds. L Hernandez A Martin X Reunin (Esa nola sobre Critolog ıa y Seguridad de la Informaci on) [8] Skrobek A 2007 Metoda rojektowania dyskretnych chaotycznych szyfrów strumieniowych oarta na krytoanalizie Phd thesis (in olish) 4