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Journal of Communications Technology and Electronics, Vol. 50, No. 12, 2005, pp. 1358­1366. Translated from Radiotekhnika i Elektronika, Vol. 50, No. 12, 2005, pp. 1466­1475. Original Russian Text Copyright © 2005 by Loskutov, Rybalko. English Translation Copyright © 2005 by MAIK "Nauka / Interperiodica" (Russia).

THEORY AND METHODS OF SIGNAL PROCESSING

Applications of Dynamical Systems with External Perturbations for Information Encoding and Hidden Data Transmission
A. Yu. Loskutov and S. D. Rybalko
Received October 20, 2002

Abstract--A new method for information processing and hidden data transmission is proposed that uses stabilized orbits of the families of dynamical systems for encoding alphabetic characters, which are put in a biunique correspondence with the periods of such orbits. An allowable noise level in the communications channel and the degree of randomness of transmitted signals are analytically estimated. The examples of encoded letter sequences are presented.

INTRODUCTION Currently, there is a particular interest in nontraditional methods for information processing (e.g., see [1-11] and references therein). This is due to a rapid recent growth of requirements for information security. Ordinarily, information security is meant as the protection of information against illegal access by unauthorized users [12­14]. A new method of encryption for the protection of information relates to the applications of the modern theory of dynamical systems, which considers chaotic signals as data carriers [3, 6, 8, 9, 11, 15­19]. Recent results obtained in this field substantially extend the range of methods available for data storage and data transmission. It is known that the behavior of chaotic systems sensitively depends on initial conditions and external perturbations. For a long time, such systems were considered unsuitable for practical applications because of their seemingly unpredictable and uncontrollable performance. However, further investigation demonstrated that such systems not only can be controlled but also can be used for practical applications. In particular, along with the recording­readout of data, chaotic systems proved to be adaptable for the hidden transmission of information. Investigations in this area are mainly based on the two following assumptions. (i) Certain periodic trajectories can be stabilized by using external perturbations [20­26]. (ii) Under specific conditions, two independent chaotic systems can be synchronized [27­31]. It is known that a dynamical system may be defined either in the form of maps or by differential equations. In the first case, the methods of trajectory stabilization are most frequently used, thus making it possible to construct fairly efficient data transmission systems (see, e.g., [5­8, 15, 18, 19]). The second variant (differential equations) is preferable for the problems con-

cerning synchronization. In this case, analog devices can be constructed (see [27, 28, 32­35, and references therein]). In this study, we propose an algorithm based on stabilization of periodic trajectories in one-dimensional (1D) maps for the hidden transmission of information. The method of stabilization is based on a known fact [22, 23, 26] that, for considerably general families of 1D maps, orbits of a certain period can be stabilized by applying external periodic perturbations. The stable orbits of a perturbed map are used to encode information. The perturbations are transmitted as a useful signal, and the mapping function presents the key for decrypting the received message. In this study, we propose an encoding algorithm that can be implemented in practice and can describe the results of applying this algorithm to a family of quadratic maps. 1. STABILIZATION OF PREDETERMINED ORBITS BY MEANS OF PARAMETRIC PERTURBATIONS Let us consider the mapping of a certain region M from Rj onto itself: Ta : x f ( x, a ) , (1)

where a is a parameter from the set of admissible values A R, x = {x1, ..., xj}, and f = {f1, ..., fj}. Let us introduce the concept of parametric perturbation. Assume that the mapping is specified with respect to a parameter in such a way that its value is defined at every instant; i.e., G : A A. Then, the perturbed map has the form x Ta : a f ( x, a ) g(a). (2)

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Perturbation will be referred to as periodic with period if function g(a) is defined only at points a1, ..., a in the following way: ai + 1 = g(ai), i = 1, ..., ­ 1, and a1 = g(a). In other words, perturbation is specified by the successive substitution of parameters into map (1). In this case, the totality of perturbations with period ^ can be put in correspondence with the set A = { a ^ A A ... A : a = (a1, ..., a), ai aj, 1 i, j , i j, a1, ..., a A}, A R. Having introduced the periodic perturbation with period for map (1), we obtain for perturbed map (2): T= T a1 : x T a2 : x T a : x f ( x, a 1 ) f f ( x, a 2 ) f
1 2 times

