TWO-FOLD MASKING OF DIFFERENT CHAOTIC ATTRACTORS, INCLUDING

MODIFIED CHAOTIC ATTRACTORS,

USING STATIC-DYNAMIC SECRET KEYS

Technical Field

The present disclosure generally relates to systems, circuits, devices, methods, and processes that utilize a set of chaotic oscillators to facilitate secure or enhanced security communication. More particularly, aspects of the disclosure relate to a secure communication system configured to utilize multiple chaotic oscillators, such as an oscillator based upon a modified Lorenz-like chaotic attractor and/or an oscillator based upon a modified five-term chaotic attractor. An embodiment of such a secure communication system can utilize fixed secret keys as well as dynamic secret keys for message recovery to facilitate enhanced information security.

Background

Entities such as private individuals, businesses, financial institutions, governmental organizations, and militaries commonly desire to communicate information from a transmitting party or source to a receiving party or destination in a manner that avoids, minimizes, or reduces a likelihood of information compromise. That is, communicating parties frequently seek to ensure that confidential information remains confidential. As a result, information privacy or security is often of great importance in signal communication situations, such as the transfer of signals over wireless communication channels.

In general, a secure or enhanced security communication system is designed to satisfy one or more secure communication objectives, depending upon communication context. For instance, a secure communication system should be capable of communicating information signals from a source to a destination in a manner that substantially reduces or acceptably reduces a likelihood of information signal interception and information extraction. A secure communication system should additionally be capable of communicating information signals in a manner that reduces the likelihood of communication disruption. In certain communication contexts, a secure communication system should further be capable of reducing or minimizing a likelihood of communication detection. Various objectives associated with secure or enhanced security communication can be facilitated through the use of unpredictable signals, such as chaotic signals, to encode or conceal information signals. In general, chaotic signals arise from nonlinear systems which, for a given set or range of operating parameters, produce outputs that exhibit aperiodic behavior. Thus, chaotic signals exhibit unpredictable values over long periods of time. Additionally, chaotic signals exhibit a strong or extreme sensitivity to initial conditions. Essentially identical chaotic systems that are subjected to slightly different initial conditions generate outputs that evolve along uncorrelated, diverging trajectories.

In general, wireless communication systems rely upon a transmitter and a receiver that are capable of operating in a linked or synchronized manner. Separate chaotic oscillators have been shown to be capable of synchronizing with each other in a manner that enables enhanced security communication. Examples of synchronized chaotic oscillators that are suitable for enhanced security communication are described in U.S. Patent No. 5,245,660 to Peccora et al., U.S. Patent 5,291,555 to Cuomo et al., and U.S. Patent No. 6,049,614 to Kim.

Present enhanced security communication systems that include chaotic oscillators can suffer from one or more drawbacks. For instance, such systems may fail to provide a sufficient or desired level of information security. Additionally, such systems may be undesirably susceptible to sources of communication disruption, such as noise. Furthermore, such systems may be implemented in a manner that is undesirably or unnecessarily complex or expensive.

Summary

In accordance with an aspect of the disclosure, a system for secure or enhanced security communication can include a transmitter having an input configured to receive a message signal and a set of outputs, where the transmitter includes a) a first drive system corresponding to a first chaotic attractor, the first drive system having a first drive system output, the first drive system configured to provide a first chaotic masking signal at the first drive system output; b) a second drive system corresponding to a second chaotic attractor that is distinct from the first chaotic attractor, the second drive system having a second drive system output, the second drive system configured to provide a second chaotic masking signal at the second drive system output; and c) a two-fold masking unit having a first masking unit input coupled to the transmitter input, a second masking unit input coupled to the first drive system output, a third masking unit input coupled to the second drive system output, and a masking unit output coupled to a first transmitter output, the two-fold masking unit configured to provide at the masking unit output a two-fold chaotically masked message signal corresponding to a sum of the message signal, the first chaotic masking signal, and the second chaotic masking signal, wherein one of the first chaotic attractor and the second chaotic attractor corresponds to one of a five-term chaotic attractor and a Lorenz-like chaotic attractor.

