ES2724598B2 - SECURE COMMUNICATION SYSTEM BASED ON MULTI-STABILITY - Google Patents

Description

DESCRIPTION

SECURE COMMUNICATION SYSTEM BASED ON

MULTI-STABILITY

OBJECT OF THE INVENTION.

The present invention is directed to a multi-stability based secure communication system.

The present invention is related to the field of secure communication systems, methods, digital electronic devices and computer programs, more specifically it refers to a system for highly secure communications based on dynamic systems with the coexistence of multiple chaotic attractors.

BACKGROUND OF THE INVENTION.

Every time a password is used, crypto is being used. Every time a financial transaction is made online, crypto is being used. If you want to have a secure voice or video call, where "secure" means that no one but the participants can attend attend, then you have to use crypto. If you want to write a message that only the recipient is able to read and no one else , it is necessary to keep a password between the participants of the information transmission.

To keep some information secret, cryptography is used. In cryptography, encryption or encryption mechanisms are used, by which an initial message, called plain text, is taken and encoded. That is, the message is changed to something that by itself cannot be read, called encrypted-text. To read the encrypted text, you need to decode this message. This process is called decoding. This should only be done by someone authorized. Otherwise the encoding is said to have been broken. Coding and decoding work using a security key that is an instrument to hide and reveal the information that you want to protect.

Using encryption, authentication and access control technologies to secure communication systems is a suitable solution to avoid spying attempts and theft of data transmission. One contribution to solving this problem is to use signals generated by components operating in the nonlinear regime, such as multi-stable chaotic oscillators.

In general, chaotic systems show sudden and dramatic changes that give rise to a complex behavior called "chaos." This behavior describes the aperiodic temporal evolution of a system. That is, it never repeats and apparently is random, but completely deterministic, with a strong dependence on initial conditions A system is said to be deterministic if the equation of time evolution, the parameters that describe the system, and the initial conditions are known.

In the state of the art of communication systems that use chaotic systems to encode information, the transmission of the key is required for decoding information. Among the patents that use these methods we can mention: U.S. Pat. No. 5,048,086 and Weiss U.S. Pat. No. 5,479,512, which consists of generating a sequence of random numbers in digital format produced by a chaotic system, the message to be encoded in digital format is attached to it and the combined signal is transmitted. The receiver extracts the digital message from the transmitted combined signal using a key that generates the same sequence of random numbers in digital format, produced by a chaotic system. The disadvantage of this method is due to decreased security as a result of key or key transmission.

One way to solve this problem using chaotic communication security systems where key transmission is not required is shown in the Carroll US Pat patents. No. 5,473,694 and Cuomo US Pat. No. 5,291,555. In summary, this method consists of modulating a parameter of a chaotic signal with a signal that carries the information or adding a signal that carries information to a chaotic signal. The resulting chaotic signal is transmitted using conventional transmission technologies from a transmitter containing an encoder to a receiver containing a decoder. The decoder in the receiver synchronizes the chaotic signal from the receiver with the original chaotic signal without the need for key exchange. Comparison of the resulting chaotic signal with the synchronized signal allows the original information to be extracted. However, these communication systems sure that they use synchronization have a deficiency, that is, if the transmission is intercepted, it is possible to reconstruct the phase space to extract the underlying dynamics of the transmission encoder allowing to extract the information signal using another non-linear chaos generator system or other technology .

Regarding proprietary technology in this field based on chaotic systems, countless patent documents have been located, for example, US 7490246 B2, US 2010 / 007445A1, US 2018 / 0032475A1, WO 2010/034728 A1, WO 2011/105972 A1, EP1129542B1, US8340295 B2, US 2009/0285395 A1, EP2359519A1, CN103957098B, CN1852089B, CN202362970U, CN106130713A, CN105827393A, CN105744294A, CN104618091A, CN104320241A, CN103532696A, CN103220127A, CN103297221B, CN101345615A, KR198150B1, KR2000009822A, CN107437266A, CN105704500A, CN105634723A, CN105429706A , CN103415009A, US20110002460A1, US20100074445A1.

However, each of the technological references found in the state of the art solves specific problems related to the points mentioned above in the Carroll U.S. patents. Pat. No. 5473694 and Cuomo U.S. Pat. No.

5291555. These secure communication systems based on chaotic signal synchronization have a weakness against synchronization attack. Synchronization-based secure communication systems use chaotic nonlinear systems coupled in a master-slave configuration, where the master represents a sender and the slave a receiver, the sender's output signal is used to encode or mask the information signal.

The brute force attack by synchronization of these communication systems includes the interception of the communication channel and the simulation of the receiver by a virtual model that adjusts all the parameters until the best synchronization or the smallest synchronization error is reached. When the sender and virtual receiver reach the same parameter values, it is when there is complete synchronization between these two systems and the extraction of the information signal is obtained by comparing the synchronized signal of the virtual receiver with the signal from the emitter.

The timing sensitivity to parameter changes of chaotic systems is very crucial for communication security. A slight variation in one of these parameters can cause the timing error to be so large that the Retrieval of information signal is impossible. In itself, we can consider two types of parameters: group of parameters related to non-linear chaotic systems (emitter-receiver) and the parameter underlying the synchronization time. The last parameter being the threshold value to achieve full synchronization. For values less than this synchronization time, the complete synchronization is not reached and the information retrieval is impossible.

When the traditional chaos communication scheme is implemented, the master chaotic oscillator and the slave chaotic oscillator must be identical or one against the other. When a small information signal m ^E ( t) is added with the signal of the master chaotic oscillator del and ^E m ^{is or} ( t) it produces s ( t ) as s ( t ) = m ^E ( t ) and ^{Em ís or} ( f í, which is then transmitted to the receiver. At the receiver, the signal s ( t ) is used to synchronize the slave chaotic oscillator with the master chaotic oscillator, the generation of the slave chaotic signal and ^{E m iSOr} ( t) it is difficult because there is an incomplete synchronization or intermittent which impossible to recover the information signal m ^R. This is a consequence of the process of adding information m ^E (t) to signal the chaotic master oscillator and ^{E m ISOR} (t), making the latter oscillator not identical to the chaotic oscillator slave, and as a result the difficulty of recovering the information signal m ^R.

Furthermore, secure communication systems based on chaotic signal synchronization have weakness against synchronization attack. The synchronization attack of these communication systems includes the interception of the communication channel and the simulation of the receiver by a virtual model that adjusts all the parameters until the best synchronization is reached or the least synchronization error is reached. Synchronization error is achieved when the synchronization time between the virtual receiver and the sender exceeds a threshold time r ^k where the synchronization error is minimal. More specifically, synchronization attacks in secure communication systems based on chaotic signals are described in the following articles: Zanin, M., Sevilla-Escoboza, JR, Jaimes-Reátegui, R., Garcia-López, JH, Huerta-Cuellar, G., & Pisarchik, AN ( 2013). Synchronization attack to chaotic communication systems. Discontinuity, Nonlinearity, and Complexity 2 ( 4) ( 2013) 333-34; García-López, JH, Jaimes-Reategui, R., Chiu-Zarate, R., Lopez-Mancilla, D., Jimenez, R., & Pisarchik, AN ( 2008). Secure computer communication based on chaotic Rossler oscillators. Open Electrical & Electronic Engineering Journal, 2, 41-44.