Indeed, the existence of the orbit entails that t j ~ ~ ~ ~ F k ( x ) = x and F k ( x ) x , 1 j < t. Consider the relationship that follows directly from the definition of Fk: f k ( Fk ( x ) ) = F p ( f k ( x ) ) , p = k + 1 ( mod ) .
n n

(6)

It can be seen readily that fk( F k ) = F p (fk). Therefore,
t t ~ ~ ~ for point x and n = t, we find F p (fk( x )) = fk( F k ( x )) = l ~ ~ fk( x ). Furthermore, when 1 l < t, we have F p (fk( x )) l ~ ~ ~ fk( x ) because, if it were F p (fk( x )) = fk( x ), we would l l ~ ~ ~ arrive at F p (fk( x )) = fk( F k ( x )) = fk( x ). However, by virtue of the uniqueness of functions fi, i = 1, ..., , it foll ~ ~ lows that fk ­ 1(fk - 2(...fk( F k ( x )))) = fk ­ 1(fk ­ 2(...fk( x ))) l+1 ~ ~ (see (4)); i.e., F k ( x ) = Fk( x ). However, this contra~ dicts the above assumption. In other words, point fk( x ) is periodic with period t for map Tp . ~ If point x is a stable periodic point of map Tk , then ~ there exists such a vicinity U x that the relationship tn ~ lim F k (x) = x holds true for each point x U. In view n

........................... f ( x, a ) f .

(3)

Let us consider functions of the form F1 = f ( f
­1

(...f 2(f 1(x))...)), (...f 3(f 2(x))...))), (4)

F2 = f 1 ( f ( f

­1

of

....................................... , F = f ­ 1 ( f ­ 2 ( ... f 1 ( f ( x ) ) ... ) ) , where x = {x1, ..., xj}, fi = { f
(1) i ( j) i (1) i

n

this leads to ~ lim f k ( F (x)) = lim F (fk(x)) = fk( x ). In other
tn k n tn p

continuity

of

functions

fk ,

, ..., f

( j) i

}, and Fi =

{ F , ..., F } are j-component functions, i = 1, 2, ..., . In terms of these functions, perturbed map (2) appears as T1 : x T2 : x T : x F 1 ( x, a 1, ..., a ) , F 2 ( x, a 1, ..., a ) , F ( x, a 1, ..., a ) , (5)

.............................. ,

with the initial conditions x1 = f1(x0) and x2 = f2(x1), ..., x - 1 = f ­ 1(x ­ 2). According to [26, 36], the maps constructed above have the following important properties. Let us assume that transformation Tk, 1 k , has a orbit with period t and functions fk(x) are continuous. Under this assumption, map Tp, p = k + 1 (mod ), also has a orbit with period t. Moreover, if the orbit of map Tk is stable, then, the periodic orbit of map Tp is stable as well and, if fk is a geomorphism, maps Tk and Tp are topologically equivalent.

words, all points from the vicinity of fk(U) tend to point t ~ fk( x ) under map T p . Topological equivalence follows directly from (6) and the definition. The essence of the above statements is that the investigation of a periodically perturbed map can be substantially simplified. Thus, instead of initial nonautonomous map (2), any one of autonomous maps T1, T2, ..., T defined by relationships (4) and (5) can be considered. Thereby, the dynamical behavior of initial map (2) is completely specified by the totality of maps (5), which act independently of one another and are coupled by only the initial conditions. It follows from this [23, 26, 36] that period t of any orbit of perturbed map (2) is a multiple of perturbation period : t = n, where n is an integer. Note that neither the construction of maps T1, ..., T by formulas (3)­(5) nor the proof of the statements required restrictions on set A. In other words, all the results are true for arbitrary set A of allowable values a of dynamical system (1) with -periodic perturbation (2). Let us form a subset Ach A of the set of parametric values such that map (1) has chaotic behavior if a Ach . In a number of studies (see, e.g., [22, 23, 36, 37]), it was analytically substantiated that, at j = 1 and j = 2, the periodic perturbations may suppress the chaos and stabilize the map orbits. It was also shown that, for cerVol. 50 No. 12 2005

