Zooming into chaos for a fast, light and reliable cryptosystem
ZZooming into chaos for a fast, light and reliable cryptosystem
Jeaneth Machicao, a) Odemir M. Bruno, b) and Murilo S. Baptista c) Scientific Computing Group. São Carlos Institute of Physics, University of São Paulo, São Carlos - SP, PO Box 369, 13560-970,Brazil. Institute for Complex Systems and Mathematical Biology, University of Aberdeen, AB24 3UX, Aberdeen,UK. (Dated: 13 January 2020)
In previous work, the k -logistic map [Machicao and Bruno, Chaos, , 053116 (2017)] was introduced as a transforma-tion operating in the k less significant digits of the Logistic map. It exploited the map’s pseudo-randomness characterthat is present in its less significant digits. In this work, we comprehensively analyze the dynamical and ergodic aspectsof this transformation, show its applicability to generic chaotic maps or sets, and its potential impact on enabling thecreation of a cryptosystem that is fast, light and reliable. Motivated by today’s huge volume of data that needs to behandled in secrecy, there is a wish to develop not only fastand light but also reliably secure cryptosystems. Chaos al-lows for the creation of pseudo-random numbers by low-dimensional transformations that need to be applied onlya small number of times. These two properties translateinto a chaos-based cryptosystem that is both fast (shortrunning time) and light (little computational effort). Thereliability of security in a chaos-based cryptosystem is sus-tained by the exponentially fast decay of the correlationof points in a chaotic trajectory. However, chaos is deter-ministic, and as such, a sufficiently long observation of thetrajectory or its symbolic encoding can reveal its past andfuture evolution, thus breaking security. That is a knownweakness of chaos in cryptography. However, this vulner-ability can be compensated by another still not much ex-plored the peculiar property of chaos. Look at the less sig-nificant digits of a chaotic trajectory and surprisingly theinformation content of past and future vanishes exponen-tially fast. What we propose here is an enhanced chaos-based cryptosystem that uses the “deep-zoom” transfor-mation on the top of a chaotic map to improve the relia-bility of security, but without compromising on the speedand weight of the cryptosystem. We use low-dimensionalchaotic maps to quickly generate numbers that have littlecorrelation, and then we quickly (“fast”) enhance secrecyby several orders (“reliability”) with very little computa-tional cost (“light”) by simply looking at the less significantdigits of the initial chaotic trajectory. This paper demon-strates this idea with rigour, making a comprehensive er-godic characterization of this procedural strategy to createpseudo-random numbers that can sustain a fast, light andreliable cryptosystem. a) Electronic mail: [email protected] b) Electronic mail: [email protected] c) Electronic mail: [email protected]
I. INTRODUCTION
The secrecy in chaos-based cryptosystems relies on mathe-matical transformations that generate a trajectory whose cor-relation decays rapidly. The correlation of chaotic trajectorieswill always decay to zero after a sufficiently long time. Thisis due to the mixing property that allows nearby points to bequickly mapped anywhere in the transformation domain, anddue to the sensibility to the initial condition chaotic transfor-mations have. In fact, the speed of correlation decay and thesensibility to the initial conditions quantified by the Lyapunovexponent are intimately connected . A chaotic system witha very large positive Lyapunov exponent is thus desirable forcryptography , since it allows for very rapid decay of corre-lations. Moreover, chaotic signals can be generated by low-powered, small area and simple integrated as well as analogcircuits operating in very large frequency bandwidths.Cryptosystems need to perform heavy calculations. For ex-ample, chaos-based block ciphers such as those that en-crypt images, movies and audio employ a series of complexmathematical transformations over too many bits of informa-tion. If one wants a light cryptosystem that can be run inany portable devices or that can be considered even for mas-sive streaming, the use of real numbers with higher precisionshould be avoided. To improve on the performance of chaos-based cryptosystems, the underlying chaotic transformationhas been discretized by considering transformations operatingon an integer domain. Discretization can preserve importantergodic properties as the mixing property and the sensibility tothe initial