DESCRIPTION OF THE INVENTION

To avoid these drawbacks, a new communication system using a changing dynamic private key operating on a private channel and another static key is proposed in the present invention. The private channel is performed by one of the sender's state variables that properly communicates or couples with the receiver and produces a dynamically changing complete synchronization between the receiver and the sender.

Furthermore, the present invention solves the drawback of synchronization attack, by implementing in the private communication channel a novel method of dynamically changing synchronization between the receiver and the sender by means of the coexisting chaotic multiple attractors in the master oscillator. This is materialized using different variations or changes in a chaotic way of the initial conditions of the variables of the same master chaotic oscillator and that work as dynamic secret keys allowing the change of the attractors in the receiver and in the emitter in a chaotic way. The chaotic change of chaotic attractors with intervals shorter than the synchronization time makes the synchronization attack impossible to break the security of this communication system.

Therefore, the main objective of the present invention is to propose a highly secure communication system, the security of which is achieved with a scheme consisting of a transmitter and a receiver based on multi-stable chaotic systems. The "multistability" of a dynamic system means that the system has the coexistence of a large number of attractors. An "attractor" is a set of numerical values towards which a system tends to evolve, given a great variety of initial conditions of the variables of the state of the system. A dynamic "system state" is a solution that remains confined to one area of its state space or attractor. Full synchronization is only possible when coupled dynamic systems are identical and are on the same attractor.

In one aspect of the present invention a multi-stability based secure communication system is disclosed comprising a sender and a receiver communicating with each other through a public channel and a private channel. In this way the sender receives an information signal to be transmitted mE ( t ), which sends encrypted s ( t), by means of coexisting chaotic masking attractors, to the receiver on the public channel, and where the sender sends a dynamic key to the receiver on the private channel for synchronization between the sender and the receiver and the decoding of the encoded signal s ( t) obtaining a msaiida ( t) signal at the receiver equal to the input mE ( t ) signal at the emitter.

In one embodiment, the emitter comprised in the multi-stability based secure communication system comprises:

• a vectoring unit configured to receive an information signal to be transmitted mE ( t), which is vectorized and divided into d mlE ( t) information packets \

• an initial conditions generating unit where the initial conditions and me are located in the interval ° \ ym.l_inicial'yml_final \;

• a multi-stable master chaotic dynamic system that generates d coexisting chaotic masking chaotic attractors that vary chaotically by changing the initial conditions ylmi of the variable and me in equations [3] to [6] through the initial conditions generating unit and solving the following differential equations:

dx _me

dt% m2% m3 [i]

d% ^m2

dt =% m2 V ^ m2 m

d% ^m3 = b - cxm3 ymíym3

dt [3]

dymí [4]

dt% mi Vmí% m2% m3

dym2

dt Vmí V ^ m2 [5]

dym3

dt = b and ^{m 3} ( y ^{m í} - c) [6]

with a = 0.2, b = 0.2 and c = 5.7;

• an encryption unit configured to combine the d mlE ( t) information packets with the d co-existing chaotic masking attractors ylmí ( t) obtaining a signal s ( t) such that s ( t) = ymí ( t) mE ( t), where the signal s ( t) is sent through a public channel to a receiver y where the signal x ^{m í} ( t) of equation [1] is sent to the receiver through a private channel.

In an embodiment of the multi-stability based secure communication emitter, the multi-stable master chaotic dynamic system comprises two Rossler oscillators with non-linear coupling between their state variables, where the behavior of the first oscillator is defined by the differential equations described above [1] to [3] and the behavior of the second oscillator is defined by the differential equations previously described [4] to [6].

In an embodiment of the multi- stability based secure communication emitter, the initial condition generating unit chaotically modulates the interval \ ym \ j niciai> ymijínai] through a chaotic logistic map defined by the following equation:

* (n + l) = ^ * n (l Xn), [Ll]

where x ⁿ is defined in the interval [0, l] and p in the interval [0,4], and the modulation of the initial conditions ymi_o in the interval [y ^{m í} j ^{n iCia i} , y ^mi j ^{in ai} ] It is defined by the following equation:

Y ^{m l_0} Y ^{m l_ in ic ia l} + X ^neither ( and ^{m i_ ^ in a i} Y ^{m í_ in ic ia l} ), [^ 2 ']

where x ^nt represents the logistic map time series (L1) in the chaotic regime and n 1 are the iterations.

In one embodiment of the issuer secure communication based on multistability, d = 751 last iterations of the time series x ^{n 1,} after the first 100 transient elapsed iterations is selected.

In an embodiment of the multi-stability based secure communication sender, each information packet d has a size v p equal to 90 iterations and a duration time t me ( t) of 18 seconds.

In one embodiment, the multi-stability based secure communication receiver comprises:

• a first decoding unit with an input connected to a public channel to receive a signal s ( t) from the transmitter;

• a second decoding unit connected to an output of the first decoding unit;

• a multi-stable slave chaotic dynamic system that generates a chaotic slave masking signal y si ( t) from a signal x mi ( t) received from a sender through a private channel, and solving the following differential equations:

where k is the coupling factor between the emitter and the receiver;

where the first decoding unit subtracts the slave chaotic masking signal and si ( t) from the signal s ( t) coming from the emitter, obtaining a recovered message signal m R ( t) that reaches the second decoding unit that subtracts a synchronization error el ( t) to the recovered message signal m R ( t) to obtain the signal m out ( t). M involves leaving the signal (t) at the output of the receiver coincides with the signal to mE (t) input emitter was desired transmitted between transmitter and receiver in encrypted form.

In an embodiment of the multi-stability based secure communication receiver, the multi-stable slave chaotic dynamic system comprises two Rossler oscillators with non-linear coupling between their state variables, where the behavior of the first oscillator is defined by the differential equations described above [7] to [9] and the behavior of the second oscillator is defined by the differential equations previously described [10] to [12].

BRIEF DESCRIPTION OF THE FIGURES

Figure 1 shows the secure communication system using chaotic signals.

Figure 2 shows the local maxima of the variable x ^{m í} as a function of the initial conditions and ^{m m} 0. Each state of the variable x ^{m í} belongs to a different coexisting chaotic attractor.

Figure 3 ( a) shows the chaotic masking state of the variable and ^{m i} ( t ) as a function of time for a chaotic attractor.

Figure 3 ( b) shows the chaotic masking state of the variable y ^{m i} ( t) as a function of time for another chaotic attractor, using different initial condition and ^m 0.

Figure 4 shows the chaotic variation of the initial conditions and ^m 0 modulated by the logistic map, equation (L1).

Figure 5 schematically represents a multi-stability based secure communication system 500 according to the present invention based on multi stable chaotic systems.

Figure 6 depicts the schematic for any digital electronic device of a multi-stability based secure communication system 500a according to the present invention.

Figure 7 ( a) shows a graph of a flat image.

Figure 7 ( b) shows an information signal packet ml ^E ( t) versus time.

Figure 7 ( c) shows a portion of the information signal m ^{E ntrada} ( t) versus time.

Figure 8 ( a) shows a chaotic masking state and l ^{m i} ( t) as a function of time generated by the master chaotic system 250a .

Figure 8 ( b) shows the transmission signal sl ( t) = yl ^{m i} ( t) m ^E ( t) which is the combination of a chaotic masking state and l ^{m i} ( t) with an information packet m ^lE ( t) performed by chaotic masking unit 150a.