In such a transmitter, the first chaotic attractor can correspond to a five-term chaotic attractor and the second chaotic attractor corresponds to a Lorenz-like chaotic attractor. The five-term chaotic attractor mathematically corresponds to the following set of Equations:

* = a(y - x),

y =— xz,

z = -b + xy

The Lorenz-like chaotic attractor mathematically corresponds to the following set of Equations:

* = a(y - x),

y = bx - dxz,

z = xy -cz

The first transmitter output is communicatively coupled to a public communication channel. In accordance with an aspect of the disclosure, the transmitter further includes a second transmitter output that is coupled to one from the group of the first drive system output and the second drive system output, the second transmitter output configured to provide a set of dynamic secret keys that is distinct from the two-fold chaotically masked message signal. The?; - set of dynamic secret keys can be generated from a time series of one of the first chaotic masking signal and the second chaotic masking signal. In such a configuration, the first transmitter output can be communicatively coupled to a public communication channel, and the second transmitter output can be communicatively coupled to a secret communication channel. In accordance with an aspect of the disclosure, a system for secure or enhanced security communication can include a receiver configured for signal communication with the transmitter, the receiver having a receiver output, a first receiver input configured to receive a transmitted version of a two-fold chaotically masked message signal, and a second receiver input configured to receive a set of dynamic secret keys that is distinct from the transmitted version of the two-fold chaotically masked message signal. The receiver can include a) a first response system corresponding to the first chaotic attractor, the first response system having a first response system input and a first response system output, the first response system input coupled to the second receiver input; and b) a second response system corresponding to the second chaotic attractor, the second response system having a second response system input and a second response system output, the second response system input coupled to the first receiver input.

The first chaotic attractor can correspond to a five-term chaotic oscillator and the second chaotic attractor can correspond to a Lorenz-like chaotic attractor. The five-term chaotic attractor mathematically corresponds to the following set of Equations:

* = a(y - x),

y = -xz,

z = -b + xy

The Lorenz-like chaotic attractor mathematically corresponds to the following set of Equations: = a(y - x),

Ϋ = bx - dxz,

z = xy -cz

Thus, the first response system can be a structural counterpart to the first drive system, and the second response system can be a structural counterpart to the second drive system. Additionally, the first response system input can be communicatively coupled to a public communication channel, and the second response system input can be communicatively coupled to a secret communication channel. In accordance with an aspect of the disclosure, the first response system is synchronizable with the first drive system, and the second response system is synchronizable with the second drive system. Synchronization can be facilitated by static secret keys that are specified, stored, or programmably defined at or within the transmitter and the receiver. A set or group of static · secret keys at the transmitter-side and a set or group of static secret keys at the receiver-side include a set of specified parameters and initial conditions for each of the chaotic attractors associated with the transmitter-side and the receiver-side, respectively. Thus, in one aspect, a first and a second set of transmitter-side static secret keys can respectively include parameters and initial conditions corresponding to a first and a second transmitter-side chaotic attractor. Similarly, a first and a second set of receiver-side static secret keys can respectively include parameters and initial conditions corresponding to a first and a second receiver-side chaotic attractor.

In one aspect, the first drive system can be configured to operate in accordance with a first set of drive static secret keys, which includes first drive system parameters and first drive system initial conditions corresponding to the first chaotic attractor. The first response system can be configured to operate in accordance with a first set of response static secret keys, which includes first response system parameters and first response system initial conditions corresponding to the first chaotic attractor. The first set of response static secret keys can facilitate synchronization between the first response system and the first drive system.

The second drive system can be configured to operate in accordance with a second set of drive static secret keys that includes second drive system parameters and second drive system initial conditions corresponding to the second chaotic attractor. The second response system can be configured to operate in accordance with a second set of response static secret keys that includes second response system parameters and second response system initial conditions corresponding to the second chaotic attractor. The second set of response static secret keys can facilitate synchronization between the second response system and the second drive system.