conditions, but might also create spurious periodiccycles of low-period , which result in correlations weaken-ing the security of cryptosystems that rely on these transfor-mations. Even chaotic transformations (such as the Bernoullishift map) acting on real numbers with finite resolution mightcreate spurious periodic cycles due to numerical errors.With recent advances, it is relatively easy today to performnumerical computation with arbitrary precision, and thus cur-rent efficient cryptosystems can rely on maps with real arith-metics of higher precision. However, any meaningful encod-ing of the chaotic trajectory that allows decoding, such asthose used to create a pseudo-random number (PRN) gener-ator or binary secret keys, would be strongly correlated withthe most significant digits of the trajectory. To create a stream a r X i v : . [ n li n . C D ] J a n cipher based on chaos , where a binary information stream isencoded by XOR transformation to a binary secret key createdby encoding a chaotic trajectory, Gerard Vidal Cassanya hasproposed the use of the less significant digits of a trajectoryobtained from a higher-dimensional chaotic system of ODEsto create the binary secret key. The idea of using the less sig-nificant digits of chaotic trajectories has appeared before inthe work of Ref. , however it was in Ref. (and other previ-ous patent applications cited within) that less significant digitswere taken by a transformation that this work claims to be op-timal to support a fast, light and reliable cryptosystem.Inspired by today huge volume of data that needs to be han-dled in secrecy, there is a desire to develop not only fast (quickrun time) and light (little computational cost) but also reliable(highly entropic, sensitive to the initial conditions, low corre-lation) cryptosystems. An important aspect of a cryptosystemis its initialization. For example, one might employ a PRNto choose parameters. Secret keys, which can be created fromPRNs, are also used to encrypt the information and represent acore operation in any cryptosystem. Any innovative inventionthat creates reliable PRNs or secret keys with optimized useof computational resources will contribute tremendously to aworld that wants to communicate massive amounts of infor-mation, but securely. PRNs are not only important for secrecyin communication. It is also fundamental to the functioningof several autonomous machines, toys, and they are essentialfor several numerical algorithms. This work demonstrates thatlooking at the less significant digits of chaotic trajectories isindeed a pathway for the creation of fast, light and reliablePRNs.The work of Ref. has analyzed the dynamics and thestatistical properties of the deep-zoom transformation to achaotic trajectory, a transformation that takes up the less sig-nificant digits of a real number. This transformation applied tothe Logistic map regarded as the k -logistic map was definedby the less significant digits located at k digits to the rightof the decimal point. It was shown that a PRN based on the k -logistic map has strong properties regarding statistical ran-domness tests DIEHARD and NIST, and thus demonstratingfrom a statistical perspective that the k -logistic map can sus-tain secure cryptosystems. The k -logistic map takes advantageof not only having trajectory points with arbitrarily large pre-cision, and thus within principle no detectable spurious cycle,but also on hiding the information about the most significantdigits, which could reveal information about the algorithm be-hind the generation of the PRNs.The interest in the present work is to understand how thedeep-zoom transformation changes a particular map ergodicproperties such as its space partition, density measure, Lya-punov exponent, Topological and Shannon’s entropies. Ourresults, mostly illustrated by how the deep-zoom transforma-tion operates into the Logistic map are valid to generic 1Dchaotic maps or a set of numbers generated by any other pro-cess. The deep-zoom transformation is a complementary op-eration to chaos-based cryptosystem: we first quickly gener-ate a chaotic trajectory by a low-dimensional map, and thenwe use the deep-zoom transformation to quickly and lightlyenhance security. This is our strategy for the creation of a fast, light and reliable chaos-based cryptosystem.Our first result is to demonstrate that the k -deep-zoom ( k -DZ) transformation to a point x is mathematically equivalentto iterating for k times the decimal shift map (DSM) .This map is well known, and it is since decades consideredto be a mathematical toy