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    • 专利摘要:
    Secure communication system based on multi-stability. A secure communication system based on multi-stability consisting of transmitter/receiver based on multi-stable chaotic oscillators and two public/private communication channels is disclosed. The private allows synchronizing the transmitter and receiver oscillators. Identical chaotic switches in emitter and receiver change the initial conditions of the oscillators discreetly with intervals shorter than the synchronization time, functioning as secret keys allowing the change of chaotic attractors simultaneously in the emitter and receiver oscillators. The information is transmitted through the public channel, adhering a very large number of information signal packets in a staggered manner to a very large number of multi-stable chaotic masking signals within the same time series. The change of the chaotic attractors faster than the synchronization time makes the synchronization attack impossible, and in addition the synchronization of chaotic signals between the sender and receiver in each attractor allows the information signal to be recovered with high stability. (Machine-translation by Google Translate, not legally binding)
    公开号:ES2724598A1
    申请号:ES201930480
    申请日:2019-05-30
    公开日:2019-09-12
    发明作者:Alexander Pisarchik；James Rider Reategui；Flores Cesar Rodriguez；Lopez Juan Hugo Garcia
    申请人:Universidad Politecnica de Madrid；Univ Guadalajara；
    IPC主号:
    专利说明:

    [0001]
    [0002]
    [0003]
    [0004] OBJECT OF THE INVENTION
    [0005]
    [0006] The present invention is directed to a secure communication system based on multi-stability.
    [0007]
    [0008] The present invention is related to the field of secure communication systems, methods, digital electronic devices and computer programs, more specifically it relates to a system for highly secure communications based on dynamic systems with coexistence of multiple chaotic attractors.
    [0009]
    [0010] BACKGROUND OF THE INVENTION
    [0011]
    [0012] Each time a password is used, cryptography is being used. Every time a financial transaction is made online, cryptography is being used. If you want to have a secure voice or video call, where "safe" means that no one but the participants can attend, then you have to use cryptography. If you want to write a message that only your recipient is able to read and nobody else , it is necessary to keep a password among the participants of the transmission of information.
    [0013]
    [0014] To keep some secret information, 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 text-encryption. To read the encrypted text, it is necessary to decode this message. This process is called decoding. This should only be done by someone authorized. Otherwise it is said that the coding has been broken. Coding and decoding work using a security key that is an instrument to hide and reveal the information you want to protect.
    [0015] Using encryption, authentication and access control technologies to secure communication systems is an appropriate solution to avoid attempts to spy and steal data transmission. A contribution to solve this problem is to use signals generated by components that operate in the non-linear regime, such as multi-stable chaotic oscillators.
    [0016]
    [0017] 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 the initial conditions A system is said to be deterministic if the temporal evolution equation, the parameters describing the system and the initial conditions are known.
    [0018]
    [0019] In the state of the art of communications 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 in generating a sequence of random numbers in digital format produced by a chaotic system, the message to be encoded in digital format is adhered to and the combined signal is transmitted. The receiver extracts the digital message from the combined signal transmitted 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 the decrease in security as a result of transmission of the key or key.
    [0020]
    [0021] One way to solve this problem by using chaotic systems for communication security where key transmission is not required is shown in 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 a signal that carries information to a chaotic signal is added. The resulting chaotic signal is transmitted using conventional transmission technologies from a transmitter that contains an encoder to a receiver that contains a decoder. The decoder in the receiver synchronizes the chaotic signal of the receiver with the original chaotic signal without the need for key exchange. The 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 .
    [0022]
    [0023] As regards patented technology in this field based on chaotic systems, innumerable 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.
    [0024]
    [0025] 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. Do not.
    [0026] 5291555. These secure communication systems based on chaotic signal synchronization have a weakness against synchronization attack. Secure communication systems based on synchronization use chaotic non-linear systems coupled in master-slave configuration, where the master represents a transmitter and the slave a receiver, the transmitter's output signal is used to encode or mask the information signal.
    [0027]
    [0028] 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 lowest synchronization error is reached. When the transmitter and the 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 of the transmitter.
    [0029]
    [0030] The sensitivity of synchronization to parameter changes in chaotic systems is very crucial for communication security. A small variation in one of these parameters can produce the synchronization error so large that the Information signal recovery is impossible. In itself, we can consider two types of parameters: group of parameters related to chaotic nonlinear systems (sender-receiver) and the parameter underlying the timing of synchronization. Being the threshold value this last primary parameter to achieve full synchronization. For values below this synchronization time, full synchronization is not achieved and information retrieval is impossible.
    [0031]
    [0032] 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 of y E m is or (t), s (t) is produced as s (t) = m E (t) and Em ís or (f í, which is then transmitted to the receiver. In the receiver, the signal s (t) is used to synchronize the chaotic slave oscillator with the master chaotic oscillator, the generation of the chaotic slave signal and E m iSOr (t) It is difficult because there is an incomplete or intermittent synchronization which makes it impossible to recover the information signal m R. The above is a consequence of the process of adding the information m E (t) to the signal of the master chaotic oscillator and E m iSOr (t), converting it to this last oscillator not identical to the chaotic slave oscillator, and as a result the difficulty of recovering the information signal m R.
    [0033]
    [0034] In addition, secure communication systems based on chaotic signal synchronization present 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 lowest synchronization error is reached. The synchronization error is achieved when the synchronization time between the virtual receiver and the transmitter 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.
    [0035] DESCRIPTION OF THE INVENTION
    [0036]
    [0037] To avoid these inconveniences, a new communication system using a dynamic changing private key operating in a private channel and another static key is proposed in the present invention. The private channel is performed by one of the transmitter's state variables that communicates or couples properly with the receiver and produces a dynamically changing complete synchronization between the receiver and the transmitter.
    [0038]
    [0039] Furthermore, the present invention solves the inconvenience of synchronization attack, by implementing in the private communication channel a novel method of dynamically changing synchronization between the receiver and the transmitter by means of the chaotic multiple attractors coexisting in the master oscillator. The above is materialized using different variations or changes in chaotic form of the initial conditions of the variables of the same chaotic master oscillator and that function as dynamic secret keys allowing the change of the attractors in the receiver and in the transmitter in chaotic form. The chaotic change of the chaotic attractors with the intervals shorter than the synchronization time makes the synchronization attack impossible to break the security of this communication system.