A receiver in accordance with an aspect of the disclosure can further include a segregation unit having a first segregation unit input, a second segregation unit input, and a segregation unit output, the first segregation unit input coupled to the first receiver input and the second segregation unit input coupled to the first response system output, the segregation unit configured to subtract the set of dynamic secret keys from the transmitted version of the twofold chaotically masked message signal.

A receiver in accordance with an aspect of the disclosure can further include a message recovery unit having a first message recovery unit input, a second message recovery unit input, and a message recovery unit output, the first message recovery unit input coupled to the segregation unit output and the second message recovery unit input coupled to the second response system output, the message recovery unit configured to provide a recovered message signal at the message recovery unit output.

In accordance with an aspect of the disclosure, a representative process for transmitter-side, encoder-side, or masking-side enhanced security communication can include a first process portion that involves defining or specifying a set or group of transmitter-side, encoder-side, or masking-side static secret keys. Such static secret keys can include a first set of transmitter- side, encoder-side, or masking-side static secret keys that corresponds to a first chaotic attractor; and a second set of transmitter-side, encoder-side, or masking-side static secret keys that corresponds to a second chaotic attractor distinct from the first chaotic attractor. One of the first and second chaotic attractors can correspond to one of a five-term chaotic attractor and a Lorenz-like chaotic attractor. In an embodiment, the first and second chaotic attractors can respectively correspond to a five-term chaotic attractor and a Lorenz-like chaotic attractor.

The process can further include a second process portion that involves generating a transmitter-side, encoder-side, or masking-side first chaotic masking signal, and a third process portion that involves generating a transmitter-side, encoder-side, or masking-side second chaotic masking signal. In an embodiment, the first and second chaotic masking signals can be generated using a transmitter-side five-term chaotic oscillator and a transmitter- side Lorenz-like chaotic oscillator, respectively.

The process additionally includes a fourth process portion that involves generating a set of dynamic secret keys, for instance, using the first chaotic masking signal. The process also includes a fifth process portion in which a received or retrieved message signal is combined with the first and second chaotic masking signals to produce a two-fold chaotically masked message signal. Finally, the process includes a sixth process portion that involves outputting, transmitting, or transferring the two-fold chaotically masked message signal, and a seventh process portion that involves outputting, transmitting, or transferring the set of dynamic secret keys.

In accordance with an aspect of the disclosure, a representative process for receiver-side, decoder-side, or unmasking-side enhanced security communication includes a first process portion that involves defining or specifying a set or group of receiver-side, decoder-side, or unmasking-side static secret keys. Such static secret keys can include a first set of receiver- side, decoder-side, or unmasking-side static secret keys that corresponds to a first chaotic attractor; and a second set of receiver-side, decoder-side, or unmasking-side static secret keys that corresponds to a second chaotic attractor distinct from the first chaotic attractor. One of the first and second chaotic attractors can correspond to one of a five-term chaotic attractor and a Lorenz-like chaotic attractor. In an embodiment, the first and second chaotic attractors can respectively correspond to a five-term chaotic attractor and a Lorenz-like chaotic attractor.

A second process portion and a third process portion include receiving or retrieving the twofold chaotically masked message signal and the set of dynamic secret keys, respectively. A fourth process portion includes generating a receiver-side, decoder-side, or unmasking-side first chaotic masking signal that is synchronized or linked to or correlated with the set of dynamic secret keys, and a fifth process portion includes generating a receiver-side, decoder- side, or unmasking-side second chaotic masking signal.

A sixth process portion includes segregating the set of dynamic secret keys from the two-fold chaotically masked message signal (e.g., using the receiver-side first chaotic masking signal) to generate a segregated chaotically masked message signal. Finally, a seventh process portion includes recovering or extracting a message signal from the segregated chaotically masked message signal using the receiver-side second chaotic masking signal.

Brief Description of the Drawings and Tables

Drawings

FIG. 1 is a schematic representation of an embodiment of a communication technique referred to as chaotic masking. FIG. 2 is a schematic illustration of a representative implementation of a communication system based upon a five-term chaotic oscillator according to an embodiment of the disclosure.

FIG. 3A is a graph of a portion of a representative message signal m(t) versus time for the communication system of FIG. 2.

