Multi-switching combination synchronization of discrete-time hyperchaotic systems for encrypted audio communication

Multi-switching combination synchronization of discrete-time hyperchaotic systems for encrypted... Abstract In this paper, encrypted audio communication based on original synchronization form is proposed for a class of discrete-time hyperchaotic systems. The new studied scheme of synchronization presents an extension of the multi-switching one to the combination synchronization, for which, the state variables of two driving systems synchronize with different state variables of the response system, simultaneously. With that in mind, at the outset, a theoretical approach for non-linear control, using aggregation techniques associated to one specific characteristic matrix description, namely, the arrow form, is developed. Then, the feasibility as well as the performance of the proposed approach of multi-switching combination synchronization is checked through its practical application in information transmission field to ensure more security of the message signal by means of hyperchaotic masking. Finally, experimental simulations are carried out in order to assess the security analysis and demonstrate that the suggested cryptosystem is large enough to resist to the noise attack thanks to its excellent encryption robustness. 1. Introduction During the past decades, hyperchaos synchronization has received a tremendous increasing interest and references therein (Baier & Klein, 1990; Belmouhoub et al., 2005; Dedieu et al., 1993). The phenomenon of synchronization is extremely widespread in nature as well as in the realm technology. The fact that various objects seek to achieve order and harmony in their behaviour, which is a characteristic of synchronization, seems to be a manifestation of the natural tendency of self organization existing in nature. Considerable attention paid to such topics is due to the potential applications of synchronization in communication engineering, using hyperchaos to mask the information-bearing signal (Djemaï et al., 2009; Grassi & Miller, 2002, 2012; Ghosh & Bhattachary, 2010; Hammami et al., 2009, 2010, 2014; Hammami, 2015; Hammami et al., 2015; Hernandez & Nijmeijer, 2000). Lately, a new scheme of synchronization was proposed (Hernández et al., 2010), in which three classic chaotic systems were made to synchronize, simultaneously, via systematically designed non-linear controls; two of which were driving a single response system, in a kind of double-driving/single response arrangement. The implication of combination synchronization proposed in (Hernández et al., 2010) for communication is such that a signal can be split into two parts, each one will be, then, loaded and transmitted between two drive systems. In order to improve the security of information transmission via synchronization, it may be required that different states of the slave system are synchronized with desired master’s ones, in a multi-switching manner (Jiang et al., 2003; Kovarev et al., 1992; Liu et al., 2013; Millerioux & Daafouz, 2003; Oppenheim et al., 1992). In such form of synchronization, a combination is adopted wherein two‐driver discrete-time hyperchaotic systems are multi-switched with a single‐response discrete-time hyperchaotic system. The possibility of realizing such a form of synchronization would present varieties of synchronization directions between variables of the driver systems and the response one, thereby ensuring better security when employed in signal transmission applications. The most important contribution, proposed throughout the present paper, consists on the original developed approach for multi-switching combination synchronization of discrete-time hyperchaotic systems by means of adapted non-linear control laws. In fact, the proposed approach developed during this work is, essentially, based on aggregation techniques for stability study associated with the arrow form matrix for system description. In addition, to prove the feasibility as well as the efficiency of this new approach, we transmit an encrypted audio message through insecure channels between two remote points. The randomness of hyperchaos is used to mix up the position of the data. Indeed, the position of the data is knotted in the order of randomness of the elements obtained from the hyperchaotic system, in encryption process, and, again, rearranged back to their original position, in decryption process. The same algorithm is tested with three non-identical discrete-time hyperchaotic systems and the performance analysis is done to put in prominent position the efficiency of the chosen maps as cryptosystems. The remainder of the paper is organized as follows: in Section 2, basic formulation of multi-switching combination synchronization is presented. In Section 3, an example of the multi-switching combination synchronization of three classic discrete-time hyperchaotic systems is formulated. In Section 4, the numerical simulation results are given and comparison of different synchronization methods is presented. The description of the communication system is discussed in Section 5. The application to secure communication with corresponding illustrations of the encrypted audio transmission are done in Section 6. The discussion of control performances is given in Section 7. The paper is recapitulated and concluded in Section 8. 2. Formulation of multi-switching combination synchronization concept Consider the following master/slave n‐dimensional discrete-time hyperchaotic systems, where the master systems are given by (Millerioux & Daafouz, 2003)   \begin{align} x_{mi} (k+1)=f_{ix} \left(x_{m1} (k),\ldots,x_{mn} (k)\right)\ \ \forall i=1,2,\ldots,n \end{align} (1) and   \begin{align} y_{mj} (k+1)=f_{jy} \left(\,y_{m1} (k),\ldots,y_{mn} (k)\right)\ \ \forall j=1,2,\ldots,n \end{align} (2) and the controlled slave system is described by   \begin{align} z_{sl} (k+1)=g_{lz} \left(z_{s1} (k),\ldots,z_{sn} (k)\right)+u_{l} (k)\ \ \forall l=1,2,\ldots,n, \end{align} (3) where $$x_{mi},\; y_{mj},\ \ z_{sl} \; \left (\forall i,j,l=1,2,\ldots ,n\right )\in \textbf{R}^{n} $$ are state space vectors of the master and slave systems, fix(.), fjy(.), glz(.) : Rn →Rn are three discrete-time non-linear vector functions, and $$u_{i} \; \left (\forall i=1,2,\ldots ,n\right ):\textbf{R}^{n} \to \textbf{R}^{n} $$ is a non-linear control function. Definition (Millerioux & Daafouz, 2003) If there exist three constant scaling matrices A, B, C ∈ Rn with C≠0, such that $$\mathop {\lim }\limits _{+\infty } \left \| Cz_{sl} \!-\!Ax_{mi} \!-\!By_{mj} \right \| =0,$$i = j ≠ l or i = l ≠ j or j = l ≠ i or i ≠ j ≠ l or i ≠ j = l or j ≠ i = l, then systems (1), (2) and (3) are said to be in multi-switching combination synchronization. To formulate the proposed procedure of multi-switching combination synchronization for the error dynamics stability study, let’s define a typical case of multi-switching synchronization error for one third order system, as:   \begin{align} \begin{cases} e_{123} (k)=\gamma_{1} z_{s1} (k)-\alpha_{2} x_{m2} (k)-\beta_{3} y_{m3} (k) \\ e_{231} (k)=\gamma_{2} z_{s2} (k)-\alpha_{3} x_{m3} (k)-\beta_{1} y_{m1} (k) \\ e_{312} (k)=\gamma_{3} z_{s3} (k)-\alpha_{1} x_{m1} (k)-\beta_{2} y_{m2} (k). \end{cases} \end{align} (4) So, it comes the following dynamical error system:   \begin{align} \begin{cases} e_{123} (k+1)=\gamma_{1} g_{1z} (.)+\gamma_{1} u_{1} (k)-\alpha_{2} f_{2x} (.)-\beta_{3} f_{3y} (.) \\ e_{231} (k+1)=\gamma_{2} g_{2z} (.)+\gamma_{2} u_{2} (k)-\alpha_{3} f_{3x} (.)-\beta_{1} f_{1y} (.) \\ e_{312} (k+1)=\gamma_{3} g_{3z} (.)+\gamma_{3} u_{3} (k)-\alpha_{1} f_{1x} (.)-\beta_{2} f_{2y} (.), \end{cases} \end{align} (5) where $$\alpha _{i},\beta _{j},\gamma _{l} \ \left(\forall i,j,l=1,2,3\right )$$ are scaling factors. Thus, the synchronization problem is reduced to the asymptotic stabilization of (5) with appropriate chosen control inputs. Here, we use the aggregation techniques associated to the arrow form matrix description, because they provide a systematic design approach for both control and synchronization, on the one hand, and guarantee global stability of the closed-loop system, on the other hand. The main feature of this approach is that the control strategy can be extended very easily to higher-dimensional systems thanks to its flexibility in terms of control laws construction. In the next part, we provide a description of a simple design procedure for the new proposed multi-switching combination synchronization. 3. Proposed non-linear control for discrete-time hyperchaotic multi-switching combination synchronization In order to generalize the concept of multi-switching combination synchronization, its application to the case of three strictly different discrete-time hyperchaotic systems, namely, Rössler, Baier–Klein and Hitzel–Zele systems, is considered. The Rössler and Baier–Klein maps afford the driving systems and are represented by the state variables xm, ym as (Pecora & Carroll, 1990; Perez & Cerdeira, 1995)   \begin{align} \begin{cases} {x_{m1} (k+1)=3.8x_{m1} (k)\left(1-x_{m1} (k)\right)-0.05\left(x_{m3} (k)+0.35\right)\left(1-2x_{m2} (k)\right)} \\ {x_{m2} (k+1)=3.78x_{m2} (k)\left(1-x_{m2} (k)\right)+0.2x_{m3} (k)} \\ {x_{m3} (k+1)=0.1\left(1-1.9x_{m1} (k)\right)\left[\left(x_{m3} (k)+0.35\right)\left(1-2x_{m2} (k)\right)-1\right]} \end{cases} \end{align} (6) and   \begin{align} \begin{cases} {y_{m1} (k+1)=-y_{m2}^{2} (k)-0.1y_{m3} (k)+1.76} \\ {y_{m2} (k+1)=y_{m1} (k)} \\ {y_{m3} (k+1)=y_{m2} (k),} \end{cases} \end{align} (7) whilst the Hitzel–Zele map is taken as the response system and its state space description, by the state variable zs is given by (Hammami et al., 2014; Perez & Cerdeira, 1995):   \begin{align} \begin{cases} {z_{s1} (k+1)=-0.3z_{s2} (k)+u_{1} } \\ {z_{s2} (k+1)=-1.07z_{s2}^{2} (k)+z_{s3} (k)+1+u_{2} } \\ {z_{s3} (k+1)=z_{s1} (k)+0.3z_{s2} (k)+u_{3}, } \end{cases} \end{align} (8) where u1, u2 and u3 are adaptive non-linear controllers to be adequately conceived. Remark 1 The hyperchaotic attractors of master systems (6) and (7) and slave system (8), Fig. 1, with the initial values $$x_{m} (0)=\left [{0.542} \quad {0.087} \quad {0.678} \right ]^{T},$$$$y_{m} (0)=\left [ {1} \quad {-1} \quad {0.5} \right ]^{T} $$ and $$z_{s} (0)=\left [ {1.5} \quad {0.2} \quad {0.1} \right ]^{T},$$ respectively, illustrate that state variables xmi(k), ymi(k) and zsi(k) are bounded (Hammami et al., 2014), such that $$\left |x_{mi} \right |<1,\ \ \left |y_{mi} \right |<2,\ \ \left |z_{si} \right |<2,\ \ \forall i=1,2,3.