model to demonstrate how a shiftinto the less significant digits results in strong chaos. De-spite its tremendous appeal due to the nice way this map dealswith decimal digits, scientists working with encryption basedon nonlinear transformations have focused their attention onother more known similar maps, such as the Bernoulli shiftmap or the Baker’s map, instead of the DSM. The main dif-ference is whereas the DSM operates by shifting the decimalnumbers, Bernoulli shift and Baker’s map shift the binary se-quence encoding the real numbers of the trajectory.Then, we demonstrate that by applying the k -DZ transfor-mation only once to generic chaotic trajectories, the mappedtrajectories will approach a uniform invariant measure fora sufficiently large but in practice small k , thus requiringmuch less computational effort to create numbers with uni-form statistics, a standard requirement for reliable PRNs. Theconvergence to the uniform invariant measure also dictates theconvergence of the Lyapunov exponent (LE) to the Topolog-ical and Shanon’s entropy of the mapped trajectories, indi-cating that the transformed points have achieved the largestsensibility to the initial conditions that is possible. Having atrajectory for which the level of chaos is the same as the levelof entropy means that uncertainty about the past and the fu-ture is as large as one could wish for the particular chaoticmap being considered. Moreover, all these quantities are lin-early proportional to k , thus implying that randomness (higherentropy) and the sensibility to the initial conditions (large LE)can be trivially increased by the resolution with which a trajec-tory is observed, and not by increasing a systems dimensionor by considering higher-order iterates of the map onto itself,operations that would require computational resources.Throughout this paper, we will show how this map amazingproperties applied to any 1D chaotic systems with finite prob-ability measure allows for a clear path to the creation of fast(quick run time, low number of iterations), light (little com-putational effort, low-dimension) and reliable (uniform statis-tics, strongly sensitive to the initial conditions, high entropy)pseudo-random numbers or symbolic secret keys, thus sup-porting fast, light and reliable chaos-based cryptosystems. II. THE k -DEEP-ZOOM (K-DZ) TRANSFORMATIONAND ITS EQUIVALENCE TO THE DECIMAL SHIFT MAP(DSM) Given a 1D map f ( x ) defined in a domain [ a , b ] and produc-ing an orbit O ( x ) = { x , x , . . . , x t } generated by the initialcondition x , with a given invariant density ρ ( x ) and prob-ability measure µ ( x ) , such that for an interval ε ∈ [ a , b ] wehave that µ ( ε ) = (cid:82) x ∈ ε ρ ( x ) dx , the k -DZ transformation φ k ( x ) was defined in by φ k ( x ) = x k − (cid:98) x k (cid:99) , (1)where (cid:98) (cid:99) stands for the floor function. In Ref. , and mo-tivated mostly for practical reasons, a parameter L was con-sidered which set the number of less significant digits for thefunction φ k ( x ) . In here, we drop this definition, and assumethat L → ∞ , or is a large number.This map can be analogously described by φ k ( x ) = k ( x , mod 10 − k ) . (2) φ k ( x ) = k x , mod 1 . (3)The DSM map is defined by D ( x ) = x , mod 1 , (4)and its k -folded version (the k -th iteration of D ) which werepresent by D k is basically D k ( x ) = k x , mod 1 , (5)which is exactly equal to Eq. (3). Thus, the k -fold DSM mapis mathematically equal to the k -DZ transformation.To illustrate the action of the DZ transformation, given thevalue x = . φ k = ( x ) = . . A. The k -DZ transformation and others maps in literature The idea of using a cryptosystem based on mod transfor-mations that extract the less significant digits of real numbersgenerated by chaotic systems was to the best of our knowledgefirst proposed in Ref. . Given a real number x n generated bya chaotic system (discrete or continuous), this work has pro-posed to cipher x by R n ≡ Ax n , mod S , (6)with A and S representing arbitrary constants.Equation (4) can be seen as a particular case of Eq. (6), butnot of the Eq. (3) because the k-DZ transformation introducesthe extra parameter k .Our work shows when this parameter can generate suit-able PRNs. However, in the work of Ref. , only the case for A = and S =
256 was studied, and without the rigour anddeepness presented in the present work to study the ergodicand dynamic manifestations of these transformations. Thechoice of S = R n into a two-dimensional gray-scale image for further process-ing. This choice, however, is not optimal for the security ofthe encryption, measured in terms of the entropy and sensibil-ity to the initial conditions. The optimal choice, demonstratedfurther, is obtained for S =