FIG. 3B is a graph illustrating results corresponding to a transmitted signal s(t) output by a transmitter of the communication system of FIG. 2.

FIG. 3C is a graph illustrating simulation results corresponding to a recovered message signal m(t) produced by a receiver of the communication system of FIG. 2.

FIG. 3D is a graph illustrating simulated steady state self-synchronization characteristics of the communication system of FIG. 2.

FIG. 4 is a schematic illustration of a representative implementation of a communication system based upon a Lorenz-like chaotic oscillator according to an embodiment of the disclosure.

FIG. 5A is a graph of a portion of a representative message signal m(t) versus time for the communication system of FIG. 4.

FIG. 5B is a graph illustrating a simulation of a transmitted signal s(t) output by a transmitter of the communication system of FIG. 4.

FIG. 5C is a graph illustrating a simulated recovered simulated message signal in(t) produced by a receiver of the communication system of FIG. 4.

FIG. 5D is a graph illustrating simulated steady state self-synchronization characteristics of the communication system of FIG. 4. FIG. 6 is a schematic illustration of a secure communication system according to an embodiment of the disclosure, based upon two-fold masking of different chaotic oscillators with static-dynamic secret keys.

FIG. 7 is a schematic illustration of a representative implementation of a secure communication system according to an embodiment of the disclosure, based upon two-fold masking of different chaotic oscillators with static-dynamic secret keys.

FIG. 8 is a graph of a portion of a representative message signal m(t) versus time for the communication system of FIG. 7.

FIG. 9 is a graph showing a portions of a second chaotic masking signal xc(t) generated by a communication system's Lorenz-like chaotic oscillator in accordance with a first representative Example of successful message signal recovery.

FIG. 10 is a graph showing a set of dynamic secret keys s ₂ (t) generated by a five-term chaotic oscillator, and which is communicated using a secret communication channel in accordance with a first representative Example of successful message signal recovery.

FIG. 11 is a graph showing portions of a two-fold chaotically masked message signal s^t) output to a public communication channel in accordance with a first representative Example corresponding to successful message signal recovery.

FIG. 12 is a graph showing portions of a successfully recovered message signal m(t) versus time in accordance with a first representative Example corresponding to successful message signal recovery.

FIG. 13 is a graph showing transient behavior of the successfully recovered message signal m(t) of FIG. 12 in accordance with a first representative Example corresponding to successful message signal recovery.

FIG. 14 is a graph showing steady state self-synchronization characteristics of a secure communication system corresponding to FIGs. 9 - 13. FIG. 15 is a graph showing portions of a second chaotic masking signal χα(ΐ) generated by a secure communication system's Lorenz-like chaotic oscillator in accordance with a second representative Example of successful message signal recovery.

FIG. 16 is a graph showing a set of dynamic secret keys s ₂ (t) generated by a five-term chaotic oscillator, and which is communicated using a secret communication channel in accordance with a second representative Example corresponding to successful message signal recovery.

FIG. 17 is a graph showing portions of a two-fold chaotically masked message signal s^t) communicated using a public communication channel in accordance with a second representative Example corresponding to successful message signal recovery.

FIG. 18 is a graph showing portions of an unsuccessfully recovered message signal m(t) versus time as a result of a transmitter-side to receiver-side static secret key mismatch in accordance with a third representative Example corresponding to unsuccessful message signal recovery.

FIG. 19 is a schematic illustration of a communication system having a receiver that is (mis)configured to generate receiver-side dynamic secret keys using an incorrect or inappropriate communication channel in accordance with a fourth representative Example corresponding to unsuccessful message signal recovery.

FIG. 20 is a graph showing portions of an unsuccessfully recovered message signal m(t) versus time as a result of a receiver-side misconfiguration with respect to the generation of dynamic secret keys in accordance with a fourth representative Example corresponding to unsuccessful message signal recovery.

FIG. 21 A is a flow diagram of a representative process for transmitter-side, encoder-side, or masking-side enhanced security communication in accordance with the present disclosure.