    [0040]
    [0041] 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 "multi-stability" 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 wide variety of initial conditions of the system state variables. A dynamic "state of a system" is a solution that remains confined in an area of its state space or attractor. Full synchronization is only possible when coupled dynamic systems are identical and are in the same attractor.
    [0042]
    [0043] In one aspect of the present invention a secure communication system based on multi-stability is disclosed comprising a transmitter and a receiver that communicate 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 coded s (t), by coexisting chaotic masking attractors, to the receiver through the public channel, and where the sender sends a dynamic key to the receiver through 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) on the receiver equal to the mE (t) signal input on the transmitter.
    [0044]
    [0045] In one embodiment, the transmitter included in the multi-stability based secure communication system comprises:
    [0046] • a vectorization unit configured to receive an information signal to be transmitted m E (t), which is vectorized and divided into d information packages m lE (t)
    [0047] • a unit generating initial conditions where the initial conditions and me are located in the interval ° ym.l_initial'yml_final ;
    [0048] • a multi-stable master chaotic dynamic system that generates coexisting chaotic attractors of chaotic masking that vary chaotically by changing the initial conditions and lmí of the variable and me in equations [3] to [6] through the condition generating unit initials and solving the following differential equations:
    [0049]
    [0050] dx m í
    [0051] d t% m2% m3 [i]
    [0052] d% m2
    [0053] d t =% m2 ^ Vm2 m
    [0054] d% m3 = b - cxm3 ymíym3
    [0055] d t [3]
    [0056] dymí [4]
    [0057] d t% me Vmí% m2% m3
    [0058] dym2
    [0059] d t Vmí ^ Vm2 [5]
    [0060] dym3
    [0061] dt = by m 3 (and m í - c) [6]
    [0062]
    [0063] with a = 0.2, b = 0.2 and c = 5.7;
    [0064] • an encryption unit configured to combine the d packets of information mlE (t) with the co-existing chaotic attractors of chaotic masking and lmí (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 and where the signal x m í (t) of equation [1] is sent to the receiver through a private channel.
    [0065]
    [0066] In an embodiment of the multistability-based secure communication transmitter, 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 described above [4] to [6].
    [0067]
    [0068] In an embodiment of the multistability-based secure communication transmitter, the initial conditions generating unit chaotically modulates the interval ym j niciai> ymijínai] through a chaotic logistic map defined by the following equation:
    [0069] * (n + l) = ^ * n (l Xn), [Ll]
    [0070]
    [0071] 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 , and mi j in ai ] is defined by the following equation:
    [0072] and m l_0 and m l_ in ic ia l + X ni (and m i_ ^ in ai and m í_ in ic ia l ), [^ 2 ']
    [0073]
    [0074] where x nt represents the time series of the logistic map (L1) in the chaotic regime and n 1 are the iterations.
    [0075]
    [0076] In an embodiment of the multistability-based secure communication sender, the last d = 751 iterations of the time series x n 1 are selected , after the first 100 transient iterations have elapsed.
    [0077]
    [0078] In an embodiment of the multistability-based secure communication sender, each information packet d has a size vp equal to 90 iterations and a duration time tme (t) of 18 seconds.
    [0079]
    [0080] In one embodiment, the multistability-based secure communication receiver comprises:
    [0081] • a first decoding unit with an input connected to a public channel to receive an s ( t ) signal from the transmitter;
    [0082] • a second decoding unit connected to an output of the first decoding unit;
    [0083] • a dynamic chaotic slave multi-stable system that generates a signal chaotic masking slave YSI (t) from a signal xm (t) received from a transmitter via a private channel, and solving the following differential equations:
    [0084]
    [0085]
    [0086]
    [0087] where k is the coupling factor between the transmitter and the receiver;
    [0088] where the first decoding unit subtracts the slave chaotic masking signal and if ( t ) to the signal s ( t ) from the sender, obtaining a recovered message signal mR ( t ) that reaches the second decoding unit that subtracts an error synchronize the ( t) to the recovered message signal mR ( t ) to obtain the msaiide signal ( t ). The msaiida ( t ) signal at the receiver's output coincides with the signal at the mE ( t) input of the transmitter that was to be transmitted between sender and receiver in an encoded manner.
    [0089]
    [0090] In an embodiment of the multistability-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 described above [10] to [12].
    [0091] BRIEF DESCRIPTION OF THE FIGURES
    [0092]
    [0093] Figure 1 shows the secure communication system using chaotic signals.
    [0094] Figure 2 shows the local maximums of the variable x m í according to the initial conditions and m í 0. Each state of the variable x m ì belongs to a different coexisting chaotic attractor.
    [0095] Figure 3 (a) shows the chaotic masking state of the variable and m í (t) as a function of time for a chaotic attractor.
    [0096] Figure 3 (b) shows the chaotic masking state of the variable and m í (t) as a function of time for another chaotic attractor, using different initial condition and m 0. 0. Figure 4 shows the chaotic variation of the initial conditions and m í 0 modulated by the logistic map, equation (L1).
    [0097] Figure 5 schematically depicts a secure multi-stability based communication system 500 in accordance with the present invention based on multi-stable chaotic systems.
    [0098] Figure 6 represents the schematic for any digital electronic device of a secure multi-stability based communication system 500a in accordance with the present invention.
    [0099] Figure 7 (a) shows a graph of a flat image.
    [0100] Figure 7 (b) shows an information signal packet ml E (t) versus time.
    [0101] Figure 7 (c) shows a portion of the information signal m E ntra da (t) versus time.
    [0102] Figure 8 (a) shows a chaotic masking state and the m (t) as a function of the time generated by the chaotic master system 250a.
    [0103] Figure 8 (b) shows the transmission signal sl (t) = yl m í (t) m E (t) which is the combination of a chaotic masking state and l m (t) with an information package m lE ( t) performed by the chaotic masking unit 150a.
    [0104] Figure 8 (c) shows the graph s l (t) versus time, where it is included; the synchronization time t s, time required for the chaotic slave system to synchronize with the chaotic master system and the transmission time r t , time required to transmit each packet of information m lE (t) masked in each multi-stable chaotic state and í ni it).
    [0105] 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 (t) and a very large number of information signals m E (t).
    [0106]
    [0107]
    [0108] Figure 9 (a) shows a chaotic masking state xlmí (t) versus time, generated by the chaotic master system, equations 250a.