1, as in Eqs. (3) or (5). The choicemade of A as an arbitrary constant is also per se not always op-timal to extract the less significant digits, unless this arbitraryconstant is of the form A = k , as in Eq. (3). III. THE LOGISTIC MAP
The logistic map has been extensively studied over the pastyears . Since it is a well know system and that producestypical chaotic behaviour , we focus the application of the k -DZ transformation to trajectory points being iterated by thelogistic map, which we refer as the k -logistic map, adoptingpreviously defined terminology. It is described by f ( x t + ) = bx t ( − x t ) , (7)where x t ∈ [ , ] . In Eq. (7), x t ∈ ℜ , and as such each tra-jectory point is assumed to have infinite precision. However,in practice, x t has finite precision, but this does not preventone from solving Eq. (7) by numerical means. The shadow-ing lemma guarantees that numerical solutions of this mapare stable even if trajectory points have finite resolution, inthe sense that the numerical trajectory will remain close to atrue trajectory for a very long time, this time depending on theresolution of the trajectory considered. IV. ANALYSISA. Phase space, partition, and topological entropy
Equation (2) is useful because it provides the key to cal-culate the location of the partition points, where the map be-comes discontinuous. The points of discontinuities happen atthe boundaries created by the mod function, so at multiplesof 10 − k , more specifically at m − k , for m ∈ N and m ≤ k .There will be then 10 k discontinuous intervals. Figure 1 showthe original Logistic map with b =4 ( k =
0, left panel), and thecorresponding k -DZ transformation for k = k = k = x ∗ where discontinuitiesappear can be obtained by solving the following equation x ∗ ( m ) = m − k . (8)We can define a topological entropy of the k -DZ transfor-mation, which is an upper bound for the Shannon-entropy, bythe Boltzmann entropy of gas measuring the entropy of it interms of the number of observable states. Here, we can definethe states as being the fall of a trajectory point into an inter-val within the partition provided by Eq. (8). Regardless ofthe value of b , and actually regardless of which kind of 1Dchaotic map is used, this number is given by the number ofpartition points of the k -DZ transformation and it is equal to10 k , which is also the number of possible symbolic sequencesthat the k -logistic map produces. It is given by H T = k ln ( ) . (9)It is useful to compare this result with the topological en-tropy of the original Logistic map, defined in terms of the FIG. 1. The k -DZ transformation applied to the trajectory points of the Logistic map. From left to right panels, for k = k = k =
2, and k = number of subintervals in its generating Markov partition, andequal to 2 o , where o ∈ N is the order of the partition represent-ing the resolution of the subintervals composing the partition(measuring 2 − o in length). That results in that H T = o ln ( ) .Here we see an advantage of the use of the k -logistic map toproduce efficient pseudo-random numbers in a light fashion,so without requiring too expensive computational resources.Assuming o and k to be of the same order, the topological en-tropy of the k -logistic map be ln ( ) / ln ( ) larger than that ofthe Logistic map.It is also worthwhile to compare the result in Eq. (9)with the topological entropy, H ( S ) T , obtained by applying Eq.(6). Defining A = k , we obtain that the topological entropyequals H ( S ) T = k ln ( ) − ln ( S ) for Eq. (6). Thus, the entropyachieved in Eq. (9) for the k -DZ transformation in Eq. (3) canonly be achieved by applying Eq. (6) to that same chaotic setif S = B. k -logistic map probability density One of the most important characteristics of a goodPRN generator is that successive output values of it, say u , u , u , . . . are independent random variables from the uni-form distribution over the interval [0, 1]. It was shown in that as k increases the probability distribution of the map be-comes more and more uniform. This is reproduced in Fig. 2,in terms of the histogram (frequency) analysis. As can be seenin this figure, for k = b =4, with ahigh probability of finding points close to 1 and 0. As k growswith k = k =
2, and k =
3, the distribution tends to becomeincreasingly uniform, as can be observed in the Figure 2(b).In (c) we show a magnification of (b) for the region close tozero.