$$ Fig. 1. View largeDownload slide Hyperchaotic attractors of the Rössler (a), Baier–Klein (b) and the Hitzel–Zele (c) maps. Fig. 1. View largeDownload slide Hyperchaotic attractors of the Rössler (a), Baier–Klein (b) and the Hitzel–Zele (c) maps. There are several possible generic switching combinations that could exist for the drive and response systems (6), (7) and (8). As far as this paper, we will, principally, focus on results for one particular switching combination, randomly selected as follows:   \begin{align} \begin{cases} {e_{112} (k)=\gamma_{1} z_{s1} (k)-\alpha_{1} x_{m1} (k)-\beta_{2} y_{m2} (k)} \\ {e_{213} (k)=\gamma_{2} z_{s2} (k)-\alpha_{1} x_{m1} (k)-\beta_{3} y_{m3} (k)} \\ {e_{311} (k)=\gamma_{3} z_{s3} (k)-\alpha_{1} x_{m1} (k)-\beta_{1} y_{m1} (k),} \end{cases} \end{align} (9) Consequently, for the considered switch (9), the error dynamics are given by   \begin{align} \begin{cases} {e_{112} (k+1)=\gamma_{1} z_{s1} (k+1)-\alpha_{1} x_{m1} (k+1)-\beta_{2} y_{m2} (k+1)} \\ {e_{213} (k+1)=\gamma_{2} z_{s2} (k+1)-\alpha_{1} x_{m1} (k+1)-\beta_{3} y_{m3} (k+1)} \\ {e_{311} (k+1)=\gamma_{3} z_{s3} (k+1)-\alpha_{1} x_{m1} (k+1)-\beta_{1} y_{m1} (k+1)}. \end{cases} \end{align} (10) Substituting zs1(k + 1), zs2(k + 1), zs3(k + 1), xm1(k + 1), ym1(k + 1), ym2(k + 1) and ym3(k + 1) by their corresponding expressions from (6), (7) and (8), the dynamical error system (10) associated to the chosen switch (9) can be rewritten as:   \begin{align}\begin{cases} {e_{112} (k+1)=\frac{7.6}{\alpha_{1} } \left(\gamma_{1} z_{s1} (k)-\beta_{2} y_{m2} (k)-0.5e_{112} (k)\right)e_{112} (k)-0.3\frac{\gamma_{1} }{\gamma_{2} } e_{213} (k)+\frac{\beta_{2} }{\beta_{1} } e_{311} (k)} \\ {\quad\quad\quad\quad\quad\quad +f(.)+\gamma_{1} u_{1} (k)} \\ {e_{213} (k+1)=-\frac{\beta_{3} }{\beta_{2} } e_{112} (k)-\frac{1.07}{\gamma_{2} } \left(e_{213} (k)+2\left(\alpha_{1} x_{m1} (k)+\beta_{3} y_{m3} (k)\right)\right)e_{213} (k)+\frac{\gamma_{2} }{\gamma_{3} } e_{311} (k)} \\ {\quad\quad\quad\quad\quad\quad +g(.)+\gamma_{2} u_{2} (k)} \\ {e_{311} (k+1)=\frac{\gamma_{3} }{\gamma_{1} } e_{112} (k)+0.3\frac{\gamma_{3} }{\gamma_{2} } e_{213} (k)} \\ {\quad\quad\quad\quad\quad\quad +h(.)+\gamma_{3} u_{3} (k)}, \end{cases} \end{align} (11) where   \begin{align} f(.)=&-0.3\frac{\gamma_{1}}{\gamma_{2}}\left(\alpha_1{}x_{m1}(k)+\beta_{1}y_{m3}(k)\right)+\frac{\beta_{1}}{\beta_2}\left(-\gamma_3 z_{s3}(k)+\alpha_1x_{m1}(k)\right)-0.035\alpha_{1}x_{m2}(k)\nonumber\\ &+0.175\alpha_{1}-0.1\alpha_{1}x_{m2}(k)x_{m3}(k)+0.05\alpha_{1}x_{m3}(k)\nonumber\\ &+\frac{3.8}{\alpha_{1}}\left(-\gamma_{1}^{2}z_{s1}^{2}(k)+2\gamma_{1}\beta_{2}z_{s1}(k)y_{m2}(k)-\beta_{2}^{2}y_{m2}^{2}(k)\right) \end{align} (12)  \begin{align} g(.) &= -\frac{1.07}{\gamma_{2} } \left(\alpha_{1} x_{m1} (k)+\beta_{3} y_{m3} (k)\right)^{2} +\frac{\gamma_{2} }{\gamma_{3} } \left(\alpha_{1} x_{m1} (k)+\beta_{1} y_{m1} (k)\right)+\gamma_{2}\nonumber\\ &\quad-\frac{\beta^{3}}{\beta_{2}}\left(\gamma_{1}z_{s1}(k)+\alpha_{1}x_{m1}(k)\right)\nonumber\\ &\quad-\alpha_{1}\left[3.8x_{1}(k)\left(1-x_{m1}(k)\right)-0.05\left(x_{m3}(k)+0.35\right)\left(1-2x_{m2}(k)\right)\right] \end{align} (13)  \begin{align} h(.)=&\frac{\gamma_{3} }{\gamma_{1} } \left(\alpha_{1} x_{m1} (k)+\beta_{2} y_{m2} (k)\right)+0.3\frac{\gamma_{3} }{\gamma_{2} } \left(\alpha_{1} x_{m1} (k)+\beta_{3} y_{m3} (k)\right)\nonumber \\ &-\alpha_{1} \left[3.8x_{m1} (k)\left(1-x_{m1} (k)\right)-0.05\left(x_{m3} (k)+0.35\right)\left(1-2x_{m2} (k)\right)\right]\nonumber \\ &-\beta_{1} \left(-y_{m2}^{2} (k)-0.1y_{m3} (k)+1.76\right). \end{align} (14) At this stage, let’s consider the non-linear control functions u1(k), u2(k) and u3(k), such that   \begin{equation} \left\{\begin{aligned} u_{1} (k)&=-\frac{1}{\gamma_{1} } \left[f(.)+\frac{7.6}{\alpha_{1} } \left(\beta_{2} y_{m2} (k)+0.5e_{112} (k)\right)\right]\\ u_{2} (k)&=-\frac{1}{\gamma_{2} } \left[g(.)-\frac{1.07}{\gamma_{2} } \left(e_{213} (k)+2\left(\alpha_{1} x_{m1} (k)+\beta_{3} y_{m3} (k)\right)\right)+0.5e_{213} (k)+\frac{\gamma_{2} }{\gamma_{3} } e_{311} (k)\right] \\ u_{3} (k)&=-\frac{1}{\gamma_{2} } \left[h(.)+0.3\frac{\gamma_{3} }{\gamma_{2} } e_{213} (k)\right] \end{aligned}\right. \end{equation} (15) which leads to the closed-loop dynamical error system described in the state space by:   \begin{align} E(k+1)=A_{c} (.)E(k) \end{align} (16) with   \begin{align} E=\left[{e_{112} } \quad {e_{213} } \quad {e_{311} } \right]^{T} \end{align} (17) and   \begin{align} A_{c} (.)=\left[\begin{array}{@{}ccc@{}} {\frac{7.6}{\alpha_{1} } \gamma_{1} z_{s1} (k)} & {-0.3\frac{\gamma_{1} }{\gamma_{2} } } & {\frac{\beta_{2} }{\beta_{1} } } \\ {-\frac{\beta_{3} }{\beta_{2} } } & {0.5} & {0} \\ {\frac{\gamma_{3} }{\gamma_{1} } } & {0} & {-0.3} \end{array}\right]. \end{align} (18) By referring to the control theory viewpoint, the drive systems (6) and (7) will achieve multi-switching combination synchronization with the response system (8) if the asymptotic stability of the dynamical error system (16) is reached. Thus, to attain this goal, let’s elaborate sufficient conditions guaranteeing the asymptotic stability of the so obtained closed-loop error system (16), by putting in prominent position the use of aggregation techniques associated with the arrow form matrix description (Hammami et al., 2014, 2015; Ramirez & Hernandez, 2001; Sun & Shen, 2016). For such a purpose, let’s consider the overvaluing system $$M\left (A_{c} (.)\right ),$$ relatively to the following vectorial norm (Ramirez & Hernandez, 2001; Sun & Shen, 2016):   \begin{align} p\left(v(k)\right)=\left[{\left|v_{1} (k)\right|} \quad {\left|v_{2} (k)\right|} \quad {\left|v_{3} (k)\right|} \right]^{T} \end{align} (19) with $$v(k)=\left [ {v_{1} (k)} \quad {v_{2} (k)} \quad {v_{3} (k)} \right ]^{T}\!,$$ described by   \begin{align} v(k+1)=M\left(A_{c} (.)\right)v(k) \end{align} (20) and $$M\left (A_{c} (.)\right )=\{m_{ij} (.)\},$$$$m_{ij} (.)=\left |a_{c_{ij} } (.)\right |,$$ ∀i, j = 1, 2, 3. Exploiting the notions that hyperchaotic signals are bounded and generated in a deterministic manner (Hammami et al., 2014), the matrix $$M\left (A_{c} (.)\right )$$ can be overvalued by an 3 × 3 matrix $$M_{o} =\left \{m_{o_{ij} } \right \},\ \ \forall i,j=1,2,3,$$ whose all elements are constant, positive and independent of state variables xm(k), ym(k) and zs(k), of both master and slave systems, such that the inequality (21):   \begin{align} p(k+1)\le M\left(A_{c} (.)\right)p(k)\le M_{o} p(k) \end{align} (21) is verified. The system described by (11), (12), (13) and (14) is, then, stabilized by the non‐linear control laws (15), if the matrix $$\left (\textbf{I}-M_{o} \right )$$ is an M −matrix, i.e.:   \begin{align} \left(\textbf{I}-M_{o} \right)\left(\begin{array}{ccc} {1} & {2} & {3} \\ {1} & {2} & {3} \end{array}\right)>0. \end{align} (22) Taking into consideration that the arrow form choice for instantaneous characteristic matrices makes sufficient asymptotic stability conditions very easy to test, we have already designed the control laws u(k), so that the characteristic overvaluing matrix Mo, associated to the closed-loop system (16), be under the arrow form, such as (Hammami et al., 2014, 2015; Ramirez & Hernandez, 2001; Sun & Shen, 2016)   \begin{align} \begin{cases} {e_{112} (k+1)=m_{o_{11} } e_{112} (k)+m_{o_{12} } e_{213} (k)+m_{o_{13} } e_{311} (k)} \\ {e_{213} (k+1)=m_{o_{21} } e_{112} (k)+m_{o_{22} } e_{213} (k)} \\ {e_{311} (k+1)=m_{o_{31} } e_{112} (k)+m_{o_{33} } e_{311} (k)}. \end{cases} \end{align} (23) Then, the following Theorem, based on the use of Kotelyanski lemma (Sun & Shen, 2016) associated to the specific arrow form matrix Mo, introduced in (23) (Hammami et al., 2014, 2015; Ramirez & Hernandez, 2001; Sun & Shen, 2016), gives sufficient conditions of multi-switching combination synchronization, relatively to slave system (8) with master ones (6) and (7). Theorem The dynamical multi-switching combination synchronization error system (16) converges towards zero if the matrix Mo, is under the arrow form and such that the diagonal elements, $$m_{o_{ii} },$$ of the constant matrix Mo satisfy   \begin{align} 1-m_{o_{ii} }>0,\ \ \forall i=2,3. \end{align} (24) there exist ε > 0 for which   \begin{align} \Delta =1-m_{o_{11} } -\sum_{i=2}^{3}\left(m_{o_{i1} } m_{o_{1i} } \left(1-m_{o_{ii} } \right)^{-1} \right)>\varepsilon. \end{align} (25) Proof. The error system (16), described by (18), is stabilized by the proposed control laws (15), if the matrix $$\left (\textbf{I}-M_{o} \right )$$ is an M‐matrix (Ramirez & Hernandez, 2001; Sun & Shen, 2016), that’s to say   \begin{align} \begin{cases} {1-m_{o_{ii} }>0,\ \ \forall i=2,3} \\ {\det \left(\textbf{I}-M_{o} \right)\ge \varepsilon >0.} \end{cases} \end{align} (26) The computation of the first member of the last inequality, announced in (26), leads to the following expression:   \begin{align} \det \left(\textbf{I}-M_{o} \right)=\Delta \prod_{i=2}^{3}\left(1-m_{o_{ii} } \right) \end{align} (27) and let us conclude that the equilibrium of the full‐dimensional system (16) is asymptotically stable, that’s to say, the global multi-switching combination synchronization has been successfully reached. This ends, easily, the proof. Now, by the use of the vectorial norm (19), the overvaluing system associated to (18) is characterized by the instantaneous matrix $$M\left (A_{c} (.)\right )$$ under the arrow form given by (28):   \begin{align} M\left(A_{c} (.)