    [0109] Figure 9 (b) shows the ^ (t) signal versus time.
    [0110] Figure 9 (c) shows the signal ^ (t) versus the signal and lmí (t).
    [0111] Figure 10 (a) shows the signal of the recovered message m jj (t) versus time.
    [0112] Figure 10 (b) represents a portion of the recovered message mR (t) = e (t) mE (t) versus time.
    [0113] Figure 11 (a) shows a message signal packet successfully recovered mLuda (t) versus time.
    [0114] Figure 11 (b) shows a portion of the message message successfully recovered msalldd (t) corresponding to a very large number of recovered signals misallda (t).
    [0115] Figure 12 graphs a portion of the signal recovered satisfactorily over (t) versus a portion of the original signal lied (t).
    [0116] Figure 13 shows flat image successfully recovered.
    [0117] 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 and im l (t), for representative example 2.
    [0118] Figure 15 (a) shows the signal of the recovered message m ^ t) versus time, for representative example 2.
    [0119] Figure 15 (b) represents a portion of the recovered message mR (t) versus time, for representative example 2.
    [0120] Figure 16 (a) shows an incorrectly recovered message signal packet msaiida (t) versus time, for representative example 2.
    [0121] Figure 16 (b) shows a portion of the incorrectly recovered message signal msalidd (t), for representative example 2.
    [0122] Figure 16 (c) represents a portion of the incorrectly recovered signal mass (t) versus a portion of the original mentured signal (t), for representative example 2.
    [0123] Figure 17 shows a flat image incorrectly recovered, for representative example 2.
    [0124] Figure 18 represents 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.
    [0125] Figure 19 (b) shows the signal ^ (t) versus the signal and lmí (t), for representative example 3.
    [0126] Figure 20 (a) shows the signal of the recovered message m jj (t) versus time, for representative example 3.
    [0127] Figure 20 (b) represents a portion of the recovered message mR (t) versus time, for representative example 3.
    [0128] Figure 21 (a) shows an incorrectly recovered message signal packet msaiida (t) versus time, for representative example 3.
    [0129] Figure 21 (b) shows a portion of the incorrectly recovered message signal msalidd (t), for representative example 3.
    [0130] Figure 21 (c) graphs a portion of the incorrectly recovered signal msality (t) versus a portion of the original lie signal (t), for representative example 3. Figure 22 shows a flat image incorrectly recovered.
    [0131]
    [0132] PREFERRED EMBODIMENT OF THE INVENTION
    [0133]
    [0134] 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 chaotic multi-stable systems. Additionally, the present invention can be implemented in digital electronic circuits as in wireless communication systems.
    [0135]
    [0136] A detailed description of the present invention is presented below. The present invention is a highly secure communication system based on multistability, whose safety is achieved with a scheme consisting of a transmitter and a receiver based on chaotic multi-stable 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 the sender and the receiver is achieved through a private channel and another public channel. The private channel is made by the state variable xmí where the dynamic keys are implemented and at the same time it is used to synchronize the sender with the receiver, while the information signal is transmitted from the sender to the receiver through the public channel, through the state variable and me.
    [0137] In general, in accordance with the present invention secure communication systems using chaos includes a transmitter and a receiver. The emitter includes at least one chaotic master oscillator. Likewise, the receiver includes a chaotic slave oscillator. In addition, the chaotic master and slave oscillators are identical or one is counterpart to the other.
    [0138]
    [0139] Both the transmitter and the receiver are comprised of two Rossler oscillators with non-linear coupling between their status variables. By varying the initial conditions in any of the state variables, the system shows a multi-stable dynamic, from periodic and / or chaotic co-existing regimes (attractors). Communication between the sender and the receiver is achieved through two channels: a private channel and a public channel. The private channel is made by the state variable xmí and is used to synchronize the transmitter with the receiver, at the same time it serves as the first dynamic key, while the information signal is transmitted from the transmitter to the receiver by the public channel, through the state variable and me. The transmission of information is done by adhering a very large number of information signal packets in a staggered manner to a very large number of chaotic masking signals with the changes between the chaotic attractors within the same time series of the state variable and me.
    [0140]
    [0141] In particular, the transmitter comprises a "chaotic master system" corresponding to two chaotic Rossler oscillators with non-linear coupling between their state variables, which generates multiple coexisting chaotic attractors. This "chaotic master system" has two outputs. A "first exit" 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 chaotic attractors coexisting through a public channel and a "second output" configured to transmit a chaotic signal through a private channel, which functions as the first dynamic key and serves to synchronize the transmitter with the receiver. The sender has a unit configured to add 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.
    [0142]
    [0143] For its part, the receiver is structurally the counterpart of the transmitter and has two inputs: a "first input" that communicates or couples through the public channel the transmitter with the receiver and a "second input" that communicates or couples through of the
    [0144]
    [0145]
    [0146] private channel the sender with the receiver. The receiver comprises a "chaotic slave system" which is the counterpart of the sender's dynamic master system and consists of two Rossler oscillators with non-linear coupling between their status variables. This chaotic dynamic slave system has an input that communicates or couples the sender and receiver through the private channel through the second output signal of the chaotic master system, which is configured to generate a very large number of chaotic attractors and serves to synchronize the receiver with the transmitter.
    [0147]
    [0148] The chaotic master system and the chaotic slave system can operate in a synchronized manner only when they are in the same attractor. This chaotic slave system has an output configured to generate a version of a very large number of chaotic masking signals through interruptions between coexisting chaotic attractors equivalent to the very large number of chaotic masking signals of the chaotic master system. This chaotic slave system has a unit for extracting a very large number of information signal packets, which is done by subtracting the very large number version of chaotic masking signal packets from the slave system from the signal transmitted by the public channel. .
    [0149]
    [0150] The emitter comprises a chaotic master system corresponding to two chaotic Rossler 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:
    [0151]
    [0152] d t% m2 ^ m3 [i]
    [0153] d% m2
    [0154] d t ^ m2 ^ Vm2 m
    [0155]
    [0156] = b - cxm3 ymíym3
    [0157] d t [3]
    [0158] dy m í
    [0159] d t ^ ml Vml% m2 ^ m3 [4]
    [0160] ^ V m2
    [0161] d t Vml ^ Vm2 [5]
    [0162] dy m.3
    [0163] dt = b ym3 (yml - c) [6]
    [0164] where the equations of [1] - [3] correspond to the first chaotic oscillator master Rossler and the equations [4] - [6] correspond to the second chaotic oscillator master Rossler.