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tain 1D and 2D chaotic maps, there are such perturba^ tions a = (a1, a2, ..., a) that perturbed map (2) becomes regular and has a stable orbit with period t = ^ n when a Ach , Ach = A ch A ch ... A ch (or g(a) Ach , see (2)). This result is proved for a wide class of maps [26, 38]. The property to display periodic dynamics under the action of external perturbations seems to be typical for a fairly wide class of maps. For practical implementation of encoding and transmission of information by means of perturbed maps, it is required to know how to find such -periodic transg(a) for map (2) that convert it into a forms G : a map having a stable orbit. Let us restrict the consideration to 1D (j = 1) maps. In this case, the theory developed in [22, 36, 37, 39] can be generalized and the method of searching for special perturbations allowing the stabilization of preassigned orbits becomes applicable for practical purposes (see Section 3). f(x, a), x M, a A possess the Let map Ta : x following properties: (i) There exists a subset M such that a value a* A that satisfies the equality f(x1, a*) = x2 can be found for any x1, x2 . (ii) There is a critical point xc such that f(x, a)/ x x = xc Dx f(xc, a) = 0 for all a A. In this case, it can be readily shown that, for all x2, x3, ..., x , there are x1 and a1, a2, ..., a such that orbit (x1, x2, ..., x) is a stable orbit of perturbed map Ta ^ for a = (a1, ..., a). Indeed, let us select arbitrary elements x1, x2, ..., x . By definition (1), the system of equations f ( x 1, a 1 ) = x 2 , f ( x 2, a 2 ) = x 3, ..., f ( x , a ) = x
1 times

remains stable and study the corresponding orbit distortions, i.e., determine xi for ( x '1 , x '2 , ..., x ' ) = (xc + x1, x2 + x2, ..., x + x). This approach can be formalized as follows. Let perturbed map Ta have a stable -periodic orbit ^ p = (x1, x2, ..., x) for a = (a1, a2, ..., a). If 1 a i a = ----------------------------------- , S a LS
­1 x



S

i x

i=1

where i = 1, 2, ..., , Sa = max D a f ( x, a ) , L =
x, a

max D x f ( x, a ) , and Sx = max D x f ( x, a ) , then this
2

map has another stable -periodic orbit p' = (xc + x1, x2 + x2, ..., x (a1 + a1, a2 In order assume that ^ + x), where |xi| x = 1/ LS x for a' = + a2, ..., a + a). to substantiate the above estimate, we all parameters ai are perturbed: a 'i = ai + ai . Let us find variation x1 = x '1 ­ xc . Here, element ' ' ' x 1 is a fixed point of map T1 (see (5)); i.e., x 1 = F1( x 1 , a '1 , a '2 , ..., a ' ). This yields xc + x1 = F1(xc, a1, a2, ..., ^ a) + DxF1(xc, a )x1 +
­1

x, a

x, a



i=1

^ D ai F1(xc, a )ai. Taking

^ into account the relationships xc = F1(xc, a ) and DxF1(xc, ^ a ) = (p) = 0, we obtain x1 = D f (xl, i=1 l = i+1 x al)Da f(xi, ai)ai. Therefore,



x1

a


i = 1l = i + 1





D x f ( x l, a l ) D a f ( x i, a i )

(7)

a S

a


i=1



(8) Sx.
i

for parameters a1, a2, ..., a has the solution in the form ^ a = (a1, a2, ..., a). Therefore, sequence (x1, x2, ..., x) = p is the -periodic orbit of map Ta with periodic pertur^ bation a = (a1, a2, ..., a). To make orbit p stable, it is sufficient to select element x1 near or at critical point xc , D f (xi, ai) and Dx f(xc, a) = 0 for because (p) i=1 x every a. This provides the stability criterion |(p)| < 1. Now, let us estimate the admissible distortions of parameters (a1, a2, ..., a) and orbit elements (x1, x2, ..., x). Let perturbation (a1, a2, ..., a) correspond to stable orbit (xc, x2, x3, ..., x). Assume that parameters ai have slightly changed to become ( a '1 , a '2 , ..., a ' ) = (a1 + a1, a2 + a2, ..., a + a), where |ai| a. Let us find the maximum value of a at which the perturbed orbit

Let us estimate the change of the multiplier of the a 'i ) orbit, (p') = D f( x 'i , = i=1 x





i=1

D f(xi, ai)

2 x



l=1 li



D x f(xl, al)xi +



i=1

D ax f(xi,

2

ai)



l=1 li

D x f(xl, al)ai. In both sums, only the first

terms are nonzero, since Dx f(x1, a1) = Dx f(xc, a1) = 0. Therefore, (p') = [ D x f(xc, a1)x1 + a1)a1] Da(Dx f(xc, a )) |x1|| D f(xc,
2 x 2



D ax f(xc,

2

l=2

D x f(xl, al). However, D ax f(xc, a1) =
a=a
1

2

= Da(0) = 0; therefore, |(p')| =
l=2

a 1) |



D x f ( x l, a l ) .
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remains stable under the condition |x1|| D x f(xc, a1)|

2



l=2

D x f ( x l, a l ) |x1| LS
­1 x

­1 x

< 1. Hence, it fol-

lows that |x1| x = 1/( LS

).