C. The natural invariant measure of the k -DZtransformation and its Shannon’s entropy To calculate the asymptotic Lyapunov exponent of the k -DZ transformation, which is independent on the choice ofthe chaotic map, we notice that the k -DZ is piecewise lin-ear, wherein each partition sub-interval the map has a con- stant derivative function. Arranging the values of x ∗ ( i ) inEq. (8) in a ranking of crescent order and indicating it by,i.e., x ∗ ( m ) ≡ x ∗ i such that x ∗ i + > x ∗ i , each partition subintervalcomprises the interval d i = [ x ∗ i , x ∗ i + [ , (10)for i ∈ N and i = [ , , . . . , k − ] .The derivative of the piecewise-linear map for each sub-interval d i can be calculated by ω i = | d i | − = − k , (11)since φ k ( d i ) =
1, where | d i | represents the length of the sub-partition d i .The evolution of an arbitrary initial probability measureto a 1D nonlinear transformation is dictated by the Perron-Frobenious operator. For piecewise linear systems, thePerron-Frobenious operator can be cast in terms of a linearsystem of equations operating in each subinterval of the mappartition. The k -DZ transformation takes as the initial mea-sure generated by the nonlinear Logistic map and then ap-plies k times the DSM . If we assume that the measure ineach subinterval of the k -DZ is uniform (which initially willbe not) and we represent it by the component [ µ ] i of the vector µ ( i = { , . . . , n } ) with n = k , and we define the density ineach interval as given by ρ i = µ i d i (12)an equation for the evolution of the non-normalized density atiteration t can be obtained . Z ρ (cid:48) t = ρ (cid:48) t + , (13)where the square matrix Z with component [ Z ] i j is the re-ciprocal of the absolute value of the slope of the map takingthe measure from the interval j to the interval i and can bedefined by a matrix with equal rows as Z = ω − ω − . . . ω − n − ω − n ω − ω − . . . ω − n − ω − n . . . . . . . . . . . . . . . . . . . . . . . . ω − ω − . . . ω − n − ω − n . F r e q uenc y F r e q uenc y x t x t x t k=0 k=1k=2k=3 k=0k=1k=2k=3 (a) (b) (c) FIG. 2. Frequency distribution curves for a) the original logistic map, b) the k − -logistic map with k = k = k = b = x t ∈ [ , ] (500 bins) and the vertical axis shows the frequency of the 10 values discarding the first 10 transientvalues. The curves represent the mean and standard deviation (shaded error bar) for sequences generated over 100 random initial conditions.c) The inset plot depicts a zoom on the windows x ∈ [ , . ] for these 4 plots. Equation (13), representing how the measure evolves con-cerning only 10 k intervals are valid if the measure and thedensity is uniform for every sub-partition d i since it has beenderived from the continuous Perron-Frobenious operator in-tegrated over intervals where the measure was assumed to beconstant. If the initial measure is not uniform for each sub-partition interval, as it is the case since the initial measurewas generated by the logistic map, we either should considerthe continuous operator (effectively described by an infinitedimension matrix) or alternatively, we can adopt a much sim-pler strategy. We take Eq. (13) and study it in the limit, when t → ∞ .Defining the vector d = { d , d , . . . , d n } and the diagonalmatrix D = I d , the element [] i j of the matrix D Z D − repre-sents the percentage of the measure in the interval d j that goesto the interval d i . The matrix Z has equal rows because thepiecewise equivalent of the k -logistic map takes measure fromeach interval to all others with the same proportion in each ofthe intervals d j .The equilibrium point of Eq. (13) is obtained when Z ρ ∗ = ρ ∗ , (14)which means that the time invariant density is a normalizedeigenvector of Z .The matrix Z is a stochastic matrix, since it is a non-negative matrix and the sum of all elements in a row totals1. This is easy to see since ∑ i ω − i = ∑ i d i = . (15)The Perron-Frobenious theorem guarantees that a squarestochastic matrix has a unique dominant real unitary eigen-value, with all other eigenvalues smaller than 1. This meansthat the density of the k -DZ transformation in the limit of k → ∞ is natural (it is unique), regardless of the initial prob-ability measure that is fed into the k -DZ transformation. Thenatural density can be recovered by proper normalization di-viding