\right)=\left[\begin{array}{{@{}ccc@{}}} {7.6\left|\frac{\gamma_{1} }{\alpha_{1} } z_{s1} (k)\right|} & {0.3\left|\frac{\gamma_{1} }{\gamma_{2} } \right|} & {\left|\frac{\beta_{2} }{\beta_{1} } \right|} \\[6pt] {\left|\frac{\beta_{3} }{\beta_{2} } \right|} & {0.5} & {0} \\[6pt] {\left|\frac{\gamma_{3} }{\gamma_{1} } \right|} & {0} & {0.3} \end{array}\right]. \end{align} (28) As it is noted in the above-cited Remark 1, state variables of the slave hyperchaotic system are bounded, and from Fig. 1, we have $$\left |z_{s1} (k)\right |<1.5;$$ thus, it comes   \begin{align} 7.6\left|\frac{\gamma_{1} }{\alpha_{1} } z_{s1} (k)\right|<11.4\left|\frac{\gamma_{1} }{\alpha_{1} } \right|. \end{align} (29) So, a new overvaluing system characterized by the constant matrix Mo, defined by   \begin{align} M_{o} =\left[\begin{array}{ccc} {11.4\left|\frac{\gamma_{1} }{\alpha_{1} } \right|} & {0.3\left|\frac{\gamma_{1} }{\gamma_{2} } \right|} & {\left|\frac{\beta_{2} }{\beta_{1} } \right|} \\[6pt] {\left|\frac{\beta_{3} }{\beta_{2} } \right|} & {0.5} & {0} \\[6pt] {\left|\frac{\gamma_{3} }{\gamma_{1} } \right|} & {0} & {0.3} \end{array}\right] \end{align} (30) which is under the arrow form, is obtained. Actually, and based on the proposed Theorem, since the multi-switching combination synchronization sufficient conditions (24) are true, it stills only to satisfy the sufficient condition (25), expressed, explicitly, as follows:   \begin{align} 1-11.4\left|\frac{\gamma_{1} }{\alpha_{1} } \right|-0.6\left|\frac{\beta_{3} \gamma_{1} }{\beta_{2} \gamma_{2} } \right|-\left|\frac{\gamma_{3} \beta_{2} }{\gamma_{1} \beta_{1} } \right|\frac{1}{0.7}>0. \end{align} (31) From various possibilities relatively to the scaling factors α1, β1, β2, β3, γ1, γ2 and γ3, let choose the following one:   \begin{align} \begin{cases}{l} {\alpha_{1} =\beta_{1} =100} \\ {\beta_{2} =10} \\ {\beta_{3} =\gamma_{1} =\gamma_{2} =\gamma_{3} =1.} \end{cases} \end{align} (32) As stated earlier, the main interest is to achieve multi-switching combination synchronization of the Rössler, Baier–Klein and Hitzel–Zele systems. The efficiency of the proposed method for designing the adapted non‐linear control laws together is proved by means of diverse numerical simulations, given below. From Fig. 2, one can see that the error dynamics of coupled master systems (6) and (7) with the slave one (8), obtained when the control laws are turned off, evolve hyperchaotically and, so, the studied systems are not yet synchronized. Fig. 2. View largeDownload slide Error dynamics evolutions when the proposed control laws are switched off. Fig. 2. View largeDownload slide Error dynamics evolutions when the proposed control laws are switched off. For the studied switch (9) and by respect to the fixed scaling factors (32), the control inputs given by (15) were programmed to turn on simultaneously. The results are shown in Fig. 3, where we can see that the property of multi-switching combination synchronization is, visibly, fulfilled. In addition, Fig. 4 illustrates the temporal behaviour of $$(z_{s1}, 100x_{m1} +10y_{m2}),$$$$\left (z_{s2},\; \text {100}x_{m1} +y_{m3} \right )$$ and $$\left (z_{s3},\; \text {100}\left (x_{m1} +y_{m1} \right )\right ),$$ in the multi-switching compound synchronization state variables, with real-time activation of the developed non‐linear control laws. Fig. 3. View largeDownload slide Multi-switching combination synchronization error states after activating the proposed control laws. Fig. 3. View largeDownload slide Multi-switching combination synchronization error states after activating the proposed control laws. Fig. 4. View largeDownload slide Temporal behaviour of the synchronizing variables. Fig. 4. View largeDownload slide Temporal behaviour of the synchronizing variables. (a) $$\left (z_{s1},\ \ \textrm {100}\ x_{m1} +10y_{m2} \right ),$$ (b) $$\left (z_{s2},\ \ \textrm {100}\ x_{m1} +y_{m3} \right )\!,$$ and (c) $$\left (z_{s3},\ \ \textrm {100}\left (x_{m1} +y_{m1} \right )\right )\!,$$ in the multi-switching compound synchronization states with simultaneous activation of the control laws. 4. Comparative study of different synchronization methods In this paragraph, the comparison amongst some synchronization methods is presented. For this purpose, following synchronization manners are used: (1) multi-switching combination synchronization, (2) hybrid synchronization and (3) projective synchronization. As far as the first synchronization method, its concept and its methodology are, accurately, discussed in the previous sections. In the case of the hybrid synchronization, some of the states of the slave system are completely synchronized and the rest of the states are anti-synchronized, with the states of the master systems. Thus, it is a combination of complete synchronization and anti-synchronization methods. Relatively, to the case of projective synchronization, the states of the slave system are projected with some scaling factor to the respective states of the master systems. The scaling factor can be constant, time-varying or function of states. Here, constant scaling factor is used; hence, generalized projective synchronization method (Vincent et al., 2015) is used and designed. The components of the hybrid synchronization error vector eHS ∈ R3×1, are defined as:   \begin{align} \begin{cases} {e_{HS_{112} } (k)=z_{s1} (k)-x_{m1} (k)-y_{m2} (k)} \\ {e_{HS_{213} } (k)=z_{s2} (k)+x_{m1} (k)-y_{m3} (k)} \\ {e_{HS_{311} } (k)=z_{s3} (k)-x_{m1} (k)-y_{m1} (k),} \end{cases} \end{align} (33) Here, the first and third states are considered for complete synchronization and the second state is considered for anti-synchronization. The average error for hybrid synchronization is shown in Fig. 5. It can be seen from the above-mentioned Fig. 5, that the average error converges to zero. The results regarding control inputs and error convergence are avoided to restrict the length of the paper. Fig. 5. View largeDownload slide Comparison of average errors for different synchronization methods. Fig. 5. View largeDownload slide Comparison of average errors for different synchronization methods. The elements of the projective synchronization error vector ePS ∈ R3×1, are obtained by respect to (34):   \begin{align} \begin{cases} {e_{PS_{112} } (k)=z_{s1} (k)-\delta \left(x_{m1} (k)+y_{m2} (k)\right)} \\ {e_{PS_{213} } (k)=z_{s2} (k)-\delta \left(x_{m1} (k)+y_{m3} (k)\right)} \\ {e_{PS_{311} } (k)=z_{s3} (k)-\delta \left(x_{m1} (k)+y_{m1} (k)\right)}. \end{cases} \end{align} (34) At this point, the scaling factor δ is fixed to 0.5. The average error of the projective synchronization is illustrated in Fig. 5, reflecting, clearly, its convergence to zero. In fact, the average errors, considered as performance measure to compare the three synchronization methods, are defined as follows:   \begin{align} \begin{cases} {\left\| e_{MSCS} (k)\right\| =\sqrt{\left(e_{112} (k)\right)^{2} -\left(e_{213} (k)\right)^{2} -\left(e_{311} (k)\right)^{2} } } \\ {\left\| e_{HS} (k)\right\| =\sqrt{\left(e_{HS_{112} } (k)\right)^{2} -\left(e_{HS_{213} } (k)\right)^{2} -\left(e_{HS_{311} } (k)\right)^{2} } } \\ {\left\| e_{PS} (k)\right\| =\sqrt{\left(e_{PS_{112} } (k)\right)^{2} -\left(e_{PS_{213} } (k)\right)^{2} -\left(e_{PS_{311} } (k)\right)^{2}, } } \end{cases} \end{align} (35) where eMSCS ∈ R3×1 is the multi-switching combination synchronization error vector, such that:   \begin{align} e_{MSCS} (k)=\left[{e_{112} (k)} {e_{213} (k)} {e_{311} (k)} \right]^{T} \end{align} (36) $$\left \| e_{MSCS} (k)\right \|,\ \ \left \| e_{HS} (k)\right \| $$ and $$\left \| e_{PS} (k)\right \| $$ denote the average errors of multi-switch combination synchronization, hybrid synchronization and projective synchronization, respectively. The results of the average synchronization errors are depicted in Fig. 5. It is observed, from the Fig. 5, that the transient and settling time of multi-switching combination synchronization is, comparatively, less than the other two synchronization methods, namely, the hybrid and the projective ones. 5. System communication description In this section, a cryptosystem based on multi-switching combination synchronization of discrete-time hyperchaotic systems is described. The aim of such cryptosystem is to transmit encrypted messages, of several forms, from transmitter A to remote receiver B, as is depicted below, in Fig. 6. Fig. 6. View largeDownload slide Hyperchaotic cryptosystem for secure transmitted signals. Fig. 6. View largeDownload slide Hyperchaotic cryptosystem for secure transmitted signals. A message m is to be transmitted over an insecure communication channel. To avoid any unauthorized receiver (intruder O) located at the mentioned channel; m is encrypted prior to transmission to generate an encrypted message c, such that   \begin{align} c=e\left(m,K\right) \end{align} (37) by using an hyperchaotic system e on transmitter A. The encrypted message c is sent to receiver B, where m is recovered as $$\hat {m}$$ from the hyperchaotic decryption d, as   \begin{align} \hat{m}=d\left(c,K\right). \end{align} (38) If e and d have used the same key K, then at receiver end B it is possible to obtain $$\hat {m}=m.$$ A secure channel represented by a dashed line, Fig. 6, is used for transmitting the keys, K. Generally, this secure communication channel is a courier and is too slow for the transmission of m. Our hyperchaotic cryptosystem is reliable, if it preserves the security of m, that’s to say, if m′≠m for even the best cryptanalytic function h, given by   \begin{align} m^{\prime}=h(c). \end{align} (39) To achieve the proposed hyperchaotic encryption scheme, we appeal to Rössler, Baier–Klein and Hitzel-Zele hyperchaotic maps for encryption and decryption purposes. All Rössler, Baier–Klein and Hitzel–Zele hyperchaotic systems have a number of parameters determining their dynamics; such parameters and initial conditions are the coding and decoding keys, K. We expect that they can perform the objective of the secure communication and the transmitting messages can be recovered at the receiver B. In order to guarantee the encryption and decryption processes, the Rössler, Baier–Klein and the Hitzel–Zele hyperchaotic maps have to achieve the so-called multi-switching combination synchronization on both hyperchaotic transmitter A and hyperchaotic receiver B. Remark 2 It is also possible to improve the security of the proposed communication scheme through the use of two asymmetric encryption and decryption keys. In fact, by referring to the inclusion method (Yang, 1999; Zheng, 2016), the considered hyperchaotic master systems (6) and (7) generate the key K used q times as a key stream to encrypt the original message m with an encryption rule e, a q − shift cipher algorithm, such as:   \begin{align} c=e\left(m,K\right)=\underbrace{F_{1} (\ldots F_{1} (F_{1} }_{q}(m,\underbrace{K),K),\ldots,K)}_{q} \end{align} (40) with   \begin{align} K=\sqrt{\left(x_{m1} +\beta_{3} y_{m3} \right)^{2} }. \end{align} (41) F1(.) is a non‐linear function defined, in this case, by   \begin{align} F_{1} (m,K)=\begin{cases} {m+K+2h,\ \ \textrm{for}\ -2h\le m+K\le -h} \\ {m+K,\ \textrm{for}\ -h<m+K<h} \\ {m+K-2h,\ \ \textrm{for}\ h\le m+K\le 2h} \end{cases} \end{align} (42) h is an encryption parameter chosen such that the transmitted message m and the key K lie within the interval $$\left [-h,h\right ]$$ The slave system (8) generates the recovered key $$\hat {K}$$ used to recover the original message, using a decryption rule d, as following:   \begin{align} \hat{m}=d(c,-\hat{K})=\underbrace{F_{1} (\ldots F_{1} (F_{1} }_{q}(c,\underbrace{-\hat{K}),-\hat{K}),\ldots,-\hat{K})}_{q} \end{align} (43) such that:   \begin{align} \hat{K}=\sqrt{\left(\gamma_{2} z_{s2} \right)^{2} }. \end{align} (44) In the next part, we will use, as it is invoked in the previous Remark, non‐identical keys for the audio signal transmission. 