Thus, we find that the orbit is stable if perturbation x1 is less than x . The relation between the maximum possible value of x1 and a is given by inequality (8). Finally, the restriction imposed on a is obtained in the form aS
a



i=1

S x = 1/( LS

i

­1 x

) or

bation) be assigned to a certain alphabetic symbol. The encoded symbol is transmitted when the assigned perturbation is sent to a receiver. The decoding procedure consists in applying the transmitted periodic perturbation to a map contained in this receiver. According to the period of the stabilized orbit in the receiver, the type of symbol that was transmitted over the communications channel can be determined. Therefore, the chosen type of the map's family is the key for deciphering the encoded information. We now consider the basic principles of information processing based on the stable orbits of a perturbed map. Let a map orbit with period arise from a perturbation. The stability of this orbit is given by the rela tionship (p) = D f(xi, ai), where x1 = f(x, a) and i=1 x xi + 1 = f(xi, ai). If |(p)| < 1, the orbit is stable. It is clear that, when f(x, a) depends smoothly on the parameter, the space of perturbations R contains a certain region U where a stable orbit is preserved.

1 a = ----------------------------------- . S a LS
­1 x



S

i x

i=1



These results validate the use of chaotic maps for the efficient encoding and hidden transmission of information. 2. USING THE STABILIZED ORBITS OF PERTURBED MAPS FOR THE ENCODING AND TRANSMISSION OF INFORMATION The original method for hidden transmission of information is based on the encoding of alphabetic symbols by the stabilized orbits of chaotic maps and can be implemented in the following manner. Let the period of each orbit stabilized (by selecting the pertur-

Let us assume an information sequence that constitutes symbols of an alphabet, where every symbol is put in correspondence with a particular natural number. Let us find all possible perturbations leading to stable orbits with periods equal to such numbers. In other words, we localize in the perturbation space a subset A such that the perturbed map has a stable orbit with the prescribed ^ period if a A . This encoding algorithm can be represented by the following scheme:

Y symbol { a 1, ..., a
n1 + 2

n1 , n2 , n3 ASCII code } , { c 1, ..., c
n3 + 2

} , { b 1, ..., b

n2 + 2

sets of parameters

}

a 1, ..., c n3 + 2 sequence of numbers

Thus, each of the symbols is associated with a set of periodic perturbations that give rise to a orbit with the desired period. As a result of transmission, perturbation in the form of a finite set of real numbers (parameters a1, a2, ..., a) arrives at the receiver. To decode (decrypt) the received symbol, this perturbation should be applied to the map used for encoding. The resulting perturbed map has a stable orbit, whose period can be determined by iteration from an arbitrary initial point. Thus, the encoded symbol is recovered. In this method, each information sequence consisting of a set of symbols is put in correspondence with the transmitted set of real numbers. Using the fact that every symbol may be encoded by a subset A rather than by a single perturba^ ^ tion a , we may apply a different perturbation a' A

to each of the equal symbols in the transmitted sequence. Moreover, for most maps, subset A is a subspace of the perturbation space and, hence, perturbation ^ a' A may be selected at random. Therefore, although the original message may include identical symbols, the transmitted sequence looks absolutely random. This excludes the possibility of the transmitted information being decoded by an unauthorized user. It is obvious that all the families of unimodal maps satisfy the conditions presented in Section 2. Since any orbit of the form (xc, x2, x3, ..., x) is stable for arbitrary xi , the encoding and decoding algorithms can be implemented practically. The determination of a1, ..., a as solutions to system of equations (7) includes the following operations:
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LOSKUTOV, RYBALKO