ρ ∗ by ∑ i [ ρ ∗ ] i d i so that the physical natural density in each interval is given by [ ρ ] i = [ ρ ∗ ] i ∑ i [ ρ ∗ ] i d i . (16)This is to guarantee that the density produces the natural mea-sure by Eq. (12).It is also easy to see that the unique unitary eigenvalue hasassociated to it a uniform eigenvector with all componentsequal to a constant value c : ρ ∗ = [ c c c . . . , c ] T , so, the piece-wise k -DZ transformation has a uniform density given by [ ρ ] i = c ∑ i [ c ] d i = . (17)This leads us to an invariant natural measure in each intervalthat equals the Lebesgue measure of the interval, and thus µ i = d i = − k . (18)So, for sufficiently large k , it is to be expected that the k -logistic map will have a uniform natural invariant density, al-though the density of the Logistic map is not uniform for eachinterval. In practice, this sufficiently large number is around k =4, when this map generates PRNs with all the good statis-tical characteristics for security . Being invariant means thatany initial probability measure will eventually evolve to thesame invariant measure. Thus, the reliability of the securityfor the PRNs generated by the k -DZ transformation is substan-tially more dependable on the properties of the DSM, than onthe statistical properties of the chaotic set of points being iter-ated by the k -DZ transformation, or also on the chaotic mapconsidered to initially generate the chaotic trajectory to be fedinto the k -DZ transformation. Since the invariant measure ofthe k -logistic map is constant (for sufficiently large k ), thismeans that any encoding supported by the partition definedin Eq. (8) will produce equiprobable symbols, this renderingcryptoanalysis based on frequency statistics to be inappropri-ate.The asymptotic Shannon’s entropy of the k -DZ transforma-tion is therefore equal to the Topological entropy: H S = − n ∑ i = µ i ln µ i = − n ∑ i = d i ln d i = H T (19) D. The Lyapunov exponent of the k -DZ transformation The Lyapunov exponent (LE) of the k -DZ transformationcan always be calculated regardless of the chaotic map be-ing used as the generator of the initial measure. This is sobecause the map is piecewise linear with constant derivativeeverywhere (except the partition points). The Lyapunov ex-ponent can be calculated by λ = (cid:90) ln (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) d φ k ( x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:33) d µ (20)where d µ = ρ ( x ) dx , represents the invariant measure of the k -DZ transformation.The chaotic map has its own domain of validity. This do-main must be normalized to fit within the domain of the k -DZtransformation. For the Logistic map, the domain is [ , ] , thesame as the domain of the k -DZ transformation. Therefore, itsLE is equal to λ = (cid:90) ln (cid:18)(cid:12)(cid:12)(cid:12) k (cid:12)(cid:12)(cid:12)(cid:19) dx = k ln ( ) = H T . (21)So, we see that for a sufficiently large k , the k -DZ transfor-mation produces a LE that approaches the topological entropywhich is also equal to Shannon’s entropy. A light cryptosys-tem that does not require much computational effort demandsthe use of transformations that can be as entropic as possibleand with the largest as possible sensibility to the initial condi-tions (which implies in a quick decay of correlation).When compared the LE of the k -DZ transformation in Eq.(3 (result in Eq. (20)) with the LE of the transformation Eq.(6) (proposed in Ref. ),assuming A = k , we notice that Eq. (6) can be equiva-lently written as R n = S (cid:32) k S x n , mod 1 (cid:33) , (22)which can be rewritten as R n = S Φ ( x n ) , (23)where Φ ( x n ) = k S x n , mod 1. Noticing that the LE of thefunction Φ ( x n ) is the same as the one obtained if Φ ( x n ) ismultiplied by a constant, then the LE of Eq. (6) is equal to λ ( S ) = ln (cid:32) k S (cid:33) = H ( S ) T . (24) Thus, the LE of Eq. (6) is only equal to the one of Eq. (3),if S =