6. Real-world application to encrypted audio communication Multi-switching combination synchronization using the new proposed approach is fulfilled successfully with the assumption that the parameters of drives and response systems are known and states of all systems are measurable. Potential applications (Hammami, 2015; Hernandez & Nijmeijer, 2000) of synchronized hyperchaotic and chaotic systems may be used in secure communication. The experimental approach is based on masking the message, to be sent, with the hyperchaotic signal at transmitter end as driving signal and recovered back the same message at receiver end, once synchronization transmitter/receiver is accomplished. At this stage, we describe the communication system based on multi-switching combination synchronization hyperchaos. We will use the discrete-time hyperchaotic systems (6) and (7) as hyperchaos generators. With this scheme, we obtain faster synchronization and higher privacy; one channel is used to send the hyperchaotic synchronizing signal $$\left (\alpha _{1} x_{m1} +\beta _{2} y_{m2} \right )$$ from the transmitters (6) and (7), with no connection with the secret audio message m. Whilst the other channel is used to transmit hidden message m which is recovered at the receiver end, by means of the comparison between the signals $$\left (c=\left (x_{m1} +\beta _{3} y_{m3} \right )+m\right )$$ and $$\left (\hat {c}=\gamma _{2} z_{s2} +\hat {m}\right ),$$ Fig. 7. Fig. 7. View largeDownload slide Proposed audio communication scheme. Fig. 7. View largeDownload slide Proposed audio communication scheme. Then, via numerical simulations, we illustrate the encrypted audio transmission. We use as transmitter the discrete-time hyperchaotic systems given in (6) and (7), and the discrete-time hyperchaotic system given in (8) is looked as a receiver. For both the encryption as well as the decryption phases, let’s take h = 0.3 and q = 15. The format of the audio signal m is Pulse-Code Modulation of 22.05 KHz, 16 Bits, monofonic channel. The mentioned audio message m is to be encrypted and transmitted to the receiver. Figure 8 shows the original transmitted audio message m to be encrypted and transmitted, Fig. 9 illustrates the encrypted audio message c, and from Fig. 10, it is clearly shown that the original audio message $$\hat {m},$$ can be, faithfully, recovered by the receiver. Fig. 8. View largeDownload slide The original transmitted audio message m. Fig. 8. View largeDownload slide The original transmitted audio message m. Fig. 9. View largeDownload slide The encrypted audio message $$\left (c=\left (x_{m1} +\beta _{3} y_{m3} \right )+m\right ).$$ Fig. 9. View largeDownload slide The encrypted audio message $$\left (c=\left (x_{m1} +\beta _{3} y_{m3} \right )+m\right ).$$ Fig. 10. View largeDownload slide The recovered audio message $$\hat {m.}$$ Fig. 10. View largeDownload slide The recovered audio message $$\hat {m.}$$ Obviously, it is easy to conclude that the information signal is recovered after a short time of transmission. Besides, the proposed method of encryption, based on multi-switching combination hyperchaos synchronization of three non‐identical systems, compared to the original chaos masking mode, can improve the security of communication. In actual fact, the main advantage of using hyperchaotic systems for synchronization study is related to the difficulty to predict all the parameters and properties of such complex systems, with more random and unpredictable behaviour than chaotic ones. At this step, the performance of the proposed encryption scheme is evaluated through several aspects. 7. Performance and evaluation of the proposed cryptosystem In this paper, the multi-switching combination synchronization of discrete-time hyperchaotic systems has been studied and an encryption system based on hyperchaotic maps has been proposed. We have shown an efficient way of constructing secure audio data transmission using a new approach stabilizing non‐linear discrete-time systems. Following, a brief discussion of the performance of such proposed cryptosystem is given. 7.1. The noise resistance effect for the proposed cryptosystem A transparency test was conducted with the transmitted audio signal with different measures in terms of signal to noise ratio, commonly abbreviated as SNR. SNR is defined as the ratio of the power of a signal or meaningful information and the power of background noise or, similarly, unwanted signal:   \begin{align} SNR=\frac{P_{signal} }{P_{noise}}, \end{align} (45) where P is the average power. Both signal and noise power must be measured at equivalent points in a system and within the same system bandwidth. Since, many signals have a very wide dynamic range, they are, habitually, expressed using the logarithmic decibel scale. So, based upon the definition of decibel, signal and noise may be expressed in decibels (dB) as following:   \begin{align} SNR=10\log \left[\frac{P_{signal} }{P_{noise} } \right]. \end{align} (46) Figure 11 shows the SNR of 100 original host audio pieces and other unwanted ones using our proposed algorithm. The average SNR using our new method is 41.12 dB, which is totally acceptable since it satisfies, clearly, the requirement announced by the international federation of the phonographic industry: SNR should be higher than 20 dB. Therefore, the proposed scheme is highly secure against statistical attacks. Fig. 11. View largeDownload slide SNR of 100 original host audio pieces and unwanted ones using our proposed communication scheme. Fig. 11. View largeDownload slide SNR of 100 original host audio pieces and unwanted ones using our proposed communication scheme. 7.2. Security analysis of the proposed cryptosystem In our proposed encryption algorithm, three hyperchaotic systems are used in order to increase the complexity of this algorithm as well as to make the key space larger. In such a way, the security which it is the main concern of one cryptosystem is improved. Hence, the key space of our encryption algorithm is large enough against potential attacks thanks to the wide range of variation, relatively to the parameters of hyperchaotic systems. 7.3. Discussion of the feasibility and usefulness attained by the obtained results In our cryptosystem, the processes of encryption and multi-switching combination synchronization are completely separated with no interference between them. So, encrypted information does not interfere with the proposed type of synchronization, therefore not increasing the sensitivity of such synchronization to external errors. As a result, the hyperchaotic communication scheme with two transmission channels gives faster synchronization between transmitter and receiver, in one hand, and high security of data transmission, in the other hand. Furthermore, the secure communication systems, considered in this work, relied on three discrete-time hyperchaotic systems; however, the obtained results can be generalized to more than three continuous-time as well as discrete-time hyperchaotic systems. This flexibility is an added advantage of the proposed contribution. In addition, many different types of messages can be transmitted that have different frequencies via simply adjusting the time scaling factor, which preserves the topological and geometrical properties of the phase diagram, relatively to the hyperchaotic transmitter. Consequently, based on the pre-cited advantages of the proposed cryptosystem and the new multi-switching combination synchronization approach, we can conclude to the high degree of security as well as the robustness against noise and several attacks, according to the developed encryption scheme. 8. Conclusion and prospect To sum up, we have studied a novel scheme of hyperchaotic synchronization that can involve numerous dynamical systems, namely multi-switching combination synchronization of three hyperchaotic systems, based on non‐linear control approach. In this new synchronization scheme, the state space variables of the three systems are multi-switched, such that their mutual synchronization takes place between different state variables. When synchronization is achieved, satisfactorily, in this manner in the communication context, it would be difficult or even impossible for an intruder to predetermine the vector space in which synchronization would occur, thereby, enhancing information security. Comparison amongst different synchronization methods is done, in order to reflect that the proposed synchronization, namely the multi-switching combination one, is better than the others. Numerical simulations, security analysis and noise interference investigation, using there the Rössler, Baier–Klein and the Hitzel–Zele maps were also employed to illustrate the effective performance of the proposed control strategy that is outlined in this paper, by demonstrating that our algorithm has large key space, high security and good immunity to noise interference. In a forthcoming work, we will be concerned with the effectiveness and feasibility of the new afforded approach of synchronization, based on aggregation techniques and arrow form matrix, by considering, in more details, the effect of delay in transmission channels. References Baier, G. & Klein, M. ( 1990) Maximum hyperchaos in generalized Hénon circuit. Physics Lett. A , 151, 281-- 284. Google Scholar CrossRef Search ADS   Belmouhoub, I., Djemaï, M. & Barbot, J. P. ( 2005) Observability quadratic normal forms for discrete-time systems. IEEE Trans. Automatic Control , 50, 1031-- 1038. Google Scholar CrossRef Search ADS   Dedieu, H., Kennedy, M. P. & Hasler, M. ( 1993) Chaos shift keying: modulation and demodulation of a chaotic carrier using selfsynchronizing chua’s circuit. IEEE Trans. Circ. Syst. II: Analog Digital Sig. Proc. , 40, 634-- 642. Cruz Hernández, C., Inzunza González, E., López Gutiérrez, R. M., Serrano Guerrero, H. & García Guerrero, E. E. 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Multi-switching combination synchronization of discrete-time hyperchaotic systems for encrypted audio communication