FIG. 21B is a flow diagram of a representative process for receiver-side, decoder-side, or unmasking-side enhanced security communication in accordance with the present disclosure. Tables

Table 1 provides a set of representative initial conditions and parameter values corresponding to a first and a second static secret key group in accordance with a first representative Example corresponding to successful message signal recovery.

Table 2 provides another set of representative initial conditions and parameter values corresponding to a first and a second static secret key group in accordance with a second representative Example corresponding to successful message signal recovery.

Table 3 provides a representative mismatched configuration of transmitter-side and receiver- side static secret keys in accordance with a third representative Example corresponding to unsuccessful message signal recovery.

Table 4 provides a representative matched configuration of transmitter-side and receiver-side static secret keys in accordance with a fourth representative Example corresponding to unsuccessful message signal recovery.

Detailed Description

Embodiments of the present disclosure are directed to aspects of secure or enhanced security communication systems that address at least one of the aforementioned drawbacks. Communication systems in accordance with the present disclosure can facilitate or enable the transfer of information or message signals in a manner that achieves a very high, high, or acceptable level of information security, and which exhibits enhanced tolerance to potential sources of communication disruption such as noise or transmitter - receiver parameter mismatch. Communication systems in accordance with the present disclosure can further be implemented using simple circuitry in a cost effective manner.

Various embodiments of enhanced security communication systems in accordance with the present disclosure can include chaotic oscillators that are based upon one or more chaotic attractors. Multiple embodiments of the present disclosure can include a chaotic oscillator based upon a modified Lorenz-like chaotic attractor. Additionally or alternatively, multiple embodiments of the present disclosure can include a chaotic oscillator based upon a modified five-term chaotic attractor. An embodiment of a secure communication system in accordance with the present disclosure can include at least one of a modified Lorenz-like chaotic oscillator and a modified five-term chaotic oscillator. As described in detail below, such a system can utilize both static keys and dynamic keys to facilitate the recovery or extraction of received messages that have been encoded or rendered obscure or indistinct prior to their transmission through the use of a set of chaotic oscillators.

Representative aspects of systems, apparatuses, devices, circuits and processes for facilitating secure or enhanced security communication in accordance with embodiments of the present disclosure are described in detail hereafter with reference to FIG. 1 to FIG. 20, in which like or analogous elements or process portions are shown numbered with like or analogous reference numerals. Relative to descriptive material corresponding to one or more of FIGs. 1 - 20, the recitation of a given reference numeral can indicate simultaneous consideration of a FIG. in which such reference numeral was previously shown. The description herein provides for embodiments that are suitable for secure or enhanced security communication, such as wireless communication. The embodiments provided by the present disclosure are not precluded from applications in which particular fundamental structural and/or operational principles present among the various embodiments described herein are desired.

In the context of the present disclosure, the term set is defined as a non-empty finite organization of elements that mathematically exhibits a cardinality of at least 1 (i.e., a set as defined herein can correspond to a singlet or single element set, or a multiple element set), in accordance with known mathematical definitions (for instance, in a manner corresponding to that described in An Introduction to Mathematical Reasoning: Numbers, Sets, and Functions, "Chapter 11: Properties of Finite Sets" (e.g., as indicated on p. 140), by Peter J. Eccles, Cambridge University Press (1998)). In general, an element of a set can include or be a structure, a device, a signal, a function or functional process, or a value depending upon the type of set under consideration.

Self-Synchronization of Chaotic Systems

In general, a chaotic communication system in accordance with the present disclosure includes a transmitter and a receiver. The transmitter includes at least one chaotic oscillator, and can be referred to as a drive or master system. The receiver correspondingly includes at least one chaotic oscillator, and can be referred to as a response or slave system. In various situations, the drive system and the response system utilize essentially identical, analogous, counterpart, or corollary types of chaotic oscillators. When a chaotic oscillator based drive system and a chaotic oscillator based response system are communicatively coupled by way of an appropriate type of common signal, the drive system and the response system can operate in a synchronized manner, even though they may initiate their operation under non-identical initial conditions. More particularly, the response system can self-synchronize or auto-synchronize its operation with that of the drive system. Such synchronization of chaotic oscillators is described by K.M. Cuomo and A.V. Oppenheim in "Circuit implementation of synchronized chaos with applications to communication," Phys. Rev. Lett., 71(1), 65-68, July 1993, which is incorporated herein by reference in its entirety.