Figure 8 ( c) shows the graph s ^l ( t) versus time, where it is included; the synchronization time ^t s, the time required for the slave chaotic system to synchronize with the master chaotic system, and the transmission time r ^t , the time required to transmit each information packet m ^lE ( t) masked in each multi-stable chaotic state and ^{í ni} it).

Figure 8 ( d) shows a portion of the transmitted signal s ( t) = y ^m ( t) m ^E ( t) , corresponding to a very large number of chaotic masking state and ^{m í} ( t) and a very large number of information signals m ^E ( t).

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Figure 9 ( a) shows a state of chaotic masking xlmi ( t) versus time, generated by the master chaotic system, equations 250a.

Figure 9 ( b) shows the signal ^ (t) versus time.

Figure 9 ( c) shows the signal ^ (t) versus the signal ylmí ( t).

Figure 10 ( a) shows the recovered message signal m jj (t) versus time.

Figure 10 ( b) represents a portion of the recovered message mR ( t) = e ( t) mE ( t) versus time.

Figure 11 ( a) shows a successfully recovered message signal packet mLuda ( t) versus time.

Figure 11 ( b) shows a portion of the successfully recovered message signal msalldd ( t) corresponding to a very large number of recovered signals misallda ( t).

Figure 12 plots a portion of the signal successfully retrieved beyond ( t) versus a portion of the original lying signal ( t).

Figure 13 shows a flat image successfully recovered.

Figure 14 ( a) shows the signal y ^ t) versus time, for representative example 2.

Figure 14 ( b) shows the signal y ^ t) versus the signal yiml ( t), for representative example 2.

Figure 15 ( a) shows the recovered message signal m ^ t) versus time, for representative example 2.

Figure 15 ( b) represents a portion of the recovered message mR ( t) versus time, for representative example 2.

Figure 16 ( a) shows an incorrectly retrieved message signal packet msaiida ( t) versus time, for representative example 2.

Figure 16 ( b) shows a portion of the incorrectly retrieved message signal msalidd ( t), for representative example 2.

Figure 16 ( c) represents a portion of the incorrectly recovered signal msaiidad ( t) versus a portion of the original lying signal ( t), for representative example 2 .

Figure 17 shows an incorrectly recovered flat image, for representative example 2.

Figure 18 depicts a secure communication system 500b that can be implemented in any digital electronic system, for representative example 3.

Figure 19 ( a) shows the signal y ^ t) versus time, for representative example 3.

Figure 19 ( b) shows the signal ^ (t) versus the signal ylmí ( t) , for representative example 3.

Figure 20 ( a) shows the recovered message signal m jj (t) versus time, for representative example 3.

Figure 20 (b) represents a portion of the recovered message mR ( t) versus time, for representative example 3.

Figure 21 ( a) shows an incorrectly retrieved message signal packet msaiida ( t) versus time, for representative example 3.

Figure 21 ( b) shows a portion of the incorrectly retrieved message signal msalidd ( t), for representative example 3.

Figure 21 ( c) plots a portion of the incorrectly recovered signal plus quality ( t) versus a portion of the original lying signal ( t), for representative example 3. Figure 22 shows a flat image recovered incorrectly.

PREFERRED EMBODIMENT OF THE INVENTION

The implementation of the present invention is directly related to the field of secure communication systems, method, digital electronic devices and computer programs, more specifically it refers to a highly secure communication system based on multi-stable chaotic systems. Additionally, the present invention can be implemented in digital electronic circuits as in wireless communication systems.

A detailed description of the present invention is presented below. The present invention is a highly secure communication system based on multi-stability, whose security is achieved with a scheme consisting of a transmitter and a receiver based on multi-stable chaotic systems. Additionally, the chaotic variation of the initial conditions in any of the state variables in the multi-stable system serves as a dynamic key. Communication between sender and receiver is achieved through a private channel and another public channel. The private channel is made by the state variable xmi where the dynamic keys are implemented and at the same time the sender is used with the receiver, while the information signal is transmitted from the sender to the receiver on the public channel, through the state variable and me.

In general, in accordance with the present invention, secure communication systems using chaos include a sender and a receiver. The emitter includes at least one master chaotic oscillator. Likewise, the receiver includes a chaotic slave oscillator. Also, the master and slave chaotic oscillators are identical or are counterparts to each other.

Both the emitter and the receiver are comprised of two Rossler oscillators with non-linear coupling between their state variables. By varying the initial conditions in any of the state variables, the system displays a multi-stable dynamic, from coexisting periodic and / or chaotic regimes (attractors). Communication between the sender and receiver is achieved through two channels: a private channel and a public one. The private channel is made by the state variable xmi and is used to synchronize the sender with the receiver, at the same time it serves as the first dynamic key, while the information signal is transmitted from the sender to the receiver on the public channel, through the state variable and me. Information transmission is accomplished by adhering a very large number of information signal packets in a staggered fashion to a very large number of chaotic masking signals with changes between chaotic attractors within the same time series of the state variable and me.

In particular, the emitter comprises a "master chaotic system" corresponding to two Rossler chaotic oscillators with non-linear coupling between their state variables, which generates multiple coexisting chaotic attractors. This "master chaotic system" has two outputs. A "first output" configured to send a large number of information signal packets and transmit it together with a large number of chaotic masking signal through interruptions between coexisting chaotic attractors on a public channel and a "second output" configured to transmit a chaotic signal through a private channel, which works as the first dynamic key and serves to synchronize the transmitter with the receiver. The emitter has a unit configured to sum a very large number of chaotic masking signal from the first output of the chaotic dynamic system with a very large number of information signal packets.

For its part, the receiver is structurally the counterpart of the sender and has two inputs: a "first input " that communicates or couples the sender with the receiver through the public channel and a "second input" that communicates or couples through of the

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private channel the sender with the receiver. The receiver comprises a "slave chaotic system" which is the counterpart to the sender's dynamic master system and consists of two Rossler oscillators with non-linear coupling between their state variables. This slave chaotic dynamic system has an input that communicates or couples the transmitter and receiver on the private channel through the second output signal of the master chaotic system, which is configured to generate a very large number of chaotic attractors and serves to synchronize the receiver with the emitter.

The chaotic master system and the chaotic slave system can operate in a synchronized manner only when they are in the same attractor. This slave chaotic system has an output configured to generate a version of a very large number of chaotic masking signals through interruptions between co-existing chaotic attractors equivalent to the very large number of chaotic masking signals of the master chaotic system. This chaotic slave system has a unit for extracting a very large number of information signal packets, which is performed by subtracting the version of a very large number of chaotic masking signal packets from the slave system from the signal transmitted by the public channel. .

The emitter comprises a master chaotic system corresponding to two Rossler chaotic oscillators with non-linear coupling between their state variables, which generates multi-stable chaotic signals and which mathematically corresponds to the following set of coupled differential equations:

dt% m2 ^ m3 [i]

d% ^m2

dt ^ m2 V ^ m2 m

= b - cxm3 ymíym3

dt [3]

dy ^me

dt ^ ml Vml% m2 ^ m3 [4]

^ V ^m2

dt Vml V ^ m2 [5]

dy ^m.3 = b ym3 ( yml - c) [6] _dt

where the equations of [1] - [3] correspond to the first Rossler master chaotic oscillator and the equations [4] - [6] correspond to the second Rossler master chaotic oscillator.