(i) Each symbol is put in correspondence with a particular map orbit . (ii) The values of x2, x3, ..., x are arbitrarily chosen. (iii) Equations (7) are used to calculate parameters ^ (a1, a2, ..., a) = a for the set of points (xc, x2, x3, ..., x). In this way, each of the symbols is associated with a ^ parametric perturbation a = (a1, a2, ..., a). Owing to arbitrary selection of points x2, x3, ..., x , each of the repeated symbols may be encoded by sequence x2, x3, ..., x that is randomly chosen from . Thus, the proposed method allows conversion of a symbol sequence into a numerical sequence, for example, MAP ... a 1, a 2, a 3, ... . (9)

..., xt), it is necessary to solve the above equations for ai: x2 a 1 = ----------------------- , x1 ( 1 ­ x1 ) x1 x3 a 2 = ----------------------- , ..., a t = --------------------- . x2 ( 1 ­ x2 ) xt ( 1 ­ xt )

(12)

It is obvious that the condition ai [0, 4] does not hold true for all xi (0, 1). However, when it does hold true, there exist parameters (a1, a2, ..., at) at which perturbed map (11) has orbit p for each p = (x1, x2, ..., xt). When |(p)| = | ti = 1 a i (1 ­ 2xi)| < 1, this orbit is stable. In this case,



In view of the randomness of elements x2, x3, ..., x and functional dependence (7), this numerical sequence is random. Now, numerical sequence (9) that is obtained is transmitted to a receiver. The decryption (decoding) of sequence (9) reduces to the extraction of sets (a1, a2, ..., a) associated with particular symbols. This can be done readily since each of such sets corresponds to the stable orbit (x1, x2, ..., x) with period and x1 = xc . Calculating the other elements x2 = f(xc, a1), x3 = f(x2, a2), ..., we arrive at a certain step to x + 1 = f(x, a) = xc . The number of this step is the orbit period corresponding to the encoded symbol. The further decoding is nothing but a repetition of the above operation starting from ~ ~ ~ step a + 1: x 2 = f(xc, a + 1), x 3 = f( x 2 , a + 2), etc. 3. NUMERICAL INVESTIGATIONS OF THE ENCODING METHOD To implement the encoding method described above, we consider the family of quadratic maps x
n+1

( p) =


i=1

t

xi + 1 --------------------- ( 1 ­ 2 x i ) xi ( 1 ­ xi ) (13) 1 ­ 2x ---------------i < 1. 1 ­ xi

=


i=1

t

Since we have (1 ­ 2xc)/(1 ­ xc) = 0 for xc = 1/2, inequality (13) can always be satisfied. The sets of quantities (x1, x2, ..., xt) at which ai [0, 4] and inequality (13) holds true form a region in space, Rt. Each point of this region corresponds to a stable orbit of the perturbed map. Using the expressions for ai, i = 1, 2, ..., t, we have no difficulty in identifying this region in parametric space Rt. Let us consider the simplest case of a perturbation with period = 2. It is obvious that the region of stable orbits in space (x1, x2) is given by the following system of inequalities: x2 0 < ----------------------- 4, x1 ( 1 ­ x1 ) x1 0 < ----------------------- 4, x2 ( 1 ­ x2 )

= ax n ( 1 ­ x n ) .

(10)

This family is commonly used to model a wide range of physical systems (see, e.g., [40­ 43] and references therein). Furthermore, any unimodal map is semiconjugate to a quadratic map. Following (2), we introduce a -periodic perturbation. In this case, it is convenient to represent perturbed map (10) in the form xn + 1 = an xn ( 1 ­ xn ) a n = a n mod ( + 1 ) . (11)

1 ­ 2 x1 1 ­ 2 x2 and ---------------- ---------------- < 1. 1 ­ x1 1 ­ x2 The first two inequalities are fulfilled for the set of all admissible orbits with a period of 2. The third inequality restricts this set to a subset where only stable orbits exist. To solve this inequality, we specify x1 (0, 1) and consider special features in the behavior of x2 . This yields the following conditions: 3 x1 ­ 2 1 0 < x 2 < ---------------- , 0 < x 1 < -- ; 3 5 x1 ­ 3 x1 0 < x 2 < ---------------- , 3 x1 ­ 1 3 1 -- < x 1 < -- ; 5 3 3 -- < x 1 < 1. 5
No. 12 2005