Chaotic Message Signal Masking and Message Recovery

FIG. 1 is a schematic representation of an embodiment of a communication technique referred to as chaotic masking. In chaotic masking, a set of chaotic signals is applied to or combined with an information bearing message signal m(t) to encode, obscure, conceal, or hide the message signal m(t). At a transmitter or drive system, a message signal m(t) and a chaotic masking signal x _t (t) are added to produce a transmitted signal s(t) = m(t) + x _t (t) that includes both the message signal and the chaotic masking signal. Thus, the drive system outputs a transmitted signal s(t) in which the message m(t) has been concealed, hidden, or rendered instinct by way of its combination with the chaotic masking signal x _t (t). The signal s(t) can therefore be defined as a chaotically masked message signal.

In general, the power level of the chaotic masking signal x _t (t) should be significantly larger than the power level of the message signal m(t) in order to enhance or maximize a likelihood that the message signal m(t) is successfully concealed by the chaotic masking signal x _t (t). As suggested by K.M. Cuomo and A.V. Oppenheim in "Synchronization of Lorenz-Based Chaotic Circuits with Applications to Communications," IEEE Trans. Circuits and Systems- II, 40(10), 626-633, October 1993, incorporated herein by reference in its entirety, the power spectra of the chaotic masking signal and the message should be highly overlapping, with an average signal-to-masking ratio of approximately -20 dB.

A receiver or response system receives the communicated signal s(t), and uses this received signal s(t) to generate a receiver-side version of the chaotic masking signal, which is defined as x _r (t). The response system removes or subtracts x _r (t) from the received signal s(t) to facilitate the "unmasking," extraction, or recovery of the message signal m(t). Therefore, the recovery or extraction of m(t) mathematically occurs by way of m(t) = s(t) - x _r (t), or fn(t) = m(t) + x _t (t) - x _r (t), which reproduces the original message signal m(t) when x _t (t) and x _r (t) are identical.

Transmitter-side chaotic oscillator parameters and initial conditions can be defined as a set of "keys" for message signal recovery. If a receiver-side chaotic oscillator and a transmitter-side chaotic oscillator synchronously operate using identical message signal recovery keys, the reproduction or extraction of the message signal m(t) will be successful, provided that the transfer of a communicated signal s(t) from the transmitter to the receiver has occurred without communication channel disruption. That is, when a receiver-side chaotic oscillator operates in accordance with chaotic oscillator parameters and initial conditions that are identical to those of a counterpart transmitter-side chaotic oscillator, the message signal m(t) can be recovered. If a receiver-side chaotic oscillator exhibits sufficiently strong or adequate self-synchronization to its counterpart transmitter-side chaotic oscillator, successful recovery of the message signal m(t) can occur even though the receiver-side chaotic oscillator has slightly different initial conditions than the transmitter-side chaotic oscillator, and/or a limited amount of communication channel disruption (e.g., due to noise) has occurred.

Aspects of Modified Chaotic Oscillators in Accordance with Embodiments of the Disclosure As previously indicated, particular embodiments of a secure or enhanced security communication system in accordance with the present disclosure can include one or both of a modified five-term chaotic oscillator and a modified Lorenz-like chaotic oscillator. Aspects of a modified five-term chaotic oscillator and a modified Lorenz-like chaotic oscillator are described in detail hereafter.

Modified Five-Term Chaotic Oscillator

A system expressed as the following set of three first-order autonomous ordinary differential equations having five terms:

* = a(y - x),

= -xz, (la) z = -b + xy, exhibits chaotic behavior when a = 5 and b > 0. Such a chaotic attractor appears accidentally to be a modified version of three existing five-term chaotic attractors as follows: (a) Case B of the simple chaotic flows described by J. C. Sprott in "Some simple chaotic flows," Physical Review E, 50(2), R647-R650, August 1994. In such a case,

* = yz,

y = x - y, (lb) z = 1 - xy which may be manipulated to be equivalent to

* = y - x,

y - xz, (lc) z = 1 - xy.