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Abstract

Abstract In this paper, encrypted audio communication based on original synchronization form is proposed for a class of discrete-time hyperchaotic systems. The new studied scheme of synchronization presents an extension of the multi-switching one to the combination synchronization, for which, the state variables of two driving systems synchronize with different state variables of the response system, simultaneously. With that in mind, at the outset, a theoretical approach for non-linear control, using aggregation techniques associated to one specific characteristic matrix description, namely, the arrow form, is developed. Then, the feasibility as well as the performance of the proposed approach of multi-switching combination synchronization is checked through its practical application in information transmission field to ensure more security of the message signal by means of hyperchaotic masking. Finally, experimental simulations are carried out in order to assess the security analysis and demonstrate that the suggested cryptosystem is large enough to resist to the noise attack thanks to its excellent encryption robustness. 1. Introduction During the past decades, hyperchaos synchronization has received a tremendous increasing interest and references therein (Baier & Klein, 1990; Belmouhoub et al., 2005; Dedieu et al., 1993). The phenomenon of synchronization is extremely widespread in nature as well as in the realm technology. The fact that various objects seek to achieve order and harmony in their behaviour, which is a characteristic of synchronization, seems to be a manifestation of the natural tendency of self organization existing in nature. Considerable attention paid to such topics is due to the potential applications of synchronization in communication engineering, using hyperchaos to mask the information-bearing signal (Djemaï et al., 2009; Grassi & Miller, 2002, 2012; Ghosh & Bhattachary, 2010; Hammami et al., 2009, 2010, 2014; Hammami, 2015; Hammami et al., 2015; Hernandez & Nijmeijer, 2000). Lately, a new scheme of synchronization was proposed (Hernández et al., 2010), in which three classic chaotic systems were made to synchronize, simultaneously, via systematically designed non-linear controls; two of which were driving a single response system, in a kind of double-driving/single response arrangement. The implication of combination synchronization proposed in (Hernández et al., 2010) for communication is such that a signal can be split into two parts, each one will be, then, loaded and transmitted between two drive systems. In order to improve the security of information transmission via synchronization, it may be required that different states of the slave system are synchronized with desired master’s ones, in a multi-switching manner (Jiang et al., 2003; Kovarev et al., 1992; Liu et al., 2013; Millerioux & Daafouz, 2003; Oppenheim et al., 1992). In such form of synchronization, a combination is adopted wherein two‐driver discrete-time hyperchaotic systems are multi-switched with a single‐response discrete-time hyperchaotic system. The possibility of realizing such a form of synchronization would present varieties of synchronization directions between variables of the driver systems and the response one, thereby ensuring better security when employed in signal transmission applications. The most important contribution, proposed throughout the present paper, consists on the original developed approach for multi-switching combination synchronization of discrete-time hyperchaotic systems by means of adapted non-linear control laws. In fact, the proposed approach developed during this work is, essentially, based on aggregation techniques for stability study associated with the arrow form matrix for system description. In addition, to prove the feasibility as well as the efficiency of this new approach, we transmit an encrypted audio message through insecure channels between two remote points. The randomness of hyperchaos is used to mix up the position of the data. Indeed, the position of the data is knotted in the order of randomness of the elements obtained from the hyperchaotic system, in encryption process, and, again, rearranged back to their original position, in decryption process. The same algorithm is tested with three non-identical discrete-time hyperchaotic systems and the performance analysis is done to put in prominent position the efficiency of the chosen maps as cryptosystems. The remainder of the paper is organized as follows: in Section 2, basic formulation of multi-switching combination synchronization is presented. In Section 3, an example of the multi-switching combination synchronization of three classic discrete-time hyperchaotic systems is formulated. In Section 4, the numerical simulation results are given and comparison of different synchronization methods is presented. The description of the communication system is discussed in Section 5. The application to secure communication with corresponding illustrations of the encrypted audio transmission are done in Section 6. The discussion of control performances is given in Section 7. The paper is recapitulated and concluded in Section 8. 2. Formulation of multi-switching combination synchronization concept Consider the following master/slave n‐dimensional discrete-time hyperchaotic systems, where the master systems are given by (Millerioux & Daafouz, 2003)   \begin{align} x_{mi} (k+1)=f_{ix} \left(x_{m1} (k),\ldots,x_{mn} (k)\right)\ \ \forall i=1,2,\ldots,n \end{align} (1) and   \begin{align} y_{mj} (k+1)=f_{jy} \left(\,y_{m1} (k),\ldots,y_{mn} (k)\right)\ \ \forall j=1,2,\ldots,n \end{align} (2) and the controlled slave system is described by   \begin{align} z_{sl} (k+1)=g_{lz} \left(z_{s1} (k),\ldots,z_{sn} (k)\right)+u_{l} (k)\ \ \forall l=1,2,\ldots,n, \end{align} (3) where $$x_{mi},\; y_{mj},\ \ z_{sl} \; \left (\forall i,j,l=1,2,\ldots ,n\right )\in \textbf{R}^{n} $$ are state space vectors of the master and slave systems, fix(.), fjy(.), glz(.) : Rn →Rn are three discrete-time non-linear vector functions, and $$u_{i} \; \left (\forall i=1,2,\ldots ,n\right ):\textbf{R}^{n} \to \textbf{R}^{n} $$ is a non-linear control function. Definition (Millerioux & Daafouz, 2003) If there exist three constant scaling matrices A, B, C ∈ Rn with C≠0, such that $$\mathop {\lim }\limits _{+\infty } \left \| Cz_{sl} \!-\!Ax_{mi} \!-\!By_{mj} \right \| =0,$$i = j ≠ l or i = l ≠ j or j = l ≠ i or i ≠ j ≠ l or i ≠ j = l or j ≠ i = l, then systems (1), (2) and (3) are said to be in multi-switching combination synchronization. To formulate the proposed procedure of multi-switching combination synchronization for the error dynamics stability study, let’s define a typical case of multi-switching synchronization error for one third order system, as:   \begin{align} \begin{cases} e_{123} (k)=\gamma_{1} z_{s1} (k)-\alpha_{2} x_{m2} (k)-\beta_{3} y_{m3} (k) \\ e_{231} (k)=\gamma_{2} z_{s2} (k)-\alpha_{3} x_{m3} (k)-\beta_{1} y_{m1} (k) \\ e_{312} (k)=\gamma_{3} z_{s3} (k)-\alpha_{1} x_{m1} (k)-\beta_{2} y_{m2} (k). \end{cases} \end{align} (4) So, it comes the following dynamical error system:   \begin{align} \begin{cases} e_{123} (k+1)=\gamma_{1} g_{1z} (.)+\gamma_{1} u_{1} (k)-\alpha_{2} f_{2x} (.)-\beta_{3} f_{3y} (.) \\ e_{231} (k+1)=\gamma_{2} g_{2z} (.)+\gamma_{2} u_{2} (k)-\alpha_{3} f_{3x} (.)-\beta_{1} f_{1y} (.) \\ e_{312} (k+1)=\gamma_{3} g_{3z} (.)+\gamma_{3} u_{3} (k)-\alpha_{1} f_{1x} (.)-\beta_{2} f_{2y} (.), \end{cases} \end{align} (5) where $$\alpha _{i},\beta _{j},\gamma _{l} \ \left(\forall i,j,l=1,2,3\right )$$ are scaling factors. Thus, the synchronization problem is reduced to the asymptotic stabilization of (5) with appropriate chosen control inputs. Here, we use the aggregation techniques associated to the arrow form matrix description, because they provide a systematic design approach for both control and synchronization, on the one hand, and guarantee global stability of the closed-loop system, on the other hand. The main feature of this approach is that the control strategy can be extended very easily to higher-dimensional systems thanks to its flexibility in terms of control laws construction. In the next part, we provide a description of a simple design procedure for the new proposed multi-switching combination synchronization. 3. Proposed non-linear control for discrete-time hyperchaotic multi-switching combination synchronization In order to generalize the concept of multi-switching combination synchronization, its application to the case of three strictly different discrete-time hyperchaotic systems, namely, Rössler, Baier–Klein and Hitzel–Zele systems, is considered. The Rössler and Baier–Klein maps afford the driving systems and are represented by the state variables xm, ym as (Pecora & Carroll, 1990; Perez & Cerdeira, 1995)   \begin{align} \begin{cases} {x_{m1} (k+1)=3.8x_{m1} (k)\left(1-x_{m1} (k)\right)-0.05\left(x_{m3} (k)+0.35\right)\left(1-2x_{m2} (k)\right)} \\ {x_{m2} (k+1)=3.78x_{m2} (k)\left(1-x_{m2} (k)\right)+0.2x_{m3} (k)} \\ {x_{m3} (k+1)=0.1\left(1-1.9x_{m1} (k)\right)\left[\left(x_{m3} (k)+0.35\right)\left(1-2x_{m2} (k)\right)-1\right]} \end{cases} \end{align} (6) and   \begin{align} \begin{cases} {y_{m1} (k+1)=-y_{m2}^{2} (k)-0.1y_{m3} (k)+1.76} \\ {y_{m2} (k+1)=y_{m1} (k)} \\ {y_{m3} (k+1)=y_{m2} (k),} \end{cases} \end{align} (7) whilst the Hitzel–Zele map is taken as the response system and its state space description, by the state variable zs is given by (Hammami et al., 2014; Perez & Cerdeira, 1995):   \begin{align} \begin{cases} {z_{s1} (k+1)=-0.3z_{s2} (k)+u_{1} } \\ {z_{s2} (k+1)=-1.07z_{s2}^{2} (k)+z_{s3} (k)+1+u_{2} } \\ {z_{s3} (k+1)=z_{s1} (k)+0.3z_{s2} (k)+u_{3}, } \end{cases} \end{align} (8) where u1, u2 and u3 are adaptive non-linear controllers to be adequately conceived. Remark 1 The hyperchaotic attractors of master systems (6) and (7) and slave system (8), Fig. 1, with the initial values $$x_{m} (0)=\left [{0.542} \quad {0.087} \quad {0.678} \right ]^{T},$$$$y_{m} (0)=\left [ {1} \quad {-1} \quad {0.5} \right ]^{T} $$ and $$z_{s} (0)=\left [ {1.5} \quad {0.2} \quad {0.1} \right ]^{T},$$ respectively, illustrate that state variables xmi(k), ymi(k) and zsi(k) are bounded (Hammami et al., 2014), such that $$\left |x_{mi} \right |<1,\ \ \left |y_{mi} \right |<2,\ \ \left |z_{si} \right |<2,\ \ \forall i=1,2,3.