If this map has a orbit p = (x1, x2, ..., xt) with the period t = , the orbit points obey the following system of equations: x2 = a1x1(1 ­ x1), x3 = a2x2(1 ­ x2), ..., xt = atxt(1 ­ xt). To solve the inverse problem, i.e., to find the parameter values at which map (11) has orbit p = (x1, x2,

x1 3 x1 ­ 2 ---------------- < x 2 < ---------------- , 5 x1 ­ 3 3 x1 ­ 1

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Thus, we obtain the domain of existence for all stable orbits p = (x1, x2) of map (11) with periods of 2 (Fig. 1a). The corresponding range of the values of parameters (a1, a2) is obtained when the domain shown in Fig. 1a is transformed by formulas (12). A simple way to do this is to divide the domain in Fig. 1a into subdomains that admit the biunique mapping of these subdomains according to (12). Then, the domain boundaries are mapped under the condition that points inside the boundary get mapped to points inside. Note that transformation of this type leads to a singularity at point (0, 0). A detailed analysis shows that, for = 2, mapping of the singularity into space (a1, a2) results in the curve a2 = 1/a1. The domain of existence of stable orbits p = (x1, x2) with periods of 2 is shown in the parametric space (a1, a2) in Fig. 1b. To reveal the details of map (12), the domain in Fig. 1a is divided by straight lines x1 = 1/2 and x2 = 1/2 into four portions denoted by different hatching. The corresponding subdomains in the space (a1, a2) are hatched similarly. Since some domains in Fig. 1 overlap, map (12) is unique but not biunique. Moreover, the presence of overlapping regions suggests that, under certain perturbations, map (11) exhibits bistability and has two coexistent stable orbits. In [39], range [3.8, 4.0] was thoroughly investigated for the possibility of chaos suppression. However, this range does not overlap the domains presented in Fig. 1b. For this reason, the orbits with periods of 2 were not discovered in map (11) at that time. In the general case, i.e., under the action of perturbation with period > 2, only the stable orbits in the form p = (xc, x2, x3, ..., xt) = (1/2, x2, x3, ..., xt) (t = ) should be selected and used for calculating parameters (a1, a2, ..., a). It is obvious that quadratic map (10) complies with all the requirements presented above. For this map, set is the segment [xb, xe], where xb and xe are the solutions to the equation xint = f(x, 4) and point xint is the intersection of curves y = 4x(1 ­ x) and y = x; i.e., [xb, xe] = [1/4, 3/4]. Thus, when a [0, 4] and x [1/4, 3/4], the admissible errors in the parameters for the quadratic map can be estimated from 1 S a = max ----- f ( x, a ) = -- , 4 x, a a S x = max ----- f ( x, a ) = 2 , x, a x L = max ------- f ( x, a ) = 8 . 2 x, a x
2

x2 1

(a) 4

1

2 0 a2 4 x1 (b)

3 1

6

5

7

0

a1

4

Fig. 1. The domain of existence for period-2 stable orbits of perturbed ( = 2) quadratic map (11) with the boundaries (1) x2 = 4x1(1 ­ x1), (2) x1 = 4x2(1 ­ x2), (3) x2 = (3x1 ­ 2)/(5x1 ­ 3), and (4) x2 = x1/(3x1 ­ 1) in plane (x1, x2) and the boundaries (5) a2 = 1/a1, (6) a2 = 8/[a1(4 ­ a1)], and (7) a1 = 8/[a2(4 ­ a2)] in plane (a1, a2).

implement this for the MS-DOS operating system.1 On IBM-compatible computers, each symbol entered from a keyboard is encoded by an integer. This number is an ASCII character and may take a value from 0 to 255. In particular, the ASCII characters from 65 to 90 and from 97 to 122 correspond to the capital and lowercase letters of the English alphabet, A­Z and a­z, respectively. The ASCII character set represents each symbol by three integers n1, n2 , and n3 satisfying the inequalities 0 n 1 2,
1

0 n 2 9,

0 n 3 9.

The proposed encoding method can successfully be applied in a system for the encryption of symbols entered from a PC keyboard. Let us consider how to

Note that the method can easily be adapted to the Windows operating system as well. Vol. 50 No. 12 2005

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1364 an 2.8 2.6

LOSKUTOV, RYBALKO a
n

2.6 2.4 2.4 2.2 2.0 1.8 1.6 1.4 2.6 2.6 2.4 2.4 2.2 2.0 1.8 1.6 1.4 0 20 40 n 60 80 100 2.2 2.0 1.8 1.6 1.4 0 20 40 n 60 80 100 2.2 2.0 1.8 1.6 1.4 2.8

Fig. 2. Variants of numerical sequences encoding the word CHAOS with the use of a quadratic map.