(b) Equations (2a) to (2c) of Gerard van der Schrier, Leo R. M. Maas, "The diffusionless Lorenz equations; ShiPnikov bifurcations and reduction to an explicit map" Physica D 141 (2000) 19-36. In such a case,

* = - y -x,

y = - xz, (Id) z = xy + b.

where b > 0.

(c) Equation (8) of J. C. Sprott in "Maximally iplex simple attractors", CHAOS 17, pp.

033124-1 -033124-6, 2007. In such a case,

* = y - x,

y = - xz, (le) z = xy - b. where b > 0.

Thus, a transmitter or drive system can include a chaotic oscillator that operates in accordance with parameters specified by the following set of equations:

For a modified five-term chaotic attractor.

y t =— x _t Z _t , (la-Tx) z _t = -90 + x _t y _t . or, for the existing five-term chaotic attractors, according to Equations (lb) and (lc),

*t = yt - fe

t = x _t z _t , (lc-Tx)

* _t = i - xtyt.

or, for the existing five-term chaotic attractors, according to Equations (Id),

t = -XtZ _t , (ld-Tx) *t = tyt + 3.4694 or, for the existing five-term chaotic attractors, according to Equations (le),

t ⁼ - tZt, (le-Tx) ¾ _t = _t y _t - 3.4694. to generate a chaotic masking signal x _t (t). The transmitter can combine a message signal m(t) with the chaotic masking signal x _t (t), and output a transmitted or communicated signal s(t) = m(t) + x _t (t).

In order to recover a message signal m(t) at a receiver or response system, the receiver can include a counterpart chaotic oscillator that generates a chaotic signal x _r (t) from a received signal s(t), such that when the receiver is synchronized with the transmitter, x _r (t) equals or approximately equals x _t (t). A receiver-side version of the message signal m(t) can be generated by way of subtracting x _r (t) from s(t). Therefore, fn(t) = x _t (t) + m(t) - x _r (t), which at least approximately equals m(t) when x _r (t) is at least approximately equal to x _t (t).

A receiver or response system can include a chaotic oscillator that operates in accordance with parameters specified by the following set of equations:

For a modified five-term chaotic attractor :

*r = 5(y _r - x _r ),

y _r = -s(t)z _r , (la-Rx) z _r = -90 + s(t)y _r . or, for the existing five-term chaotic attractors, according to Equations (lb) and (lc),

*r = y _r - Xr» y _r = s(t)z _r , (lc-Rx)

* _r = l - s(t)y _r . or, for the existing five-term chaotic attractors, according to Equations (Id),

-y _r - r,

-s(t)z _t , (ld-Rx) s(t)y, + 3.4694

·, for the existing five-term chaotic attractors, according to Equations

r = y _r - r,

y _r = -s(t)z _r , (le-Rx) i _r = s(t)y _t - 3.4694. to generate x _r (t), which can be subtracted from a communicated signal s(t) received from an associated transmitter to facilitate the generation of m(t) and correspondingly the recovery, extraction, or regeneration of the message signal m(t).

FIG. 2 is a schematic illustration of a representative implementation of a communication system 100 based upon a five-term chaotic oscillator according to an embodiment of the disclosure. The communication system 100 includes a transmitter 110 and a receiver 150 that correspond to Equation sets (la-Tx) and (la-Rx), respectively. The transmitter 110 is configured to receive an input or original message m(t); combine the message signal m(t) with a chaotic masking signal x _t (t) generated in accordance with Equation set (la-Tx); and output a transmitted signal s(t) that is communicated to the receiver 150. The receiver 150 is configured to detect or receive the communicated signal s(t); generate a receiver-side version of the chaotic masking signal x _r (t) in accordance with Equation set (la-Rx); and subtract the receiver-side chaotic masking signal x _t (t) from the communicated signal s(t) to generate a recovered message signal m(t).