$$ Fig. 1. View largeDownload slide Hyperchaotic attractors of the Rössler (a), Baier–Klein (b) and the Hitzel–Zele (c) maps. Fig. 1. View largeDownload slide Hyperchaotic attractors of the Rössler (a), Baier–Klein (b) and the Hitzel–Zele (c) maps. There are several possible generic switching combinations that could exist for the drive and response systems (6), (7) and (8). As far as this paper, we will, principally, focus on results for one particular switching combination, randomly selected as follows:   \begin{align} \begin{cases} {e_{112} (k)=\gamma_{1} z_{s1} (k)-\alpha_{1} x_{m1} (k)-\beta_{2} y_{m2} (k)} \\ {e_{213} (k)=\gamma_{2} z_{s2} (k)-\alpha_{1} x_{m1} (k)-\beta_{3} y_{m3} (k)} \\ {e_{311} (k)=\gamma_{3} z_{s3} (k)-\alpha_{1} x_{m1} (k)-\beta_{1} y_{m1} (k),} \end{cases} \end{align} (9) Consequently, for the considered switch (9), the error dynamics are given by   \begin{align} \begin{cases} {e_{112} (k+1)=\gamma_{1} z_{s1} (k+1)-\alpha_{1} x_{m1} (k+1)-\beta_{2} y_{m2} (k+1)} \\ {e_{213} (k+1)=\gamma_{2} z_{s2} (k+1)-\alpha_{1} x_{m1} (k+1)-\beta_{3} y_{m3} (k+1)} \\ {e_{311} (k+1)=\gamma_{3} z_{s3} (k+1)-\alpha_{1} x_{m1} (k+1)-\beta_{1} y_{m1} (k+1)}. \end{cases} \end{align} (10) Substituting zs1(k + 1), zs2(k + 1), zs3(k + 1), xm1(k + 1), ym1(k + 1), ym2(k + 1) and ym3(k + 1) by their corresponding expressions from (6), (7) and (8), the dynamical error system (10) associated to the chosen switch (9) can be rewritten as:   \begin{align}\begin{cases} {e_{112} (k+1)=\frac{7.6}{\alpha_{1} } \left(\gamma_{1} z_{s1} (k)-\beta_{2} y_{m2} (k)-0.5e_{112} (k)\right)e_{112} (k)-0.3\frac{\gamma_{1} }{\gamma_{2} } e_{213} (k)+\frac{\beta_{2} }{\beta_{1} } e_{311} (k)} \\ {\quad\quad\quad\quad\quad\quad +f(.)+\gamma_{1} u_{1} (k)} \\ {e_{213} (k+1)=-\frac{\beta_{3} }{\beta_{2} } e_{112} (k)-\frac{1.07}{\gamma_{2} } \left(e_{213} (k)+2\left(\alpha_{1} x_{m1} (k)+\beta_{3} y_{m3} (k)\right)\right)e_{213} (k)+\frac{\gamma_{2} }{\gamma_{3} } e_{311} (k)} \\ {\quad\quad\quad\quad\quad\quad +g(.)+\gamma_{2} u_{2} (k)} \\ {e_{311} (k+1)=\frac{\gamma_{3} }{\gamma_{1} } e_{112} (k)+0.3\frac{\gamma_{3} }{\gamma_{2} } e_{213} (k)} \\ {\quad\quad\quad\quad\quad\quad +h(.)+\gamma_{3} u_{3} (k)}, \end{cases} \end{align} (11) where   \begin{align} f(.)=&-0.3\frac{\gamma_{1}}{\gamma_{2}}\left(\alpha_1{}x_{m1}(k)+\beta_{1}y_{m3}(k)\right)+\frac{\beta_{1}}{\beta_2}\left(-\gamma_3 z_{s3}(k)+\alpha_1x_{m1}(k)\right)-0.035\alpha_{1}x_{m2}(k)\nonumber\\ &+0.175\alpha_{1}-0.1\alpha_{1}x_{m2}(k)x_{m3}(k)+0.05\alpha_{1}x_{m3}(k)\nonumber\\ &+\frac{3.8}{\alpha_{1}}\left(-\gamma_{1}^{2}z_{s1}^{2}(k)+2\gamma_{1}\beta_{2}z_{s1}(k)y_{m2}(k)-\beta_{2}^{2}y_{m2}^{2}(k)\right) \end{align} (12)  \begin{align} g(.) &= -\frac{1.07}{\gamma_{2} } \left(\alpha_{1} x_{m1} (k)+\beta_{3} y_{m3} (k)\right)^{2} +\frac{\gamma_{2} }{\gamma_{3} } \left(\alpha_{1} x_{m1} (k)+\beta_{1} y_{m1} (k)\right)+\gamma_{2}\nonumber\\ &\quad-\frac{\beta^{3}}{\beta_{2}}\left(\gamma_{1}z_{s1}(k)+\alpha_{1}x_{m1}(k)\right)\nonumber\\ &\quad-\alpha_{1}\left[3.8x_{1}(k)\left(1-x_{m1}(k)\right)-0.05\left(x_{m3}(k)+0.35\right)\left(1-2x_{m2}(k)\right)\right] \end{align} (13)  \begin{align} h(.)=&\frac{\gamma_{3} }{\gamma_{1} } \left(\alpha_{1} x_{m1} (k)+\beta_{2} y_{m2} (k)\right)+0.3\frac{\gamma_{3} }{\gamma_{2} } \left(\alpha_{1} x_{m1} (k)+\beta_{3} y_{m3} (k)\right)\nonumber \\ &-\alpha_{1} \left[3.8x_{m1} (k)\left(1-x_{m1} (k)\right)-0.05\left(x_{m3} (k)+0.35\right)\left(1-2x_{m2} (k)\right)\right]\nonumber \\ &-\beta_{1} \left(-y_{m2}^{2} (k)-0.1y_{m3} (k)+1.76\right). \end{align} (14) At this stage, let’s consider the non-linear control functions u1(k), u2(k) and u3(k), such that   \begin{equation} \left\{\begin{aligned} u_{1} (k)&=-\frac{1}{\gamma_{1} } \left[f(.)+\frac{7.6}{\alpha_{1} } \left(\beta_{2} y_{m2} (k)+0.5e_{112} (k)\right)\right]\\ u_{2} (k)&=-\frac{1}{\gamma_{2} } \left[g(.)-\frac{1.07}{\gamma_{2} } \left(e_{213} (k)+2\left(\alpha_{1} x_{m1} (k)+\beta_{3} y_{m3} (k)\right)\right)+0.5e_{213} (k)+\frac{\gamma_{2} }{\gamma_{3} } e_{311} (k)\right] \\ u_{3} (k)&=-\frac{1}{\gamma_{2} } \left[h(.)+0.3\frac{\gamma_{3} }{\gamma_{2} } e_{213} (k)\right] \end{aligned}\right. \end{equation} (15) which leads to the closed-loop dynamical error system described in the state space by:   \begin{align} E(k+1)=A_{c} (.)E(k) \end{align} (16) with   \begin{align} E=\left[{e_{112} } \quad {e_{213} } \quad {e_{311} } \right]^{T} \end{align} (17) and   \begin{align} A_{c} (.)=\left[\begin{array}{@{}ccc@{}} {\frac{7.6}{\alpha_{1} } \gamma_{1} z_{s1} (k)} & {-0.3\frac{\gamma_{1} }{\gamma_{2} } } & {\frac{\beta_{2} }{\beta_{1} } } \\ {-\frac{\beta_{3} }{\beta_{2} } } & {0.5} & {0} \\ {\frac{\gamma_{3} }{\gamma_{1} } } & {0} & {-0.3} \end{array}\right]. \end{align} (18) By referring to the control theory viewpoint, the drive systems (6) and (7) will achieve multi-switching combination synchronization with the response system (8) if the asymptotic stability of the dynamical error system (16) is reached. Thus, to attain this goal, let’s elaborate sufficient conditions guaranteeing the asymptotic stability of the so obtained closed-loop error system (16), by putting in prominent position the use of aggregation techniques associated with the arrow form matrix description (Hammami et al., 2014, 2015; Ramirez & Hernandez, 2001; Sun & Shen, 2016). For such a purpose, let’s consider the overvaluing system $$M\left (A_{c} (.)\right ),$$ relatively to the following vectorial norm (Ramirez & Hernandez, 2001; Sun & Shen, 2016):   \begin{align} p\left(v(k)\right)=\left[{\left|v_{1} (k)\right|} \quad {\left|v_{2} (k)\right|} \quad {\left|v_{3} (k)\right|} \right]^{T} \end{align} (19) with $$v(k)=\left [ {v_{1} (k)} \quad {v_{2} (k)} \quad {v_{3} (k)} \right ]^{T}\!,$$ described by   \begin{align} v(k+1)=M\left(A_{c} (.)\right)v(k) \end{align} (20) and $$M\left (A_{c} (.)\right )=\{m_{ij} (.)\},$$$$m_{ij} (.)=\left |a_{c_{ij} } (.)\right |,$$ ∀i, j = 1, 2, 3. Exploiting the notions that hyperchaotic signals are bounded and generated in a deterministic manner (Hammami et al., 2014), the matrix $$M\left (A_{c} (.)\right )$$ can be overvalued by an 3 × 3 matrix $$M_{o} =\left \{m_{o_{ij} } \right \},\ \ \forall i,j=1,2,3,$$ whose all elements are constant, positive and independent of state variables xm(k), ym(k) and zs(k), of both master and slave systems, such that the inequality (21):   \begin{align} p(k+1)\le M\left(A_{c} (.)\right)p(k)\le M_{o} p(k) \end{align} (21) is verified. The system described by (11), (12), (13) and (14) is, then, stabilized by the non‐linear control laws (15), if the matrix $$\left (\textbf{I}-M_{o} \right )$$ is an M −matrix, i.e.:   \begin{align} \left(\textbf{I}-M_{o} \right)\left(\begin{array}{ccc} {1} & {2} & {3} \\ {1} & {2} & {3} \end{array}\right)>0. \end{align} (22) Taking into consideration that the arrow form choice for instantaneous characteristic matrices makes sufficient asymptotic stability conditions very easy to test, we have already designed the control laws u(k), so that the characteristic overvaluing matrix Mo, associated to the closed-loop system (16), be under the arrow form, such as (Hammami et al., 2014, 2015; Ramirez & Hernandez, 2001; Sun & Shen, 2016)   \begin{align} \begin{cases} {e_{112} (k+1)=m_{o_{11} } e_{112} (k)+m_{o_{12} } e_{213} (k)+m_{o_{13} } e_{311} (k)} \\ {e_{213} (k+1)=m_{o_{21} } e_{112} (k)+m_{o_{22} } e_{213} (k)} \\ {e_{311} (k+1)=m_{o_{31} } e_{112} (k)+m_{o_{33} } e_{311} (k)}. \end{cases} \end{align} (23) Then, the following Theorem, based on the use of Kotelyanski lemma (Sun & Shen, 2016) associated to the specific arrow form matrix Mo, introduced in (23) (Hammami et al., 2014, 2015; Ramirez & Hernandez, 2001; Sun & Shen, 2016), gives sufficient conditions of multi-switching combination synchronization, relatively to slave system (8) with master ones (6) and (7). Theorem The dynamical multi-switching combination synchronization error system (16) converges towards zero if the matrix Mo, is under the arrow form and such that the diagonal elements, $$m_{o_{ii} },$$ of the constant matrix Mo satisfy   \begin{align} 1-m_{o_{ii} }>0,\ \ \forall i=2,3. \end{align} (24) there exist ε > 0 for which   \begin{align} \Delta =1-m_{o_{11} } -\sum_{i=2}^{3}\left(m_{o_{i1} } m_{o_{1i} } \left(1-m_{o_{ii} } \right)^{-1} \right)>\varepsilon. \end{align} (25) Proof. The error system (16), described by (18), is stabilized by the proposed control laws (15), if the matrix $$\left (\textbf{I}-M_{o} \right )$$ is an M‐matrix (Ramirez & Hernandez, 2001; Sun & Shen, 2016), that’s to say   \begin{align} \begin{cases} {1-m_{o_{ii} }>0,\ \ \forall i=2,3} \\ {\det \left(\textbf{I}-M_{o} \right)\ge \varepsilon >0.} \end{cases} \end{align} (26) The computation of the first member of the last inequality, announced in (26), leads to the following expression:   \begin{align} \det \left(\textbf{I}-M_{o} \right)=\Delta \prod_{i=2}^{3}\left(1-m_{o_{ii} } \right) \end{align} (27) and let us conclude that the equilibrium of the full‐dimensional system (16) is asymptotically stable, that’s to say, the global multi-switching combination synchronization has been successfully reached. This ends, easily, the proof. Now, by the use of the vectorial norm (19), the overvaluing system associated to (18) is characterized by the instantaneous matrix $$M\left (A_{c} (.)\right )$$ under the arrow form given by (28):   \begin{align} M\left(A_{c} (.)