The receiver is structurally the counterpart of the emitter and comprises a chaotic slave system corresponding to two Rossler chaotic oscillators with non-linear coupling between their state variables, which generates the coexistence of multiple chaotic attractors and mathematically corresponds to the following set of coupled differences equations :

_dt ~ xs2 xs3 k ( Xm- ± Xsl)

dxs2

= xs2 ays2

dxs3

_dt = b - cxs3 and sly s3 [9]

where equations [7] to [9] correspond to the first Rossler slave chaotic oscillator and equations [10] to [12] correspond to the second Rossler slave chaotic oscillator.

The first Rossler chaotic slave oscillator is configured to receive a set of secret dynamic keys through the term k ( xmí - xsl) in equation [7], where k coupling factor of the between the emitter and the receiver and works as a static key .

As a first preliminary process, the generation of multiple attractors is performed in the initial conditions interval [ymn niCiai> ym ± final] corresponding to the state variable ymi of the second master chaotic oscillator.

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As a second previous process, the coupling factor k and the synchronization time rk are determined in the private channel k ( x ^{m í} - xsl) in equation [7], where the receiver is fully synchronized with the emitter in each coexisting attractor .

As a third previous process, the interval of initial conditions [y ^{m í _ln lc la i} , and ^{m i_ f in ai} ] is chaotically modulated through a chaotic logistic map.

As a fourth previous process, the information signal is vectorized in information packets equivalent to the number of the chaotic attractors.

When the master-slave chaotic oscillators are communicated or coupled by a common signal, these oscillators can operate in a synchronized manner and even when both oscillators are initially operating under different initial conditions.

In a traditional chaos communication system Figure 1, a small information signal m ^E ( t) is combined with a chaotic signal and ^E mi ^sor ( t) to mask, hide, encrypt, encode or cover up the information signal m ^E ( t). In the master chaotic transmitter or system the information signal m ^E ( t) and the chaotic signal or masking signal and ^{E m iso r} ( t) are added together to produce the transmission signal s ( t) = m ^E ( t) and ^{E m iso r} ( t). The signal s ( t) is known as the chaotically masked information signal.

In the receiver or chaotic slave system the signal s ( t) is received and uses this signal to generate the chaotic signal and ^{R eceptor r} ( t) of the receiver by synchronization between the transmitter and the receiver. In the receiver signal and ^{R EPEC to r} (t) of the signal s (t) to unmask, decode, decrypt, extract or recover the m signal ^R signal is subtracted.

Mathematically it is represented as m ^R = s ( t) - y ^{R ecep tor} o m ^R = m ^E ( t) y ^{E m ísor} y ^{R ecep to r} ' which produces T ^ ^R ^ ^{■ E} when y ^{E m isor} e V ^{R eceptor} are identical.

In multi-stable Rossler dynamic systems with non-linear coupling, both the emitter and the receiver comprise two Rossler chaotic oscillators with non-linear coupling between their state variables. These chaotic oscillators are expressed by the following set of differential equations:

1

where xltx2, x3, and lty2, y3 are state variables, with a = 0.2, b = 0.2 and c = S.7 are system parameters operating in the chaotic regime when Rossler oscillators are decoupled and at the same time serve as static keys , in the present invention.

To demonstrate a large number of coexisting or multi-stability attractors, the initial condition of a singular variable ymi 0 is varied and the final state is found by numerically solving the system of equations [1] to [6]. This process can be implemented electronically or numerically.

In Figure 2, the local maxima of the variable xmi are shown as a function of the initial conditions and mi 0, which belong to different coexisting attractors. In this Figure 2, regimes such as period-3, period-4, period-5, period-8 and chaos are observed. The entire graph stands for a bifurcation diagram, but we emphasize that the axis of the abscissa is not a control parameter but are initial conditions of the state variable and me of the coupled oscillators. In Figure 2 we can distinguish two labels ymí initiai and ymí f inai, which represent initial and final condition, where the system of differential equations [1] to [6] shows the coexistence of multiple chaotic attractors or multi-stability.

The emitter based on multi-stable chaotic systems comprises a master chaotic system based on two Rossler oscillators with non-linear coupling between their state variables and is described by the following system of differential equations [1] - [6]:

dt = -Xm2 ^ m3 [i]

dt% m2 O-V-ml m

dXm.3

b CXm 3 ym.iym.3 _dt [3]

dym i

dt ^ ml y-mi m ^ .2 ^ m3 [4]

V ^ m2

dt y ^-mí + ^ V ^m2 [5]

dym.3

_dt b ym3 ( ymi - c), [6]

where xmi, xm2, xm3 are state variables of the first master chaotic oscillator and ymi, ym ² , ym ³ are state variables of the second master chaotic oscillator, with a = 0.2, b = 0.2 and c = 5.7 are operating system parameters in the chaotic regime when Rossler oscillators are decoupled and simultaneously serve as static keys, in the present invention.

The signal xmi ( t) of equation [1] is used to communicate, pair or synchronize the receiver with the transmitter and is carried out through the private channel, which in turn serves as the first dynamic key. While the chaotic masking signal and me ( t) of equation [4], is used to encrypt, encode, hide or hide the information signal mE ( t). The transmission signal s ( t) results from the combination of the information signal mE ( t) and the chaotic masking signal and me ( t). The signal transmitted from the sender to the receiver is

s ( t) = yml ( t) mE ( t).

The emitter includes a chaotic generation and modulation process of the initial conditions and mi 0 of the state variable and mi of equation [4]. Changing the initial conditions and me 0 in the interval [ymlJniciai, ymijinai] in a chaotic manner results in interruptions between the multiple coexisting chaotic attractors and me ( t).

Two of these states are represented in Figures 3 ( a) and ( b).

The chaotic generation and modulation process of the initial conditions ymí 0 of the state variable ymí is carried out by the initial conditions generator unit 175, (see Figure 5), and more specifically by a logistic map that defines the following equation in differences:

1

X ( n + 1) = M * n (l Xn), [L l]

where xn is defined in the interval [0,1] and p in the interval [0,4], when p = 4 the logistic map shows chaotic behavior.

The change of the initial conditions ymi_o in the interval [ymí initiai, y mi final] is defined by the following equation:

y ^{ml _0} y ^{m l_ in ic ia l} + X ^nl \ y ^{m l_ end al} y ^{m l_ in ic ia l} ), [L2]

where x ^{n 1} represents the time series of the logistic map [L1] in the chaotic regime and n 1 are the iterations. In the present invention, the last d = 751 iterations of the time series x ^{nt were selected} , after the transitory iterations n ^{tra ns lto r to s} = 100 had elapsed. Likewise, in the present invention it is configured so that the equations [ L1] and [L2] can be implemented in any digital electronic system. The chaotic variation of the initial conditions and ^{m 0} is shown in Figure 4.

In the implementation of the present invention, equation [L2] represents the initial conditions generating unit 175 (see Figure 5). This unit 175 (see Figure 5) generates d = 751 initial conditions y ^{m í} 0, which are the results of modulating the interval of initial conditions [y ^{m í in iCiai} , ^mifinal ] by the time series x ^{n 1} of the map chaotic logistics [L1]. In turn, the variable y ^{m í} uses these y ^m 0 initial conditions for the master chaotic dynamic system, equations [1] - [6], to generate d = 751 chaotic masking signals through interruptions between multiple chaotic attractors coexisting; y ^{m i} ( t) and x ^{m i} ( t) are transmitted from the sender to the receiver by the public and private channels respectively.