1. In the result of Eq. (24), we have assumed thatthe speed of convergence of the probability density measure of Eq. (6) is the same as the one of Eq. (3). This is to beexpected, since the second largest eigenvalue of the matrix Z regulating the evolution of the density measure for Eq. (3) isthe same as the one for this matrix regulating the evolution ofthe density measure for Eq. (6), and both are equal to zero. E. Enhancement of sensibility to the initial conditions of the k -logistic map The LE of the k -DZ transformation does not depend on thechoice of the chaotic map generating the measure. It is never-theless interesting to understand how much chaos is enhancedby the application of the k -DZ transformation into a chaoticmap. Considering this chaotic map to be the logistic map (Eq.(7)), we then want to understand how much chaos is enhancedif the DZ transformation with k = φ k ( f ( x t )) , whose state space ( φ k ( f ( x t )) × x t ) is shown in Fig.3. Additionally, Fig. 4 show a colored version of this previouspicture for parameters in region b ∈ [ . , ] .This map is described by φ k ( f ( x )) = k ( f ( x t ) , mod 10 − k ) . (25)Its LE can be calculated by λ = (cid:90) ln (cid:32)(cid:12)(cid:12)(cid:12)(cid:12) d φ k ( f ( x )) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:33) d µ (26)where d µ = ρ ( x ) dx now represents the measure of the Logis-tic map.The first derivative of the map in Eq. (25) is d φ k ( x ) dx = k b ( − x ) , (27)whereas its density for b = ρ ( x ) = π − [ x ( − x )] / (28)Placing Eqs. (27) and (28) in Eq. (26) and integrating overthe map domain ( x ∈ [ , ] ), we obtain that λ = (cid:90) k ln 10 + ln 4 + ln | ( − x ) | π (cid:112) x ( − x ) dx = k ln ( ) + ln ( ) , (29)since (cid:90) ln | ( − x ) | π (cid:112) x ( − x ) dx = ln 2 . FIG. 3. The k -logistic map state space. From left to right panels are shown k = k = ... , k = b =
4. The horizontal andvertical axes show the state space of x kt against φ k ( f ( x t )) . Each orbit contains 10 points starting from random initial conditions.FIG. 4. The k -logistic map state space. From left to right panels are shown k = k = ... , k = b ∈ [ . , ] . The horizontal andvertical axes show the state space of x kt against φ k ( f ( x t )) . Each orbit contains 5 × points starting from random initial conditions. So, the first thing to notice is that the LE of the map in Eq.(25) is equal to the LE of the 1-logistic map plus the LE of theoriginal logistic map for b = ( ) ). Thistells us that when creating a cryptosystem based on a chaoticmap, more entropy and sensibility to the initial conditions canbe achieved by a simple inspection to the least k significantdigits, instead of doing more iterations in the chaotic map gen-erating the initial chaotic sequence. This analysis can be easily extended to the logistic map op-erating under any parameter b that produces chaotic motion.The Lyapunov exponent of the map in Eq. (25) can be cal-culated using the time approach by λ ( b ) = lim T → ∞ T T ∑ t = ln (cid:12)(cid:12)(cid:12) k b ( − x i ) (cid:12)(cid:12)(cid:12) , (30)which lead us to λ ( b ) = k ln 10 + lim T → ∞ T T ∑ t = ln (cid:12)(cid:12) b ( − x i ) (cid:12)(cid:12) , (31)and finally to λ ( b ) = k ln 10 + λ ( b ) , (32)where λ ( b ) is just the Lyapunov exponent of the Logisticmap for the parameter b . Thus, here it is obvious that the gainfor sensibility to the initial conditions is trivially achieved byjust choosing a sufficiently large k . V. PSEUDO-RANDOM NUMBERS AND SYMBOLICSECRET KEYS
Once the output of the k -DZ transformation φ k ( x ) generatesreal points in the unit interval, these values can be considereddirectly as a pseudo-random number that can be re-scaled asrequired. The security analysis of the so-called k -logistic mapPRN was analysed in , showing high-quality pseudo-randomnumbers for k ≥ and NIST .Another strategy to generate PRNs is by means of the sym-bolic representation of the trajectory of the k -DZ transforma-tion. Thus, a partition that is not the natural partition of the k -DZ transformation needs to be considered. This natural par-tition is given by d whose borders are defined by Eq. (8).Then, for a given k , there will be 10 k symbols for the naturalpartition. The point φ k ( x i ) ∈ [ d i , d i + ] is encoded by the i -thsymbol of the alphabet ( i = { , , . . . , k − } ), representedby s i . A transformed trajectory of length L represented by { φ k ( x ) , φ k ( x ) , . . . , φ k ( x L ) } will have the symbolic represen-tation s = { s , s , . . . , s L } , where s i ∈ [ , k − ] . The vector s fully represents the information about the location of thepoints x i being mapped (within the resolution of the partitioncells), and therefore should be avoided for the creation of thesecret key. The partition to create a secret key should have aminimal number of intervals, for example a binary partitionwhere φ k ( x i ) < . φ k ( x i ) ≥ . x i ∈ [ , ] will beencoded with equal probabilities for ‘0’ and ‘1’. VI. CONCLUSIONS