    [0165]
    [0166] The receiver is structurally the counter-part of the transmitter and comprises a chaotic slave system corresponding to two chaotic Rossler 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 equations coupled differences :
    [0167]
    [0168] dt ~ xs2 xs3 k ( Xm- ± Xsl)
    [0169] dxs2
    [0170] = xs2 ays2
    [0171] dxs3
    [0172] dt = b - cxs3 and sly s3 [9]
    [0173]
    [0174]
    [0175]
    [0176]
    [0177] where equations [7] to [9] correspond to the first chaotic slave oscillator Rossler and equations [10] to [12] correspond to the second chaotic oscillator slave Rossler.
    [0178]
    [0179] The first chaotic slave oscillator Rossler is configured to receive a set of secret dynamic keys through the term k (xmí - xsl) in equation [7], where k coupling factor between the emitter and the receiver and functions as a static key .
    [0180]
    [0181] As the first previous process, the generation of multiple attractors is carried out in the range of initial conditions [ymn niCiai> ym ± final] corresponding to the state variable ymí of the second master chaotic oscillator.
    [0182]
    [0183]
    [0184] As a second prior process, the coupling factor k and the synchronization time rk, where the receiver synchronizes completely with the emitter at each coexisting attractor, is determined in the private channel k (x m í - xsl) in equation [7].
    [0185]
    [0186] As a third prior process, the range of initial conditions [and m l _ ln la i , and m i_ f in ai ] is chaotically modulated through a chaotic logistic map.
    [0187]
    [0188] As a fourth process, the information signal is vectorized in packets of information equivalent to the number of chaotic attractors.
    [0189]
    [0190] When chaotic master-slave oscillators are communicated or coupled by a common signal, these oscillators can operate in a synchronized manner and even when initially both oscillators are operating in different initial conditions.
    [0191]
    [0192] In a traditional system of communication with chaos Figure 1, a small information signal m E (t) is combined with a chaotic signal and E my 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 to produce the transmission signal s (t) = m E (t) and E m iso r (t). The signal s (t) is known as a chaotically masked information signal.
    [0193]
    [0194] In the receiver or chaotic slave system the signal s (t) is received and uses this signal to generate the chaotic signal and Receiver to r (t) of the receiver by synchronization between the sender and the receiver. In the receiver the signal is subtracted and R ecep to r (t) of the signal s (t) to unmask, decode, decrypt, extract or recover the signal the signal m R. Mathematically it is represented as m R = s (t) - and R ecep tor om R = m E (t) and E m ísor and R ecep to r 'which produces T R ^ ^ ■ E when y E m isor e V Receiver are identical.
    [0195]
    [0196] In multi-stable Rossler dynamic systems with non-linear coupling, both the sender and the receiver comprise two chaotic Rossler oscillators with non-linear coupling between their state variables. These chaotic oscillators are expressed by the following set of differential equations:
    [0197]
    [0198]
    [0199]
    [0200]
    [0201] where xlt x2, x 3, ylt y2, and 3 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 uncoupled and at the same time serve as static keys, in the present invention.
    [0202]
    [0203] To demonstrate a large number of coexisting attractors or multi stability, the initial condition of a singular variable and my 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.
    [0204]
    [0205] Figure 2 shows the local maximums of the variable xmi as a function of the initial conditions and m0, 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 whole graph shows a bifurcation diagram, but we emphasize that the axis of the abscissa is not a control parameter but they 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.
    [0206]
    [0207] The emitter based on multi-stable chaotic systems comprises a chaotic master 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]:
    [0208] dt = -Xm2 ^ m3 [i]
    [0209]
    [0210] d t% m2 O-V-ml m
    [0211] dXm.3
    [0212] b CXm 3 ym.iym.3 dt [3]
    [0213] dym i
    [0214] d t ^ ml y-mi ^ m.2 ^ m3 [4]
    [0215] ^ Vm2
    [0216] dty -mí + ^ V m2 [5] dym.3
    [0217] dt b ym3 (ymi - c), [6]
    [0218]
    [0219] where xm, xm2, xm3 are state variables of the first master chaotic oscillator and ymi, ym 2 , and m 3 are state variables of the second master chaotic oscillator, with a = 0.2, b = 0.2 and c = 5.7 are system parameters operating in the chaotic regime when the Rossler oscillators are uncoupled and at the same time serve as static keys, in the present invention.
    [0220]
    [0221] The xmí (t) signal in equation [1] is used to communicate, couple or synchronize the receiver with the transmitter and is carried out through the private channel, which also serves as the first dynamic key. While the chaotic masking signal and my (t) of equation [4], is used to encrypt, code, 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
    [0222] s (t) = ym l (t) mE (t).
    [0223]
    [0224] The issuer includes a chaotic generation and modulation process of the initial conditions and my 0 of the state variable and my equation [4]. The change in the initial conditions ymí 0 in the interval [ymlJniciai, ym ijinai] 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).
    [0225]
    [0226] The chaotic generation and modulation process of the initial conditions and me 0 of the state variable ymí is carried out by the generating unit of initial conditions 175, (see Figure 5), and more specifically by a logistic map that defines the following equation in differences:
    [0227]
    [0228]
    [0229] X (n + 1) = M * n (l Xn), [L l]
    [0230]
    [0231] where xn is defined in the interval [0,1] and p in the interval [0,4], when p = 4 the logistic map shows a chaotic behavior.
    [0232]
    [0233] The change of the initial conditions ymi_o in the interval [and my initial, and my final] is defined by the following equation:
    [0234]
    [0235] and ml _0 and m l_ in ic ia l + X nl y m l_ end al and m l_ in ic ia l ), [L2]
    [0236]
    [0237] 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 transitional iterations have elapsed after the s = 100. 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.
    [0238]
    [0239] In the implementation of the present invention, equation [L2] represents the generating unit of initial conditions 175 (see Figure 5). This unit 175 (see Figure 5) generates d = 751 initial conditions and m 0, which are the results of modulating the interval of initial conditions [and my iCiai , and mifinal ] by the time series x n 1 of the map chaotic logistics [L1]. In turn, the variable and m í uses these and m t 0 initial conditions for the chaotic dynamic system master, equations [1] - [6], generate d = 751 signals chaotic masking via interruptions between multiple chaotic attractors coexisting; and m í (t) and x m í (t) are transmitted from the sender to the receiver through the public and private channel respectively.