\right)=\left[\begin{array}{{@{}ccc@{}}} {7.6\left|\frac{\gamma_{1} }{\alpha_{1} } z_{s1} (k)\right|} & {0.3\left|\frac{\gamma_{1} }{\gamma_{2} } \right|} & {\left|\frac{\beta_{2} }{\beta_{1} } \right|} \\[6pt] {\left|\frac{\beta_{3} }{\beta_{2} } \right|} & {0.5} & {0} \\[6pt] {\left|\frac{\gamma_{3} }{\gamma_{1} } \right|} & {0} & {0.3} \end{array}\right]. \end{align} (28) As it is noted in the above-cited Remark 1, state variables of the slave hyperchaotic system are bounded, and from Fig. 1, we have $$\left |z_{s1} (k)\right |<1.5;$$ thus, it comes   \begin{align} 7.6\left|\frac{\gamma_{1} }{\alpha_{1} } z_{s1} (k)\right|<11.4\left|\frac{\gamma_{1} }{\alpha_{1} } \right|. \end{align} (29) So, a new overvaluing system characterized by the constant matrix Mo, defined by   \begin{align} M_{o} =\left[\begin{array}{ccc} {11.4\left|\frac{\gamma_{1} }{\alpha_{1} } \right|} & {0.3\left|\frac{\gamma_{1} }{\gamma_{2} } \right|} & {\left|\frac{\beta_{2} }{\beta_{1} } \right|} \\[6pt] {\left|\frac{\beta_{3} }{\beta_{2} } \right|} & {0.5} & {0} \\[6pt] {\left|\frac{\gamma_{3} }{\gamma_{1} } \right|} & {0} & {0.3} \end{array}\right] \end{align} (30) which is under the arrow form, is obtained. Actually, and based on the proposed Theorem, since the multi-switching combination synchronization sufficient conditions (24) are true, it stills only to satisfy the sufficient condition (25), expressed, explicitly, as follows:   \begin{align} 1-11.4\left|\frac{\gamma_{1} }{\alpha_{1} } \right|-0.6\left|\frac{\beta_{3} \gamma_{1} }{\beta_{2} \gamma_{2} } \right|-\left|\frac{\gamma_{3} \beta_{2} }{\gamma_{1} \beta_{1} } \right|\frac{1}{0.7}>0. \end{align} (31) From various possibilities relatively to the scaling factors α1, β1, β2, β3, γ1, γ2 and γ3, let choose the following one:   \begin{align} \begin{cases}{l} {\alpha_{1} =\beta_{1} =100} \\ {\beta_{2} =10} \\ {\beta_{3} =\gamma_{1} =\gamma_{2} =\gamma_{3} =1.} \end{cases} \end{align} (32) As stated earlier, the main interest is to achieve multi-switching combination synchronization of the Rössler, Baier–Klein and Hitzel–Zele systems. The efficiency of the proposed method for designing the adapted non‐linear control laws together is proved by means of diverse numerical simulations, given below. From Fig. 2, one can see that the error dynamics of coupled master systems (6) and (7) with the slave one (8), obtained when the control laws are turned off, evolve hyperchaotically and, so, the studied systems are not yet synchronized. Fig. 2. View largeDownload slide Error dynamics evolutions when the proposed control laws are switched off. Fig. 2. View largeDownload slide Error dynamics evolutions when the proposed control laws are switched off. For the studied switch (9) and by respect to the fixed scaling factors (32), the control inputs given by (15) were programmed to turn on simultaneously. The results are shown in Fig. 3, where we can see that the property of multi-switching combination synchronization is, visibly, fulfilled. In addition, Fig. 4 illustrates the temporal behaviour of $$(z_{s1}, 100x_{m1} +10y_{m2}),$$$$\left (z_{s2},\; \text {100}x_{m1} +y_{m3} \right )$$ and $$\left (z_{s3},\; \text {100}\left (x_{m1} +y_{m1} \right )\right ),$$ in the multi-switching compound synchronization state variables, with real-time activation of the developed non‐linear control laws. Fig. 3. View largeDownload slide Multi-switching combination synchronization error states after activating the proposed control laws. Fig. 3. View largeDownload slide Multi-switching combination synchronization error states after activating the proposed control laws. Fig. 4. View largeDownload slide Temporal behaviour of the synchronizing variables. Fig. 4. View largeDownload slide Temporal behaviour of the synchronizing variables. (a) $$\left (z_{s1},\ \ \textrm {100}\ x_{m1} +10y_{m2} \right ),$$ (b) $$\left (z_{s2},\ \ \textrm {100}\ x_{m1} +y_{m3} \right )\!,$$ and (c) $$\left (z_{s3},\ \ \textrm {100}\left (x_{m1} +y_{m1} \right )\right )\!,$$ in the multi-switching compound synchronization states with simultaneous activation of the control laws. 4. Comparative study of different synchronization methods In this paragraph, the comparison amongst some synchronization methods is presented. For this purpose, following synchronization manners are used: (1) multi-switching combination synchronization, (2) hybrid synchronization and (3) projective synchronization. As far as the first synchronization method, its concept and its methodology are, accurately, discussed in the previous sections. In the case of the hybrid synchronization, some of the states of the slave system are completely synchronized and the rest of the states are anti-synchronized, with the states of the master systems. Thus, it is a combination of complete synchronization and anti-synchronization methods. Relatively, to the case of projective synchronization, the states of the slave system are projected with some scaling factor to the respective states of the master systems. The scaling factor can be constant, time-varying or function of states. Here, constant scaling factor is used; hence, generalized projective synchronization method (Vincent et al., 2015) is used and designed. The components of the hybrid synchronization error vector eHS ∈ R3×1, are defined as:   \begin{align} \begin{cases} {e_{HS_{112} } (k)=z_{s1} (k)-x_{m1} (k)-y_{m2} (k)} \\ {e_{HS_{213} } (k)=z_{s2} (k)+x_{m1} (k)-y_{m3} (k)} \\ {e_{HS_{311} } (k)=z_{s3} (k)-x_{m1} (k)-y_{m1} (k),} \end{cases} \end{align} (33) Here, the first and third states are considered for complete synchronization and the second state is considered for anti-synchronization. The average error for hybrid synchronization is shown in Fig. 5. It can be seen from the above-mentioned Fig. 5, that the average error converges to zero. The results regarding control inputs and error convergence are avoided to restrict the length of the paper. Fig. 5. View largeDownload slide Comparison of average errors for different synchronization methods. Fig. 5. View largeDownload slide Comparison of average errors for different synchronization methods. The elements of the projective synchronization error vector ePS ∈ R3×1, are obtained by respect to (34):   \begin{align} \begin{cases} {e_{PS_{112} } (k)=z_{s1} (k)-\delta \left(x_{m1} (k)+y_{m2} (k)\right)} \\ {e_{PS_{213} } (k)=z_{s2} (k)-\delta \left(x_{m1} (k)+y_{m3} (k)\right)} \\ {e_{PS_{311} } (k)=z_{s3} (k)-\delta \left(x_{m1} (k)+y_{m1} (k)\right)}. \end{cases} \end{align} (34) At this point, the scaling factor δ is fixed to 0.5. The average error of the projective synchronization is illustrated in Fig. 5, reflecting, clearly, its convergence to zero. In fact, the average errors, considered as performance measure to compare the three synchronization methods, are defined as follows:   \begin{align} \begin{cases} {\left\| e_{MSCS} (k)\right\| =\sqrt{\left(e_{112} (k)\right)^{2} -\left(e_{213} (k)\right)^{2} -\left(e_{311} (k)\right)^{2} } } \\ {\left\| e_{HS} (k)\right\| =\sqrt{\left(e_{HS_{112} } (k)\right)^{2} -\left(e_{HS_{213} } (k)\right)^{2} -\left(e_{HS_{311} } (k)\right)^{2} } } \\ {\left\| e_{PS} (k)\right\| =\sqrt{\left(e_{PS_{112} } (k)\right)^{2} -\left(e_{PS_{213} } (k)\right)^{2} -\left(e_{PS_{311} } (k)\right)^{2}, } } \end{cases} \end{align} (35) where eMSCS ∈ R3×1 is the multi-switching combination synchronization error vector, such that:   \begin{align} e_{MSCS} (k)=\left[{e_{112} (k)} {e_{213} (k)} {e_{311} (k)} \right]^{T} \end{align} (36) $$\left \| e_{MSCS} (k)\right \|,\ \ \left \| e_{HS} (k)\right \| $$ and $$\left \| e_{PS} (k)\right \| $$ denote the average errors of multi-switch combination synchronization, hybrid synchronization and projective synchronization, respectively. The results of the average synchronization errors are depicted in Fig. 5. It is observed, from the Fig. 5, that the transient and settling time of multi-switching combination synchronization is, comparatively, less than the other two synchronization methods, namely, the hybrid and the projective ones. 5. System communication description In this section, a cryptosystem based on multi-switching combination synchronization of discrete-time hyperchaotic systems is described. The aim of such cryptosystem is to transmit encrypted messages, of several forms, from transmitter A to remote receiver B, as is depicted below, in Fig. 6. Fig. 6. View largeDownload slide Hyperchaotic cryptosystem for secure transmitted signals. Fig. 6. View largeDownload slide Hyperchaotic cryptosystem for secure transmitted signals. A message m is to be transmitted over an insecure communication channel. To avoid any unauthorized receiver (intruder O) located at the mentioned channel; m is encrypted prior to transmission to generate an encrypted message c, such that   \begin{align} c=e\left(m,K\right) \end{align} (37) by using an hyperchaotic system e on transmitter A. The encrypted message c is sent to receiver B, where m is recovered as $$\hat {m}$$ from the hyperchaotic decryption d, as   \begin{align} \hat{m}=d\left(c,K\right). \end{align} (38) If e and d have used the same key K, then at receiver end B it is possible to obtain $$\hat {m}=m.$$ A secure channel represented by a dashed line, Fig. 6, is used for transmitting the keys, K. Generally, this secure communication channel is a courier and is too slow for the transmission of m. Our hyperchaotic cryptosystem is reliable, if it preserves the security of m, that’s to say, if m′≠m for even the best cryptanalytic function h, given by   \begin{align} m^{\prime}=h(c). \end{align} (39) To achieve the proposed hyperchaotic encryption scheme, we appeal to Rössler, Baier–Klein and Hitzel-Zele hyperchaotic maps for encryption and decryption purposes. All Rössler, Baier–Klein and Hitzel–Zele hyperchaotic systems have a number of parameters determining their dynamics; such parameters and initial conditions are the coding and decoding keys, K. We expect that they can perform the objective of the secure communication and the transmitting messages can be recovered at the receiver B. In order to guarantee the encryption and decryption processes, the Rössler, Baier–Klein and the Hitzel–Zele hyperchaotic maps have to achieve the so-called multi-switching combination synchronization on both hyperchaotic transmitter A and hyperchaotic receiver B. Remark 2 It is also possible to improve the security of the proposed communication scheme through the use of two asymmetric encryption and decryption keys. In fact, by referring to the inclusion method (Yang, 1999; Zheng, 2016), the considered hyperchaotic master systems (6) and (7) generate the key K used q times as a key stream to encrypt the original message m with an encryption rule e, a q − shift cipher algorithm, such as:   \begin{align} c=e\left(m,K\right)=\underbrace{F_{1} (\ldots F_{1} (F_{1} }_{q}(m,\underbrace{K),K),\ldots,K)}_{q} \end{align} (40) with   \begin{align} K=\sqrt{\left(x_{m1} +\beta_{3} y_{m3} \right)^{2} }. \end{align} (41) F1(.) is a non‐linear function defined, in this case, by   \begin{align} F_{1} (m,K)=\begin{cases} {m+K+2h,\ \ \textrm{for}\ -2h\le m+K\le -h} \\ {m+K,\ \textrm{for}\ -h<m+K<h} \\ {m+K-2h,\ \ \textrm{for}\ h\le m+K\le 2h} \end{cases} \end{align} (42) h is an encryption parameter chosen such that the transmitted message m and the key K lie within the interval $$\left [-h,h\right ]$$ The slave system (8) generates the recovered key $$\hat {K}$$ used to recover the original message, using a decryption rule d, as following:   \begin{align} \hat{m}=d(c,-\hat{K})=\underbrace{F_{1} (\ldots F_{1} (F_{1} }_{q}(c,\underbrace{-\hat{K}),-\hat{K}),\ldots,-\hat{K})}_{q} \end{align} (43) such that:   \begin{align} \hat{K}=\sqrt{\left(\gamma_{2} z_{s2} \right)^{2} }. \end{align} (44) In the next part, we will use, as it is invoked in the previous Remark, non‐identical keys for the audio signal transmission. 