For example, letter A with the ASCII character number 65 is represented by n1 = 0, n2 = 6, and n3 = 5; letter z with the ASCII character number 122, by n1 = 1, n2 = 2, and n3 = 2. Now, we can formulate how the encryption system works: (a) First, triad n1, n2, and n3 is assigned to each symbol in accordance with its ASCII number. (b) Each number in the triad is associated with a sequence of parameters stabilizing the orbit with period ni + 2. This algorithm is schematically represented by the flow chart below. Y symbol
t T stable cycle

t number set of perturbations

In the last step, there is a transition from the separate sets of parameters to a continuous sequence, which is transmitted to a receiver with a decipherer. The decipherer converts this sequence into a sequence of integers corresponding to the orbit periods. Diminishing each of the integers by two and grouping the sequence into sets of three, we easily reconstruct the ASCII character of a symbol and the symbol itself. It is seen that any symbol can be encoded by using the stabilized orbits with periods from = t = 0 + 2 = 2 to = t = 9 + 2 = 11. Thus, the conditions for the applicability of the method are fulfilled (see Section 3) if the parameters are calculated with an error a 10­8; hence, x 10-4. In practice, owing to the inaccuracy of the parameter calculation, the exit condition xc = xk + 1 = f(xk, ak) is substituted for inequality |xc ­ f(xk, ak)| x . The intermediate points of a orbit x2, ..., xk are chosen to satisfy the condition xi [xc ­ x, xc + x], i = 2, ..., k. The results of encryption are presented in Figs. 2 and 3. Figure 2 shows four different parameter
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APPLICATIONS OF DYNAMICAL SYSTEMS WITH EXTERNAL PERTURBATIONS an 2.6 2.4 2.2 2.0 1.8 1.6 1.4

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Fig. 3. Numerical sequence of encoding the ten letters cccccccccc with the use of a quadratic map.

sequences encoding the word CHAOS. Figure 3 depicts the parameter sequence encoding the text string of ten symbols cccccccccc. The proposed method can be combined successfully with the keyboard entry of text messages and easily adapted to any other PC operating system. CONCLUSIONS A new effective method is developed for hidden transmission of information encoded by preassigned stabilized orbits of 1D maps. The proposed method bears a partial resemblance to a well-known encryption method that uses a preassigned text source as the key in which each letter of the secret message is located and encoded by the numbers of the page, line, and column. However, the use of maps with strong chaotic properties makes the encryption process much more reliable. Furthermore, the capabilities of encryption based on a preassigned text source are always restricted by the size of the source, thereby allowing the decryption of information. On the contrary, the use of chaotic maps offers a theoretically infinite choice of the parameters. In a particular implementation of the proposed method, when a family of unimodal maps is specified, it is possible to select perturbations that would stabilize the orbit passing through previously given points. In this case, the encoding, transmitting, and decoding of information can be automated easily, which is a significant advantage of the method. In this study, the algorithm is implemented on an IBM PC for the family of quadratic maps used as an example. The main advantages of the method proposed for the hidden transmission of information include the following:

(i) A sufficiently wide class of maps, including multidimensional maps, may be used in practice. (ii) No part of the information sequence itself appears in the communication channel; it is only the signal required for further processing that is transmitted. (iii) The transmitted signal exhibits a purely random behavior, which provides for a high degree of data security. (iv) No preliminary synchronization between a transmitter and a receiver is required. (v) The method is stable against external interferences. (vi) There is not only one signal sequence that may be assigned to the input information. Theoretically, the number of encoding variants is infinite. Thus, (ii), (iii), and (iv) ensure a high degree of message security during transmission and (i), (v), and (vi) afford the widespread use of the method proposed. Moreover, since each symbol of a transmitted data sequence is associated with an entire region in the space of parameters, this method can be employed in the design of noise-immune information-processing systems. The practical implementation of the hidden-transmission method also seems possible since the family of unimodal maps is readily modeled by means of standard electronic components. Thus, an operable integrated circuit (a chip) can be designed for practical purposes. REFERENCES
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