    [0240]
    [0241] The receiver based on multi-stable chaotic systems of the present invention comprises a chaotic slave system and is structurally the counterpart of the chaotic master system. This chaotic slave system is based on two chaotic Rossler 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:
    [0242] yes
    [0243] dt = X s 2 - xs3 k (xmí - xsí) [7]
    [0244]
    [0245]
    [0246]
    [0247]
    [0248] where xsVxs2, xs3 are state variables of the first slave chaotic oscillator and yst, yS 2 , and S3 are state variables of the second chaotic slave oscillator, with a = 0.2, b = 0.2 and c = 5.75 system parameters operating in the chaotic regime when Rossler oscillators are uncoupled and at the same time serve as static keys in the present invention.
    [0249]
    [0250] The chaotic slave system of the receiver 450 (see Figure 5) comprises a first Rossler oscillator, which is configured to receive on the private channel, the signal xmi (.t) of the transmitter through the term k (xmí (t) - xsí (t)) in equation [7], where k coupling factor between the transmitter and the receiver. That is, the receiver's chaotic slave system uses the master's chaotic masking signal xmí (t) to generate the version of the chaotic slave masking signal and s l (t) by synchronization between the sender and the receiver. Additionally, the receiver subtracts ys l (t) from the information signal s (t) to unmask, decode, decrypt, extract or recover the mR (t) signal. Mathematically it is expressed as:
    [0251]
    [0252]
    [0253]
    [0254] which produces
    [0255]
    [0256] when ymí (t) and ysl are identical, with e (t) being the signal of the synchronization error.
    [0257]
    [0258] Figure 5 is a schematic representation of the secure multi-stability based communication system 500 in accordance with the present invention based on chaotic multi-stable systems. The system 500 comprises a transmitter 200 and a receiver 400 that are configured to transmit and recover over a public channel 300 a very large number of masked information packets with a very large number of coexisting chaotic attractors, and also transmits keys
    [0259]
    [0260]
    [0261] dynamic through a private channel 350 different from the public channel 300. The sender 200 is configured to receive an information signal mInput (t) = mE (t), and the receiver 400 is configured to retrieve an information signal mSalidad (t). In addition, the transmitter 200 comprises: a multi-stable master chaotic dynamic system 250, an information-dividing vector unit 120 and an initial conditions generating unit 175. In itself, the receiver 400 comprises a chaotic slave dynamic system 450 which is the counter-part of the multi-stable master chaotic dynamic system 250. Additionally, the receiver comprises a first decoding or unmasking unit 470 and a second decoding or unmasking unit 480.
    [0262]
    [0263] In the implementation of the present invention, the multi-stable master dynamic chaotic system 250 of the emitter 200 generates a very large number of multi-stable chaotic masking signals and my (t) operating in accordance with the system of equations [1] - [6]. In addition, the multistable slave chaotic dynamic system 450 of the transmitter 400 is coupled by a private channel with the signal xmí (t) of the dynamic master system 250 through the expression k (xmí - xsl) equation [7] and which by synchronization between these two slave systems-450 and master-250, generate on the side of the slave system 450 a very large number of chaotic masking signals ysi (t). To itself, the chaotic slave system 450 is in accordance with equations [7] to [12].
    [0264]
    [0265] In the implementation of the present invention, the transmitter 200 comprises an information vectorization unit 120, which vectorizes and divides the information signal mE (t), at d = 751 mlE information packages (t) and each package comprises a vp size = 90 iterations and a duration time tme (t) = 18 s.
    [0266]
    [0267] The emitter 200 is configured to chaotically mask each information package mlE (t), using the signal of each state of each chaotic masking attractor and imi (t) of the chaotic master system 250. In addition, the transmitter 200 combines each information packet mlE (t), with each state of each chaotic masking attractor and lmí (t) to produce the transmission signal sl (t) = and lmí (t) m lE (t) of each information packet.
    [0268] The chaotic master system 250 of the emitter 200 generates d = 751 states of coexisting attractors of chaotic masking and lmí (t) chaotically varying the initial conditions of the variable and lmí (t) equation [4] through the generating unit of initial conditions 175. Each of these states of different attractors and lmí (t) comprise a duration time of t state = 70 s, and this in turn has a size of Mestated = 360 iterations. To itself, each of the attractors and lmí (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 chaotic slave system 450 of the receiver 400 to synchronize with the chaotic master system 250 of the transmitter 200, b) second section or section subsequent to the first, corresponds to a time rt = 18 s equivalent to vp = 90 iterations, being rt the transmission time of each mlE (t) information packet.
    [0269]
    [0270] The generating unit of initial conditions 175 of the emitter 200 generates d = 751 initial conditions and which serve as dynamic keys. These initial conditions or dynamic keys are used in the variable and my equation [4] of the chaotic master system 250. These initial conditions are chaotically selected by the logistic map operated by the equation [L1]. This amount of initial conditions d = 751 allows generating d = 751 states of chaotic masking and lmí (t) different.
    [0271]
    [0272] The sender 200 comprises an encryption or masking unit 150 that is configured to combine each mlE (t) information packet with each signal from the coexisting attractor for chaotic masking 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 chaotic masking signal and lmí (t) in a coexisting attractor. The masking unit 150 combines the information signal mlE (t) with the chaotic masking signal and lmí (t) as follows. The information signal mlE (t) that has vp = 90 iterations or a duration time of rt = 18 s, is added to the second leg of the chaotic masking signal and lmí (t) that also has vp = 90 iterations and a duration time of rt = 18 s. Subsequently this process is repeated. 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) = y m í (t) m E (t) is transmitted from sender 200 to receiver 400 by public channel 300.
    [0273]
    [0274] The combination or sum of each information signal m lE (t) with the chaotic masking signal and lm í (t) is always the second leg of the chaotic masking signal and lm í (t), that is, after The synchronization time elapsed t s = 52 s equivalent at mx = 270 iterations.
    [0275]
    [0276] The change of each initial condition in unit 175 is made after each chaotic masking signal and lm í (t) has completed M state = 360 iterations
    [0277]
    [0278]
    [0279]
    [0280] 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 xmi (t) through the private channel 350. The chaotic dynamic slave system 450 of the receiver 400 uses this signal xmi (t ) to generate the slave chaotic masking signal and if (t) by synchronization between the receiver 400 and the transmitter 200.