6. Real-world application to encrypted audio communication Multi-switching combination synchronization using the new proposed approach is fulfilled successfully with the assumption that the parameters of drives and response systems are known and states of all systems are measurable. Potential applications (Hammami, 2015; Hernandez & Nijmeijer, 2000) of synchronized hyperchaotic and chaotic systems may be used in secure communication. The experimental approach is based on masking the message, to be sent, with the hyperchaotic signal at transmitter end as driving signal and recovered back the same message at receiver end, once synchronization transmitter/receiver is accomplished. At this stage, we describe the communication system based on multi-switching combination synchronization hyperchaos. We will use the discrete-time hyperchaotic systems (6) and (7) as hyperchaos generators. With this scheme, we obtain faster synchronization and higher privacy; one channel is used to send the hyperchaotic synchronizing signal $$\left (\alpha _{1} x_{m1} +\beta _{2} y_{m2} \right )$$ from the transmitters (6) and (7), with no connection with the secret audio message m. Whilst the other channel is used to transmit hidden message m which is recovered at the receiver end, by means of the comparison between the signals $$\left (c=\left (x_{m1} +\beta _{3} y_{m3} \right )+m\right )$$ and $$\left (\hat {c}=\gamma _{2} z_{s2} +\hat {m}\right ),$$ Fig. 7. Fig. 7. View largeDownload slide Proposed audio communication scheme. Fig. 7. View largeDownload slide Proposed audio communication scheme. Then, via numerical simulations, we illustrate the encrypted audio transmission. We use as transmitter the discrete-time hyperchaotic systems given in (6) and (7), and the discrete-time hyperchaotic system given in (8) is looked as a receiver. For both the encryption as well as the decryption phases, let’s take h = 0.3 and q = 15. The format of the audio signal m is Pulse-Code Modulation of 22.05 KHz, 16 Bits, monofonic channel. The mentioned audio message m is to be encrypted and transmitted to the receiver. Figure 8 shows the original transmitted audio message m to be encrypted and transmitted, Fig. 9 illustrates the encrypted audio message c, and from Fig. 10, it is clearly shown that the original audio message $$\hat {m},$$ can be, faithfully, recovered by the receiver. Fig. 8. View largeDownload slide The original transmitted audio message m. Fig. 8. View largeDownload slide The original transmitted audio message m. Fig. 9. View largeDownload slide The encrypted audio message $$\left (c=\left (x_{m1} +\beta _{3} y_{m3} \right )+m\right ).$$ Fig. 9. View largeDownload slide The encrypted audio message $$\left (c=\left (x_{m1} +\beta _{3} y_{m3} \right )+m\right ).$$ Fig. 10. View largeDownload slide The recovered audio message $$\hat {m.}$$ Fig. 10. View largeDownload slide The recovered audio message $$\hat {m.}$$ Obviously, it is easy to conclude that the information signal is recovered after a short time of transmission. Besides, the proposed method of encryption, based on multi-switching combination hyperchaos synchronization of three non‐identical systems, compared to the original chaos masking mode, can improve the security of communication. In actual fact, the main advantage of using hyperchaotic systems for synchronization study is related to the difficulty to predict all the parameters and properties of such complex systems, with more random and unpredictable behaviour than chaotic ones. At this step, the performance of the proposed encryption scheme is evaluated through several aspects. 7. Performance and evaluation of the proposed cryptosystem In this paper, the multi-switching combination synchronization of discrete-time hyperchaotic systems has been studied and an encryption system based on hyperchaotic maps has been proposed. We have shown an efficient way of constructing secure audio data transmission using a new approach stabilizing non‐linear discrete-time systems. Following, a brief discussion of the performance of such proposed cryptosystem is given. 7.1. The noise resistance effect for the proposed cryptosystem A transparency test was conducted with the transmitted audio signal with different measures in terms of signal to noise ratio, commonly abbreviated as SNR. SNR is defined as the ratio of the power of a signal or meaningful information and the power of background noise or, similarly, unwanted signal:   \begin{align} SNR=\frac{P_{signal} }{P_{noise}}, \end{align} (45) where P is the average power. Both signal and noise power must be measured at equivalent points in a system and within the same system bandwidth. Since, many signals have a very wide dynamic range, they are, habitually, expressed using the logarithmic decibel scale. So, based upon the definition of decibel, signal and noise may be expressed in decibels (dB) as following:   \begin{align} SNR=10\log \left[\frac{P_{signal} }{P_{noise} } \right]. \end{align} (46) Figure 11 shows the SNR of 100 original host audio pieces and other unwanted ones using our proposed algorithm. The average SNR using our new method is 41.12 dB, which is totally acceptable since it satisfies, clearly, the requirement announced by the international federation of the phonographic industry: SNR should be higher than 20 dB. Therefore, the proposed scheme is highly secure against statistical attacks. Fig. 11. View largeDownload slide SNR of 100 original host audio pieces and unwanted ones using our proposed communication scheme. Fig. 11. View largeDownload slide SNR of 100 original host audio pieces and unwanted ones using our proposed communication scheme. 7.2. Security analysis of the proposed cryptosystem In our proposed encryption algorithm, three hyperchaotic systems are used in order to increase the complexity of this algorithm as well as to make the key space larger. In such a way, the security which it is the main concern of one cryptosystem is improved. Hence, the key space of our encryption algorithm is large enough against potential attacks thanks to the wide range of variation, relatively to the parameters of hyperchaotic systems. 7.3. Discussion of the feasibility and usefulness attained by the obtained results In our cryptosystem, the processes of encryption and multi-switching combination synchronization are completely separated with no interference between them. So, encrypted information does not interfere with the proposed type of synchronization, therefore not increasing the sensitivity of such synchronization to external errors. As a result, the hyperchaotic communication scheme with two transmission channels gives faster synchronization between transmitter and receiver, in one hand, and high security of data transmission, in the other hand. Furthermore, the secure communication systems, considered in this work, relied on three discrete-time hyperchaotic systems; however, the obtained results can be generalized to more than three continuous-time as well as discrete-time hyperchaotic systems. This flexibility is an added advantage of the proposed contribution. In addition, many different types of messages can be transmitted that have different frequencies via simply adjusting the time scaling factor, which preserves the topological and geometrical properties of the phase diagram, relatively to the hyperchaotic transmitter. Consequently, based on the pre-cited advantages of the proposed cryptosystem and the new multi-switching combination synchronization approach, we can conclude to the high degree of security as well as the robustness against noise and several attacks, according to the developed encryption scheme. 8. Conclusion and prospect To sum up, we have studied a novel scheme of hyperchaotic synchronization that can involve numerous dynamical systems, namely multi-switching combination synchronization of three hyperchaotic systems, based on non‐linear control approach. In this new synchronization scheme, the state space variables of the three systems are multi-switched, such that their mutual synchronization takes place between different state variables. When synchronization is achieved, satisfactorily, in this manner in the communication context, it would be difficult or even impossible for an intruder to predetermine the vector space in which synchronization would occur, thereby, enhancing information security. Comparison amongst different synchronization methods is done, in order to reflect that the proposed synchronization, namely the multi-switching combination one, is better than the others. Numerical simulations, security analysis and noise interference investigation, using there the Rössler, Baier–Klein and the Hitzel–Zele maps were also employed to illustrate the effective performance of the proposed control strategy that is outlined in this paper, by demonstrating that our algorithm has large key space, high security and good immunity to noise interference. 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IMA Journal of Mathematical Control and InformationOxford University Press

Published: Jan 29, 2018

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