The multi-stable chaotic system based receiver of the present invention comprises a slave chaotic system and is structurally the counterpart to the master chaotic system. This chaotic slave system is based on two Rossler chaotic oscillators with non-linear coupling between their state variables, which generates multi-stability and is mathematically defined by the following set of coupled differential equations:

dxyes

_dt = X ^{s 2} - xs3 k ( xmí - xsí) [7]

1

where xsVxs2, xs3 are state variables of the first slave chaotic oscillator and yst, yS ² , and ^S3 are state variables of the second slave chaotic oscillator, with a = 0.2, b = 0.2 and c = 5.75 system parameters operating in the regime chaotic when Rossler oscillators are decoupled and simultaneously serve as static keys in the present invention.

The chaotic slave system of the receiver 450 (see Figure 5) comprises a first Rossler oscillator, which is configured to receive by the private channel, the signal xmi ( .t) from the transmitter through the term k ( xmí ( t) - xsí ( t)) in equation [7], where k coupling factor between the emitter and the receiver. That is, the slave chaotic masking system uses the master chaotic masking signal xmi ( t) to generate the version of the slave chaotic masking signal ys l (t) by synchronization between the sender and receiver. Additionally, ys l (t) is subtracted from the information signal s ( t) at the receiver to unmask, decode, decrypt, extract or recover the mR ( t) signal . Mathematically it is expressed as:

mR ( t) = s ( t) - ysí or mR ( t) = mE ( t) ymí ( t) - ysí ( t) which produces

mR ( t) = e ( t) mE ( t)

when ymi ( t) and ysl are identical, where e ( t) is the signal of the synchronization error.

Figure 5 is a schematic representation of the multi-stability based secure communication system 500 in accordance with the present invention based on multi-stable chaotic systems. System 500 comprises a transmitter 200 and a receiver 400 that are configured to transmit and retrieve over a public channel 300 a very large number of masked information packets with a very large number of coexisting chaotic attractors, and also transmit keys

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Dynamics on a private channel 350 different from public channel 300. Sender 200 is configured to receive an information signal mEntrance ( t) = mE ( t), and receiver 400 is configured to retrieve an information signal mHealth (t). In addition, emitter 200 comprises: a multi-stable master chaotic dynamic system 250, an information vectoring-dividing unit 120, and an initial condition generating unit 175. itself, receiver 400 comprises a chaotic dynamic slave system 450 which is the counterpart of the 250 multistable master chaotic dynamic system. Additionally, the receiver comprises a first decoding or unmasking unit 470 and a second decoding or unmasking unit 480.

In the implementation of the present invention, the multi-stable master chaotic dynamic system 250 of emitter 200 generates a very large number of multi-stable chaotic masking signals and me ( t) operating in accordance with the system of equations [1] - [6]. Furthermore, the multistable slave chaotic dynamic system 450 of the emitter 400 is coupled by a private channel with the signal xmí ( t) of the master dynamic system 250 through the expression k ( xmí - xsl) equation [7] and that by synchronization between These two slave systems-450 and master-250, generate on the side of slave system 450 a very large number of chaotic masking signals ysi (t). Itself, the slave chaotic system 450 is in accordance with equations [7] to [12].

In the implementation of the present invention, the emitter 200 comprises an information vectoring unit 120, which vectorizes and divides the information signal mE ( t), into d = 751 mlE ( t) information packets and each packet comprises a size of v p = 90 iterations and a duration time t me ( t) = 18 s.

Emitter 200 is configured to chaotically mask each mlE ( t) information packet , using the signal of each state of each chaotic yimí ( t) masking attractor of master chaotic system 250. In addition, emitter 200 combines each mlE information packet. ( t), with each state of each chaotic masking attractor ylmí ( t) to produce the transmission signal sl ( t) = ylmí ( t) mlE ( t) of each information packet.

The master chaotic system 250 of the emitter 200 generates d = 751 states of coexisting chaotic masking attractors ylmí ( t) by chaotically varying the initial conditions of the variable ylmí ( t ) equation [4] through the initial conditions generating unit 175. each of these states different ylmí attractors (t) comprise a duration of 70 s = tested, and this in turn has a size of Mestaado = 360 iterations. To itself, each of the attractors ylmí ( t) has two sections; a) first section corresponds to a time ts = 52 s equivalent to tmx = 270 iterations, ts being the time required by the slave chaotic system 450 of the receiver 400 to synchronize with the master chaotic system 250 of the emitter 200, b) second section or section subsequent to the first, corresponds to a time rt = 18 s equivalent to vp = 90 iterations, with rt being the transmission time of each information packet mlE ( t).

The initial condition generating unit 175 of the emitter 200 generates d = 751 initial conditions that serve as dynamic keys. These initial conditions or dynamic keys are used in the variable ymi equation [4] of the master chaotic system 250. These initial conditions are selected chaotically by the logistic map operated by the equation [L1]. This quantity of initial conditions d = 751 allows to generate d = 751 different chaotic and lmí ( t) masking states.

Emitter 200 comprises an encryption or masking unit 150 that is configured to combine each mlE ( t) information packet with each signal from the coexisting chaotic masking attractor and ^ iCO. That is, an initial condition is generated by the initial condition generating unit 175 and is used by the master chaotic system 250 to generate a signal of the chaotic masking ylmí ( t) in a coexisting attractor. The masking unit 150 combines the information signal mlE ( t) with the chaotic masking signal and lmí ( t) in the following way. The information signal mlE ( t) that has vp = 90 iterations or a duration time of rt = 18 s, is added to the second section of the chaotic masking signal and lmí ( t) that also has vp = 90 iterations and a time duration of rt = 18 s. Subsequently this process is repeated again. That is, a new initial condition is generated by unit 175 and this new initial condition is used by the master chaotic system 250 to generate a new chaotic masking signal yí + í ( t) in another coexisting attractor and then the masking unit. 150 combines the new information signal mi + 1 ( t) with the new chaotic masking signal and ^lf it), so on. The combined signal s ( t ) = and ^{m i} ( t) m ^E ( t) is transmitted from sender 200 to receiver 400 on public channel 300.

The combination or sum of each information signal m ^lE ( t) with the chaotic masking signal and ^{lm í} ( t ), is always the second section of the chaotic masking signal and ^{lm í} ( t ), that is, after Synchronization time ^t s = 52 s equivalent to t ^mx = 270 iterations elapsed.

The change of each initial condition in unit 175 is made after each chaotic masking signal and ^lm ( t) has completed M ^state = 360 iterations

The change of the initial conditions or dynamic keys is transmitted from the transmitter 200 to the receiver 400 by a second chaotic masking signal x mi ( t) through the private channel 350. The chaotic dynamic slave system 450 of the receiver 400 uses this signal x mi ( t) to generate the slave chaotic masking signal and si ( t) by synchronization between receiver 400 and emitter 200 .

The chaotic master 250 and the chaotic slave 450 are configured in full synchronization regimes, based on the values of their parameters a = 0.2, b = 0.2 and c = 5.7 that operate as static keys and facilitate the recovery of signal signals. information transmitted m E ( t).