    [0281]
    [0282] The chaotic master system 250 and the chaotic slave system 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 the information signals transmitted mE (t).
    [0283]
    [0284] The receiver 400 is configured to receive the signal transmitted s (t) by the public channel 300 in addition, it is additionally configured to receive the initial conditions or dynamic keys by a second chaotic masking signal xmi (t) through the private channel 350. The receiver 400 uses the chaotic slave system 450 to generate the version of the chaotic slave masking signal ysi (t) and eliminates the initial conditions or dynamic keys from the transmitted signal s (t) by subtracting ysi (t) from s (t) through the first unmasking or decryption unit 470. For proper suppression of the initial conditions or dynamic keys, the chaotic slave system 450 and the chaotic master system 250 must be fully synchronized by means of the static keys, based on the values of its parameters a = 0.2, b = 0.2 and c = 5.7. As a result of the subtraction of ysi (t) to s (t) by means of the first unit of
    [0285]
    [0286]
    [0287] Unmasking or decoding 470, the transmitted chaotic masked message signal s (t) is transformed into the recovered message signal mR (t).
    [0288]
    [0289] Specifically, each signal of the message 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 tranche subsequent to the first, a corresponding transmission signal mlE (t), ie mlR (t) = the (t) mlE (t).
    [0290]
    [0291] In summary, the operation of the transmitter 250 and the receiver 450 can be represented by the following mathematical equations:
    [0292] • s (t) = ym l (t) mE (t),
    [0293] • xm i (.t): synchronization signal of transmitter 250 and receiver 450,
    [0294] • mR (t) = mE (t) and my (t) - ysí (t), where mR (t) = e (t) mE (t) when ymi (t) = ysi (t),
    [0295] where:
    [0296] • s (t): chaotic masked message signal, transmitted by public channel 300,
    [0297] • xmí (t): initial condition signal or dynamic key transmitted by private channel 350,
    [0298] • mR (t): signal recovered,
    [0299] • e (t): synchronization error.
    [0300]
    [0301] Additionally, the receiver 400 includes a second unmasking or decoding unit 480, which is configured to recover the msaiida information signal (t), by the subtraction e (t) of mR (t), that is,
    [0302] mSaiity (t) = mR (t) - e (t),
    [0303] ™ Health (t) = e (t) mE (t) - e (t), d ° ncle msalidad (t) = mE (t) ° mSalidad (t) = WEntrada (0 ■
    [0304]
    [0305] Figure 6 is a schematic representation of a digital electronic device that implements the system of the present invention. In general terms, the digital electronic device 500a comprises a transmitter 200a and a receiver 400a that are configured to transmit and recover over a public channel 300a a very large number of masked information packets with a very large number of coexisting chaotic attractors, and in addition transmit the dynamic keys through a channel private 350th different from public channel 300th. The sender 200a is configured to receive an mInput information signal (t), and the receiver 400a is configured to retrieve an msaiida information signal (t). In addition, the emitter 200a comprises: a chaotic multi-stable master system 250a, a vectorizing-dividing information unit 100a and an initial conditions generating unit 175a. In itself, the receiver 400a comprises a chaotic slave system 450a which is the counter-part of the multi-stable master chaotic dynamic system 250a. In addition, the receiver 400a incorporates the first unmasking or decoding unit 470a, which is configured to suppress the initial conditions or dynamic keys. At the same time, receiver 400a comprises the second unmasking or decoding unit 480a, which is configured to recover the output information signal (t).
    [0306]
    [0307] Representative examples of the chaotic coding-decoding system of information signal based on multi-stable systems
    [0308]
    [0309] 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 ntra da (t) = m E (t) or vectorized signal. Figure 7 (b) shows an information signal packet mE (t) versus time, performed by the information-dividing vector unit 120a. Figure 7 (c) shows a portion of the information signal m E ntra da (.t) versus time, corresponding to a very large number of information signal packet m lE (t).
    [0310]
    [0311] The following are representative examples that illustrate particular cases, under which, there are correct and incorrect recoveries of the information signal m E ntra da (t). These representative examples correspond to the secure communication system based on multi-stability 500a in accordance with the present invention.
    [0312]
    [0313] Representative example 1: message mEntry (t), successfully retrieved
    [0314]
    [0315] Masking or encryption process
    [0316]
    [0317] In this representative example, the chaotic master system 250a of the transmitter 200a and the chaotic slave system 450a of the receiver 400a are configured to be identical
    [0318]
    [0319]
    [0320] or similar. The respective static keys of the transmitter 200a and the receiver 400a are shown in Table I.
    [0321]
    [0322]
    [0323]
    [0324]
    [0325] Table I: Static keys of the transmitter 200a and the receiver 400a, corresponding to the parameters of the systems of equations [1] - [6] and [7] - [12], for example, representative 1: input message (t), recovered correctly.
    [0326]
    [0327] Figure 8 (a) shows a chaotic masking state and the m (t) as a function of the time generated by the master chaotic system 250a, due to the change of an initial condition produced in the generating unit of initial conditions 175a. Figure 8 (b) shows the combination of a chaotic masking state and the m (t) with an information package m lE (t) performed by the chaotic masking unit 150a, to produce a transmission signal sl (t) = yl m í (t) ml E (t), which is transmitted from sender 200a to receiver 400a through public channel 300a.
    [0328]
    [0329] To clarify how the process of encryption of 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
    [0330] t s = 52 s equivalent to the time required by the chaotic slave system to synchronize with the chaotic master system. b) Second section: signal s l (t), it corresponds to a time r t = 18s equivalent to the transmission time of each packet of information m lE (t) is masked, encrypted in each coexisting chaotic attractor and í ni it) . In this Figure 8 (c) the total time t state = 70 s of each coexisting chaotic attractor and í ni it) is also observed, equivalent to state = t s + r t .
    [0331]
    [0332] 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 attractors of chaotic masking and m (t) and a very large number of information signals m E (t), which are transmitted from sender 200a to receiver 400a.