Cryptography relies on the application of several transfor-mations to eliminate all existing correlations between the mes-sage and its ciphered version. A preliminary requirement forachieving this relies on the use of highly entropic and non-correlated pseudo-random numbers. The sensitivity to the ini-tial conditions property chaotic systems have is the key to thisgoal. The interest today is to be able to accomplish such a taskfor reliable encryption but by relying on transformations thatrequire little computational effort (light) and quick runningtime (fast). In this work, we characterize the properties of the so-calledthe k -Deep Zoom ( k -DZ) to support reliable cryptosystemsthat uses pseudo-random numbers or secret keys that were cre-ated fast and lightly. Besides the Decimal Shift Map (DSM)is not conceptually equivalent to the k-DZ, we show that thek-DZ is mathematically equivalent to the DSM map iteratedk times. More than that, we show that the k -fold DSM canbe rewritten into a form completely equivalent to the k -DZtransformation. So, all the good properties of the DSM mapsuch as uniform statistics, high entropy, and sensibility to theinitial conditions are inherited by the k -DZ. There is a seman-tic difference between both maps. Whereas the k -DZ trans-formation effectively represents an algorithm that simply ex-tracts the less significant digits of a real number, the DSM is amap that transforms a point into another point. This semanticinterpretation of the DZ-transformation can be in the futureexploited for the creation of dedicated electronic chips oper-ating at the hardware level that only work with less significantdigits, thus potentially bringing the encryption process to thephysical level. We show that the entropy and the Lyapunovexponent is linearly proportional to k . This means that thetrivial and light task of peeking onto the sequence of less sig-nificant digits positioned k digits to the decimal floating-pointis sufficient to drastically increase the entropy and thereforethe uncertainty past and future numbers, at a minimal compu-tational cost.Several of the properties of the k -DZ transformation dependonly on the map itself, not on the chaotic system being consid-ered as the generator of the original trajectory being encoded,or any other set of numbers being generated by any other pro-cess (e.g. stochastic processes). Thus, one might wonder whyto use the k -DZ transformation into a chaotic set of numbersafter all? The reason is that chaotic trajectories have severaladvantages. They are easy to be generated and do not requirethe use of higher-dimensional systems, in both digital or ana-log domains, they require less algorithmic complexity, less-power electronics, less CPU dedication and can be generatedat impressive large bandwidths. Chaos, however, is determin-istic and correlation does decay quickly, but not as quickly asone would wish. The additional application of the k -DZ trans-formations to chaotic trajectories fast and lightly enhances thealready existing wished properties of chaos to cryptography.A transformation that optimizes essential ingredients to a se-cure cryptosystem, but with minimal computational effort.Our strategy to create pseudo-random numbers or secretkeys requires the use of a chaotic system whose simulated tra-jectory is guaranteed to be chaotic for a long period, and thatcan be additionally generated using minimal computationalresources. For this reason, the Logistic map is a good candi-date. The k -DZ transformation is then applied a single timeto this stable chaotic trajectory. Our claim is that this strategyquickly generates secure and light PRNs. Another strategyto generate secure PRNs, which might increase the computa-tional cost to some extent, was proposed in Ref. , where anapproximate true trajectory of the Bernoulli map is calculateddirectly using real algebraic numbers. ACKNOWLEDGMENTS
J. M. acknowledges a scholarship from the National Coun-cil for Scientific and Technological Development (CNPqgrant
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