Receiver 400 is configured to receive the signal transmitted s ( t) on public channel 300 , and is additionally configured to receive initial conditions or dynamic keys by a second chaotic masking signal x mi ( t) through private channel 350 . the receiver 400 uses the slave chaotic system 450 to generate the release signal chaotic masking slave and if (t) and removes the initial conditions or dynamic keys from the transmitted signal s (t) by subtracting and if (t) of s ( t) through the first unmasking or decryption unit 470 . For adequate suppression of the initial conditions or dynamic keys, the slave chaotic system 450 and the master chaotic system 250 must be completely synchronized by means of the static keys, based on the values of their parameters a = 0.2, b = 0.2 and c = 5.7. As a result of subtraction of y si ( t) from s ( t) by means of the first unit of

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unmasking or decoding 470, the transmitted chaotic masked message signal s ( t) is transformed into the recovered message signal mR ( t).

Specifically, each message signal recovered mlR is made up of two sections: a) First section corresponds to the synchronization error, the ( t) equivalent to the time ts = 52 s or tmx = 270 iterations. b) Second section subsequent to the first one, corresponding to a transmission signal mlE ( t), that is, mlR ( t) = el ( t) mlE ( t).

In summary, the operation of transmitter 250 and receiver 450 can be represented by the following mathematical equations:

• s ( t) = yml ( t) mE ( t),

• xmi ( .t): synchronization signal of transmitter 250 and receiver 450,

• mR ( t) = mE ( t) ymí ( t) - ysí ( t), where mR ( t) = e ( t) mE ( t) when ymi ( t) = ysi ( t),

where:

• s ( t): chaotic masked message signal, transmitted by public channel 300,

• xmi ( t): initial conditions signal or dynamic key transmitted by private channel 350 ,

• mR ( t): recovered signal,

• e ( t): synchronization error.

Additionally, receiver 400 includes a second unmasking or decoding unit 480, which is configured to retrieve the information signal msaiida ( t), by subtraction e ( t) from mR ( t), i.e.

mSaiidad ( t) = mR ( t) - e ( t),

™ SaUdad ( t) = e ( t) mE ( t) - e ( t), d ° ncle msalidad ( t) = mE ( t) ° mSalidad ( t) = WEntrada ( 0 ■

Figure 6 is a schematic representation of a digital electronic device that implements the system of the present invention. Generally speaking, the digital electronic device 500a comprises a transmitter 200a and a receiver 400a which are configured to transmit and retrieve on a public channel 300a a very large number of masked information packets with a very large number of coexisting chaotic attractors, and furthermore transmits dynamic keys through a channel private 350a different from public channel 300a. Sender 200a is configured to receive an information signal m Input ( t), and receiver 400a is configured to retrieve an information signal m output ( t). Furthermore, the emitter 200a comprises: a multi-stable master chaotic system 250a, an information vectoring-dividing unit 100a and an initial condition generating unit 175a. The receiver 400a itself comprises a slave chaotic system 450a which is the counterpart of the multi-stable master chaotic dynamic system 250a. In addition, receiver 400a incorporates the first unmasking or decoding unit 470a, which is configured to suppress initial conditions or dynamic keys. At the same time, the receiver 400a comprises the second unmasking or decoding unit 480a, which is configured to retrieve the information signal m output ( t).

Representative examples of the chaotic information signal encoding-decoding system based on multi-stable systems

Figure 7 ( a) is a graph of a flat image and by conventional techniques, the pixels of this image are converted to an information signal m ^{E ntrada} ( t) = m ^E ( t ) or vectorized signal. Figure 7 ( b) shows an information signal packet m ^lE ( t) versus time, performed by the information vectoring-dividing unit 120a. Figure 7 ( c) shows a portion of the information signal m ^{E ntrada} ( .t) versus time, corresponding to a very large number of information signal packet m ^lE ( t).

Representative examples that illustrate particular cases are described below, under which there are correct and incorrect recoveries of the information signal m ^{E ntrada} ( t). These representative examples correspond to the multi-stability based secure communication system 500a in accordance with the present invention.

Representative example 1: message m Entry ( t) , successfully recovered

Masking or encryption process

In this representative example, the master chaotic system 250a of the emitter 200a and the slave chaotic system 450a of the receiver 400a are configured to be identical

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or the like. The respective static keys of transmitter 200a and receiver 400a are shown in Table I.

Table I: Static keys of the emitter 200a and the receiver 400a, corresponding to the parameters of the systems of equations [1] - [6] and [7] - [12], for example, representative 1: message mEntrance ( t), recovered correctly.

Figure 8 ( a) shows a chaotic masking state and l ^{m i} ( t ) as a function of time generated by the master chaotic system 250a, due to the change of an initial condition produced in the initial conditions generating unit 175a. Figure 8 ( b) shows the combination of a chaotic masking state and l ^{m í} ( t) with an information packet m ^lE ( t) performed by the chaotic masking unit 150a, to produce a transmission signal sl ( t) = yl ^{m i} ( t) m ^E ( t), which is transmitted from sender 200a to receiver 400a on public channel 300a.

To clarify how the process of encrypting information with coexisting chaotic attractors is, Figure 8 ( c) shows s ^l ( t) versus time. The signal s ^l ( t) includes two sections: a) First section: signal s ^l ( t), which corresponds to a time

^t s = 52 s equivalent to the time required by the slave chaotic system to synchronize with the master chaotic system. b) Second section: signal s ^l ( t) , it corresponds to a time r ^t = 18s equivalent to the transmission time of each information packet m ^lE ( t) is masked, encrypted in each coexisting chaotic attractor and ^{í ni} it) . This Figure 8 ( c) also shows the total time t ^state = 70 s of each coexisting chaotic attractor and ^{í ni} it), equivalent to t ^{state = t} s ⁺ r ^t .

Figure 8 ( d) shows a portion of the transmitted signal s ( t) = y ^m ( t) m ^E ( t) , corresponding to a very large number of coexisting chaotic masking attractors and ^{m i} ( t) and a very large number of information signals m ^E ( t), which are transmitted from sender 200a to receiver 400a.

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Recovery or decryption process

Figure 9 ( a) shows a chaotic masking state xlmi ( t ) versus time, generated by the master chaotic system, equations 250a, due to the change of an initial condition produced in the initial conditions generating unit 175a. The signal xlmi ( t) is transmitted from the transmitter 200a to the receiver 400a on the private channel 350a and the chaotic slave system 450a uses the xlmi ( t) by synchronization to generate the ylsi ( t) signal . Figure 9 ( b) shows the signal ^ (t) versus time. While Figure 9 ( c) shows the signal ^ (t) versus the signal ylmí ( t) and in which a strong complete synchronization can be observed between these signals after the synchronization time ^t s has elapsed .

Figure 10 ( a) shows the recovered message signal m jj (t) versus time, as a result of subtraction of ^ (t) from sl ( t) by means of the first unmasking or decoding unit 470a. The signal of the recovered message m jj (t) is made up of a first section corresponding to the synchronization error el ( t) and a second section corresponding to an information signal packet mlE ( t), that is mlR ( t) = el ( t) mlE ( t). Figure 10 ( b) represents a portion of the recovered message mR ( t) = e ( t) mE ( t) versus time, corresponding to a very large number of synchronization error signals el ( t) and a very large number of mlE ( t) information packages .