    [0333]
    [0334]
    [0335] Recovery or decryption process
    [0336]
    [0337] Figure 9 (a) shows a chaotic masking state xlmí (t) versus time, generated by the master chaotic system, equations 250a, due to the change of an initial condition produced in the generating unit of initial conditions 175a. The signal x lmí (t) is transmitted from the sender 200a to the receiver 400a by the private channel 350a and the chaotic slave system 450a uses the xlmí (t) by synchronization to generate the signal and lsí (t). Figure 9 (b) shows the signal ^ (t) versus time. While Figure 9 (c) shows the signal ^ (t) versus the signal and lmí (t) and in which a strong complete synchronization between these signals can be observed after the synchronization time t s has elapsed.
    [0338]
    [0339] Figure 10 (a) shows the signal of the recovered message m jj (t) versus time, as a result of the 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 formed by a first section corresponding to the synchronization error on (t) and a second section corresponding to an information signal packet mlE (t), that is mlR (t) = the (t) m lE (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 on (t) and a very large number of mlE (t) information packages.
    [0340]
    [0341] Figure 11 (a) shows a message signal packet successfully recovered mlsalida (t) versus time, as a result of the subtraction on (t) of m ^ t) by means of the second unmasking or decoding unit 480a. Figure 11 (b) shows a portion of the message message successfully recovered msalidd (t) corresponding to a very large number of signals recovered misallda (t).
    [0342]
    [0343] Figure 12 depicts a portion of the successfully recovered signal mass (t) versus a portion of the original lie signal (t). This graph shows in the status space the complete synchronization between the recovered signal and the original signal. The conversion of the successfully recovered signal higher (t) to flat image is shown in Figure 13.
    [0344] Figures 8-1 show a high correlation between the masking process and recovery of the information signal, which makes the secure communication system 500a of the present invention a highly robust secure communication system.
    [0345]
    [0346] Representative example 2: message m r „tr„ * „(t '), incorrectly retrieved when the static keys are improperly implemented
    [0347]
    [0348] Masking or encryption process
    [0349]
    [0350] In this representative example, the chaotic master system 250a of the transmitter 200a and the chaotic slave system 450a of the receiver 400a are configured to be completely different in at least one parameter a. The respective static keys of the transmitter 200a and the receiver 400a are different and are shown in Table II.
    [0351]
    [0352]
    [0353]
    [0354]
    [0355] Table II: Static keys of sender 200a and receiver 400a, corresponding to the parameters of the systems of equations [1] - [6] and [7] - [12], for representative example 2: message mInput (t), recovered incorrectly.
    [0356]
    [0357] Figure 8 (a) shows a chaotic and lm masking state generated by the chaotic master system 250a, due to the change of an initial condition produced in the generating unit of initial conditions 175a. Figure 8 (b) shows the combination of a chaotic masking attractor and lmí (t) with a package of information miE (t) made by the chaotic masking unit 150a, to produce a transmission signal sl (t) = y im l (t) m iE (t), which is transmitted from sender 200a to receiver 400a through 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 signals of mE (t) information, which are transmitted from sender 200a to receiver 400a.
    [0358]
    [0359]
    [0360] Recovery or decryption process
    [0361]
    [0362] Figure 9 (a) shows a chaotic masking state xlmí (t) versus time, generated by the chaotic master dynamic system, 250a, due to the change of an initial condition produced in the generating unit of initial conditions 175a. The signal xlmí (t) is transmitted from sender 200a to receiver 400a through private channel 350a and the chaotic slave system 450a uses the x lmí (t) by synchronization to generate the signal and imi (t).
    [0363]
    [0364] Figure 14 (a) shows the ^ (t) signal versus time. While Figure 14 (b) shows the signal y ^ i (t) versus the signal and lmí (t) in which a weak synchronization can be observed between these signals whose state space y ^ i (t) vs y lm l (t) occupies a wide region, which represents a poor correlation between these signals, that is, and ^ i (t) e and lmí (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 chaotic master dynamic system 250a by a = 0.2 and in the chaotic slave system 450a by a = 0.3. This difference of the static keys in the transmitter 200a and the receiver 400a makes it difficult to recover the original image.
    [0365]
    [0366] Figure 15 (a) shows the signal of the recovered message m jj (t) versus time, as a result of the subtraction of y ^ t) to 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 the (t) has very large values and a second section which is a completely different signal than the information signal mlE (t). Figure 15 (b) represents a portion of the recovered message mR (t) versus time, corresponding to a very large number of incorrectly retrieved information packets.
    [0367]
    [0368] Figure 16 (a) shows an incorrectly recovered message signal packet msaiida (t) versus time, as a result of the subtraction on (t) of m ^ t) by means of the second unmasking or decryption unit 480a. Figure 16 (b) shows a portion of the incorrectly recovered message signal msalidd (t) corresponding to a very large number of incorrectly recovered signals mlsalida (t).
    [0369]
    [0370]
    [0371] 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 trada (t), in which, a weak synchronization between these can be observed signals and whose space of states m sa i id ad (t) vs m in tr ada (t) occupies a wide region, representing a poor correlation between these signals, that is, m sa l id ad (t) and m in tr ada (t) they are completely different.
    [0372]
    [0373] The conversion of the incorrectly recovered signal m sa l ld ad (t) to flat image is shown in Figure 17, which is completely different from the original flat image Figure 7 (a).
    [0374]
    [0375] 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, a small difference from the static keys in the transmitter 200a and the receiver 400a makes it impossible to recover the original image.
    [0376]
    [0377] Representative example 3: mInput message (t) retrieved incorrectly when dynamic keys are improperly implemented
    [0378]
    [0379] In the present representative example 3, the dynamic keys are improperly implemented. Figure 18 represents a secure communication system 500b that can be implemented in any digital electronic system. In this system, the private communication channel that communicates the transmitter 200b with the receiver 400b is incorrectly 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 system of Secure communication 500a of the present invention, this private channel 350a does exist or is well configured as illustrated in Figure 6.
    [0380]
    [0381] In addition, the chaotic slave system 450b of the transmitter 400b cannot use the xm í (t) signal of the dynamic master system 250b and by synchronization generate the JsiW signal. In the absence of synchronization between 200b transmitter and receiver 400b, the signal chaotic masking and m i (t) of the transmitter 200b and the signal chaotic ys l masking (t) the receiver 400b are completely different, namely and m t ( t) ^ and st (t).
    [0382] In the secure communication system 500b, the combined signal s (t) = ymí (t) mE (t) is transmitted from sender 200b to receiver 400b via public channel 300b, where ymí (t) is the chaotic masking signal and mE (t) is the information signal. In addition, in the receiver 400b the message recovered mR (t), as a result of the subtraction of ysi (t) as (t) by means of the first unmasking or decoding unit 470a is performed unsatisfactoryly or incorrectly, this is because ymí (t) ^ ysi (t) and mathematically is represented by
    [0383]
    [0384] mR (t) = s (t) - and yes,
    [0385] mR (t) = ym l (t) mE (t) - ysU
    [0386]
    [0387] Because ymí (t) ^ and yes (t), the recovered signal mR (t) is different from the signal mE (t). In this representative example 3, the omission of the dynamic key xmí (t) or lack of the private communication channel that synchronizes the sender 200b with the receiver 400b causes the original message to be unsuccessfully recovered by the receiver 400b of the communication system safe 500b.
    [0388]
    [0389] Masking or encryption process
    [0390]
    [0391] In this representative example, the chaotic master system 250b of the transmitter 200b and the dynamic chaotic slave system 450b of the receiver 400b are configured to be identical. The respective static keys of the transmitter 200b and the receiver 400b are similar and are shown in Table I.
    [0392]
    [0393] Figure 8 (a) shows a chaotic and lm masking state generated by the chaotic master dynamic system 250b, due to the change of an initial condition-i produced in the generating unit of initial conditions 175a. Figure 8 (b) shows the combination of a chaotic-i and lmí (t) masking state with a mlE (t) information packet performed by the chaotic masking unit 150b, to produce a transmission signal sl (t) = y lmí (t) m lE (t), which is transmitted from sender 200b to receiver 400a through 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 signals of mE (t) information, which are transmitted from sender 200b to receiver 400b.
    [0394]
    [0395]
    [0396] Recovery or decryption process
    [0397]
    [0398] Figure 19 (a) shows the ^ (t) signal versus time. While Figure 19 (b) shows the signal ^ (t) versus the signal yl m í (t) in which we can observe a weak synchronization between these signals whose space of states ^ (t) vs and lm í (t) It occupies a wide region, which represents a poor correlation between these signals, that is, ^ (t) e and lm í (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 chaotic master system which synchronizes transmitter 200b with receiver 400b. This lack of synchronization of the transmitter 200b with the receiver 400b performed by the dynamic keys xl m í (t) through a private communication channel makes it difficult to recover the original image.
    [0399]
    [0400] Figure 20 (a) shows the signal of the recovered message m jj (t) versus time, as a result of the subtraction of y ^ t) as 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 which is a completely different signal than the information signal m lE (t) . Figure 20 (b) represents a portion of the recovered message m R (t) versus time, corresponding to a very large number of incorrectly retrieved information packets.
    [0401]
    [0402] Figure 21 (a) shows an incorrectly recovered message signal packet m sai id a (t) versus time, as a result of the subtraction e l (t) of m ^ t) by means of the second unmasking unit or 480b decoding. Figure 21 (b) shows a portion of the incorrectly recovered message signal m salt id d (t) corresponding to a very large number of incorrectly recovered signals more than id a (t).
    [0403]
    [0404] Figure 21 (c) represents a portion of the incorrectly recovered signal m sa i id ad (t) versus a portion of the original signal m in triad (t), in which we can observe a weak synchronization between these signals and whose space of states m sa i id ad (t) vs m in tr ada (t) occupies a wide region, representing a poor
    [0405]
    [0406]
    [0407] Correlation between these signals, that is, misality (t) and lying (t) are completely different.
    [0408]
    [0409] The conversion of the incorrectly recovered signal quality (t) to flat image is shown in Figure 22, which is completely different from the original flat image Figure 7 (a).
    [0410]
    [0411] 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.
    权利要求:
    Claims (9)
    [1]
    1.- Transmitter (200) for secure communication based on multi-stability, characterized in that it comprises:
    • a vectorization unit (120) configured to receive an information signal to be transmitted mE (t), which is vectorized and divided into d information packages mlE (t)
    • a unit generating initial conditions (175) where the initial conditions and lmX are located in the range [yml_iniciai> ymijinai ;
    • a dynamic multi-stable master system (250) that generates coexisting chaotic attractors of chaotic masking that vary chaotically by changing the initial conditions and lmí of the ymi variable in equations [3] - [6] through the generating unit of initial conditions (175) and solving the following differential equations:

    [2]
    2. - Emitter (200) for secure communication based on multi-stability, according to claim 1, characterized in that the multi-stable 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]
    3. - Emitter (200) of secure communication based on multi-stability, according to claim 1, characterized in that the generating unit of initial conditions (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 , and mi j in ai ] is defined by the following equation:
    yml_0 yml_inicial + ^ n + l {yml_final yml_inicial), [L2]
    where xn + t represents the time series of the logistic map (L1) in the chaotic regime and n 1 are the iterations.
    [4]
    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 transient iterations have elapsed.
    [5]
    5. - Secure communication transmitter (200) based on multi-stability, according to claim 1, characterized in that each information packets d has a vp size equal to 90 iterations and a duration time tme (t) of 18 seconds.
    [6]
    6. - Receiver (400) for secure communication based on multi-stability, 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 multi-stable chaotic dynamic slave system ( 450) that generates a slave chaotic masking signal and if ( t) from an x mi ( t) signal received from a transmitter (200) via a private channel (350), and solving the following differential equations:

    [7]
    7. - Receiver (400) for secure communication based on multi-stability, 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 status 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]
    8. - Secure communication system based on multi-stability, characterized in that it comprises a transmitter (200) defined in any one of claims 1 to 5, a receiver ( 400 ) defined in any one of claims 6 to 8, wherein the The sender (200) communicates with the receiver (400) through a public channel (300) and a private channel (400).
    [9]
    9. A 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 sends coded s (t), by means of coexisting attractors of chaotic masking, to the receiver through the public channel, and where the sender sends a dynamic key to the receiver through 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).
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    同族专利:
    公开号 | 公开日
    ES2724598B2|2020-03-19|
    引用文献:
    公开号 | 申请日 | 公开日 | 申请人 | 专利标题
    US5048086A|1990-07-16|1991-09-10|Hughes Aircraft Company|Encryption system based on chaos theory|
    US20110002460A1|2009-07-01|2011-01-06|Harris Corporation|High-speed cryptographic system using chaotic sequences|
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