Figure 11 (a) shows a packet message signal successfully recovered mlsalida (t) versus time as a result of subtracting the (t) of m ^ t) by the second unit unmasking or decoding 480a. Figure 11 ( b) shows a portion of the successfully recovered message signal msalidd ( t) corresponding to a very large number of recovered signals misallda ( t).

Figure 12 depicts a portion of the signal successfully recovered msaidad ( t) versus a portion of the original lying signal ( t). This graph shows in the state space the complete synchronization between the recovered signal and the original signal. The conversion of the successfully recovered signal plus ( t) to flat image is shown in Figure 13.

Figures 8-1-3 show a high correlation between the masking and retrieval process of the information signal, making the secure communication system 500a of the present invention a highly robust secure communication system.

Representative example 2: message mr „tr„ * „ ( t '), incorrectly retrieved when static keys are improperly implemented

Masking or encryption process

In this representative example, the master chaotic system 250a of the emitter 200a and the slave chaotic system 450a of the receiver 400a are configured to be completely different in at least one parameter a . The respective static keys of transmitter 200a and receiver 400a are different and are shown in Table II.

Table II: Static keys of transmitter 200a and receiver 400a, corresponding to the parameters of the systems of equations [1] - [6] and [7] - [12], for representative example 2: message mEntrance ( t), incorrectly recovered.

Figure 8 ( a) shows a chaotic and limiting masking state generated by the master chaotic system 250a, due to the change of an initial condition produced in the initial condition generating unit 175a. Figure 8 ( b) shows the combination of a chaotic masking attractor ylmí ( t) with an information package miE ( t ) performed by the chaotic masking unit 150a, to produce a transmission signal sl ( t) = yiml ( t) miE ( t), which is transmitted from transmitter 200a to receiver 400a on public channel 300a. Figure 8 ( d) shows a portion of the transmitted signal s ( t) = ym ( t) mE ( t), corresponding to a very large number of chaotic masking attractors ym ( t) and a very large number of signal signals. information mE ( t) , which are transmitted from sender 200a to receiver 400a.

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Recovery or decryption process

Figure 9 ( a) shows a chaotic masking state xlmi ( t ) versus time, generated by the master chaotic dynamic system, 250a, due to the change of an initial condition produced in the initial condition generating unit 175a.

The signal xlmi ( t) is transmitted from the transmitter 200a to the receiver 400a on the private channel 350a and the chaotic slave system 450a uses the xlmi ( t) by synchronization to generate the yimi ( t) signal .

Figure 14 ( a) shows the signal ^ (t) versus time. While Figure 14 ( b) shows the signal y ^ i (t) versus the signal ylmí ( t) in which a weak synchronization can be observed between these signals whose state space y ^ i (t) vs ylml ( t ) occupies a wide region, which represents a poor correlation between these signals, that is, y ^ i (t) and ylmí ( t) are completely different. In accordance with the present invention, representative example 2 shows that the secure communication system 500a of the present invention is very sensitive to small variations of its static keys; represented in the 250a master chaotic dynamic system by a = 0.2 and in the 450a slave chaotic system by a = 0.3. This difference of the static keys in the emitter 200a and the receiver 400a makes original image recovery difficult.

Figure 15 ( a) shows the recovered message signal m jj (t) versus time, as a result of subtracting y ^ t) from sl ( t) by means of the first unmasking or decoding unit 470a. The signal of the recovered message m jj (t) is made up of a first section corresponding to the synchronization error el ( t) has very large values and a second section that is a completely different signal than the information signal mlE ( t). Figure 15 ( b) represents a portion of the retrieved message mR ( t) versus time, corresponding to a very large number of incorrectly retrieved information packets.

Figure 16 ( a) shows an incorrectly retrieved message signal packet msaiida ( t) versus time, as a result of subtraction el ( t) from m ^ t) by means of the second unmasking or decryption unit 480a. Figure 16 ( b) shows a portion of the incorrectly retrieved message signal msalidd ( t) corresponding to a very large number of incorrectly retrieved signals msalidd ( t).

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Figure 16 ( c) shows a portion of the incorrectly recovered signal m ^{sa i id ad} ( .t ) versus a portion of the original signal m ^{in tr ada} ( t ), in which a weak synchronization between these can be observed. signals and whose state space m ^{sa iíd ad} ( t) vs m ^{en tr ada} ( t) occupies a wide region, representing a poor correlation between these signals, that is, m ^{sa l id ad} ( t) and m ^{en tr ada} ( t) are completely different.

The conversion of the incorrectly recovered signal m ^{sa ldd} ( t) to flat image is shown in Figure 17, which is completely different from the original flat image Figure 7 ( a).

Representative Example 2 shows that the secure communication system 500a of the present invention is very sensitive to small variations of its static keys. That is to say, a small difference of the static keys in the emitter 200a and the receiver 400a makes it impossible to recover the original image.

Representative example 3: message m Entry ( t) incorrectly retrieved when dynamic keys are improperly implemented

In the present representative example 3, dynamic braces are improperly implemented. Figure 18 depicts a secure communication system 500b that can be implemented in any digital electronic system. In this system, the private communication channel that communicator 200b communicates with receiver 400b is poorly configured, is not considered, or does not exist. In the communication system 500b there is no signal x ^{m í} ( t) or dynamic keys generated by the chaotic master system 250b to be transmitted by a private channel that communicates the transmitter 200b with the receiver 400b , but unlike in the Secure communication 500a of the present invention, this private channel 350a does exist or is well configured as illustrated in Figure 6 .

Furthermore, the chaotic slave system 450b of the emitter 400b cannot use the signal x ^{m í} ( t) of the master dynamic system 250b and by synchronization generate the JsiW signal. As there is no synchronization between the emitter 200b and receiver 400b, the chaotic masking signal y ^{m í} ( t) of the emitter 200b and the chaotic masking signal y s (t) of the receiver 400b are completely different, that is, y ^{m í} ( t) ^ and ^st ( t).

In the secure communication system 500b, the combined signal s ( t ) = ymí ( t ) mE ( t ) is transmitted from sender 200b to receiver 400b on public channel 300b, where ymí ( t) is the chaotic masking signal and mE ( t) is the information signal. Furthermore, at receiver 400b the recovered message mR ( t ), as a result of the subtraction of ysi (t) from s ( t) by means of the first unmasking or decoding unit 470a is unsuccessfully or incorrectly performed, this is due to that ymí ( t) ^ ysi (t) and mathematically is represented by

mR ( t) = s ( t) - and yes,

mR ( t) = yml ( t) mE ( t) - ysU

Because ymí ( t) ^ ysí ( t ), the recovered signal mR ( t) is different from the signal mE ( t). In this representative example 3, the omission of the dynamic key xmi ( t) or lack of the private communication channel that synchronizes the sender 200b with the receiver 400b causes the original message signal to be unsuccessfully retrieved by the receiver 400b of the communication system safe 500b.

Masking or encryption process

In this representative example, the master chaotic system 250b of emitter 200b and the chaotic dynamic slave system 450b of receiver 400b are configured to be identical. The respective static keys of transmitter 200b and receiver 400b are similar and are shown in Table I.

Figure 8 ( a) shows a chaotic and limiting masking state generated by the master chaotic dynamic system 250b, due to the change of an initial-i condition produced in the initial conditions generating unit 175a. Figure 8 ( b) shows the combination of a chaotic masking state-i and lmí ( t) with a mlE ( t) information packet performed by the chaotic masking unit 150b, to produce a transmission signal sl ( t) = ylmí ( t) mlE ( t), which is transmitted from the transmitter 200b to the receiver 400a on the public channel 300a. Figure 8 ( d) shows a portion of the transmitted signal s ( t) = ym ( t) mE ( t), corresponding to a very large number of chaotic masking attractors ym ( t) and a very large number of signal signals. mE ( t) information, which are transmitted from sender 200b to receiver 400b.

1

Recovery or decryption process

Figure 19 ( a) shows the signal ^ (t) versus time. While Figure 19 ( b) shows the signal ^ (t) versus the signal y l ^{m í} ( t) in which we can observe a weak synchronization between these signals whose state space ^ (t) vs and ^{lm í} ( t) It occupies a wide region, which represents a poor correlation between these signals, that is, ^ (t) e and ^lmí ( t) are completely different. In accordance with the present invention, representative example 3 shows that the secure communication system 500b is very sensitive to the non-existence of dynamic keys, that is, there is no signal x ^{lm í} ( t) or dynamic keys of the master chaotic system which synchronizes transmitter 200b with receiver 400b. This lack of synchronization of the transmitter 200b with the receiver 400b made by the dynamic keys xl ^{m í} ( t) through a private communication channel makes it difficult to recover the original image.

Figure 20 ( a) shows the recovered message signal m jj (t) versus time, as a result of subtracting y ^ t) from s ^l ( t) by means of the first unmasking or decoding unit 470b. The signal of the recovered message m jj (t) is made up of a first section corresponding to the synchronization error and ^l ( t) has very large values and a second section that is a completely different signal than the information signal m ^lE ( t) . Figure 20 ( b) represents a portion of the retrieved message m ^R ( t) versus time, corresponding to a very large number of incorrectly retrieved information packets.

Figure 21 ( a) shows an incorrectly retrieved message signal packet m ^{sai d a} ( t) versus time, as a result of subtraction e ^l ( t) from m ^ t) by means of the second unmasking unit o 480b decoding . Figure 21 ( b) shows a portion of the incorrectly recovered message signal m ^{sal id d} ( t) corresponding to a very large number of signals incorrectly recovered m ^{ls to id a} ( t).

Figure 21 ( c) represents a portion of the incorrectly recovered signal m ^{sa i dd} ( t) versus a portion of the original signal m ^{in tr ada} ( t), in which we can observe a weak synchronization between these signals and whose state space m ^{sa i id ad} ( t) vs m ^{en tr ada} ( t) occupies a wide region, representing a poor

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correlation between these signals, that is, msality ( t) and lying ( t) are completely different.

The conversion of the incorrectly recovered signal plus quality ( t) to flat image is shown in Figure 22, which is completely different from the original flat image Figure 7 ( a).

Representative example 3 shows that the secure communication system 500b of the present invention is very sensitive to the lack of its dynamic keys, that is, the lack of synchronization between the transmitter 200b and the receiver 400b makes it impossible to recover the original image.

Claims (9)

1.- Secure communication transmitter ( 200) based on multi-stability, characterized in that it comprises: • a vectoring unit ( 120) configured to receive an information signal to be transmitted mE ( t ), which is vectorized and divided into d mlE ( t) information packets \ • an initial conditions generating unit ( 175) where the initial conditions ylmX are located in the interval [yml_iniciai> ymijinai \; • a multi-stable master dynamic system ( 250) that generates d coexisting chaotic masking chaotic attractors that vary chaotically by changing the initial conditions ylmi of the variable ymi in equations [3] - [6] through the condition generating unit initials ( 175) and solving the following differential equations:

with a = 0.2, b = 0.2 and c = 5.7; • an encryption unit ( 150) configured to combine the d mlE ( t) information packets with the d co-existing chaotic masking attractors ylmí ( t) obtaining a signal s ( t) such that s ( t) = ymi ( t ) mE ( t), where said signal s ( t) is sent through a public channel ( 300) to a receiver ( 400) and where the signal xmi ( t) of equation [1] is sent to the receiver (400 ) through a private channel (350). 4 2. - Multi-stability based secure communication emitter ( 200) , according to claim 1, characterized in that the multistable master chaotic dynamic system ( 250) comprises two Rossler oscillators with non-linear coupling between their state variables, where the behavior of the first oscillator is defined by equations [1] to [3] and the behavior of the second oscillator is defined by equations [4] to [6]. 3. - Multi-stability based secure communication emitter (200), according to claim 1, characterized in that the initial conditions generating unit ( 175) chaotically modulates the interval [ymlJniciai, and mljmai ] through a defined chaotic logistic map by the following equation: xn +! = pxn ( l xn), [Ll] where x ⁿ is defined in the interval [0, l] and p in the interval [0,4], and the modulation of the initial conditions ymi ^_0 in the interval {y ^m n ^{n icia i} , y ^mi j ^{in ai} ] It is defined by the following equation: yml_0 yml_inicial + ^ n + l {yml_final yml_inicial ), [L2] where xn + t represents the logistic map time series (L1) in the chaotic regime and n 1 are the iterations. 4. - Secure communication transmitter (200) based on multi-stability, according to claim 1, characterized in that d = 751 iterations of the time series xn + t are selected, after the first 100 transitory iterations have elapsed. 5. - Multi-stability based secure communication transmitter (200), according to claim 1, characterized in that each information packet d has a size vp equal to 90 iterations and a duration time tme ( t) of 18 seconds. 6. - Multi-stability based secure communication receiver ( 400) , characterized in that it comprises: • a first decoding unit ( 470) with an input connected to a public channel ( 300) to receive a signal s ( t) from a transmitter (200); • a second decoding unit ( 480) connected to an output of the first decoding unit (470); • a dynamic slave chaotic multi-stable system (450) generating a signal chaotic masking YSI slave (t) from a xm (t) signal received from a transmitter (200) through a private channel (350), and solving the following differential equations:

where k is the coupling factor between the emitter (200) and the receiver (400); where the first decoding unit ( 470) subtracts the slave chaotic masking signal ysi ( t) from the signal s ( t) coming from the emitter (200), obtaining a recovered message signal mR ( t) that reaches the second unit decoding ( 480) which subtracts a timing error el ( t) from the recovered message signal mR ( t) to obtain the signal msaiida ( t). 7. - Multi-stability based secure communication receiver (400), according to claim 6, characterized in that the multi-stable slave chaotic dynamic system ( 450) comprises two Rossler oscillators with non-linear coupling between their state variables, where the behavior of the first oscillator is defined by equations [7] to [9] and the behavior of the second oscillator is defined by equations [10] to [12]. 8. - Multi-stability based secure communication system, characterized in that it comprises an emitter (200) defined in any one of claims 1 to 5, a receiver ( 400 ) defined in any one of claims 6 to 8, wherein the sender (200) communicates with receiver (400) through a public channel (300) and a private channel (400). 9.- Secure communication system based on multi-stability, according to claim 8, characterized in that the transmitter ( 200) receives an information signal to be transmitted mE ( t), which it sends encoded s ( t), using coexisting attractors of chaotic masking, to the receiver on the public channel, and where the sender sends a dynamic key to the receiver on the private channel for synchronization between the sender and the receiver and the decoding of the encoded signal s ( t) obtaining a msaiida signal ( t ) equal to mE ( t).