In the communication system 100, the transmitter 110 and the receiver 150 can be implemented using essentially identical or analogous types of circuits, thereby facilitating improved transmitter - receiver synchronization. In the representative chaotic communication system 100 of FIG. 2, the receiver 150 differs from the transmitter 110 in that at the receiver 150, a) the receiver-side oscillator loop is open; b) each of y _r and ± _r incorporates the communicated signal s(t) rather than x _t (t) or x _r (t); and c) x _r (t) is generated from * _r (f). Results of a numerical simulation of the communication system 100 using a simplified representative message signal m(t) are provided hereafter. In the numerical simulation, initial conditions (x, y, z) at the transmitter 110 are defined to be (5, 10, 5), and initial conditions at the receiver 150 are differently defined to be (3, 10, 5).

FIG. 3A is a graph of a portion of a representative message signal m(t) versus time, which is input to the transmitter 110 of FIG. 2. For purpose of illustration and to aid understanding, m(t) is a simple sinusoid defined by m(t) = 0.2sin(2 ft), where the frequency f is defined to be 1 kHz.

FIG. 3B is a graph illustrating simulation results corresponding to a transmitted signal s(t) output by the transmitter 110 of FIG. 2, where s(t) equals the sum of m(t) of FIG. 3 A plus a chaotic masking signal x _t (t) that is generated by the transmitter 110 in accordance with Equation set (la-Tx).

FIG. 3C is a graph illustrating simulation results corresponding to a recovered message signal m(t) that is generated by the receiver 150. The receiver 150 operates in accordance with Equation set (la-Rx) to generate a counterpart chaotic masking signal x _r (t), which is subtracted from the communicated or received signal s(t) to generate the recovered message signal m(t). Given that x _r (t) = x _t (t), it can be seen that m(t) -≡m(t).

FIG. 3D is a graph illustrating simulated steady state self-synchronization characteristics of the communication system 100 of FIG. 2. In the context of the present disclosure, synchronization can be defined as a tendency or process by which the state variables of a receiver-side chaotic oscillator converge to or toward the state variables of a corresponding, counterpart, or corollary transmitter-side chaotic oscillator, or a linear function of such transmitter-side state variables. As indicated in FIG. 3D, variations in an original message signal m(t) give rise to corresponding variations in a recovered message signal m(t) in accordance with a linear relationship, which implies that the receiver-side five-term chaotic oscillator exhibits a strong, robust, or significant tendency to self-synchronize or maintain synchronization with the transmitter-side five-term chaotic oscillator. Modified Lorenz-Like Chaotic Oscillator

A Lorenz-like system can be expressed as a set of three first-order autonomous differential equations as follows:

* = a(y - x),

Ϋ = bx - dxz, (2a) z = xy -cz.

This Lorenz-like system exhibits chaotic behavior when a = 5.8, b = 16, and c = 1.8. The parameter d can be defined an attractor scale factor, and may take on various values, for instance, d can be defined as (b / c) in an embodiment, which equals (16/1.8) using the above chaotic oscillation parameters. Such a chaotic attractor appears accidentally to be a modified version of two existing Lorenz-like chaotic attractors as follows :

(a) Li X-F, Chu Y-D, Zhang J-G and Chang Y-X., "Nonlinear dynamics and circuit implementation for a new Lorenz-like attractor", Chaos, Solitons & Fractals, 41, 2360- 2370, 2009. In such a case,

* = a(y -x),

= abx - axz, (2b) z = xy - cz,

where a= 5, b = 4 and c = 2.

(b) Tigan, G, Dumitru O., "Analysis of a 3D chaotic system," Chaos Solitons & Fractals, 36, 1315-1319, 2008. In such a case,

* = a(y -x),

V = (c - a)x - axz, (2c) z = xy - bz, where a= 2.1, b = 0.6 and c = 30.

A transmitter or drive system can include a chaotic oscillator that operates in accordance with parameters specified by the following set of equations:

For a modified Lorenz-Like chaotic attractor :

*, = 5.8(y _t - x _t ), _t = 16x _t - (16/1.8)x _t z _t , (2a-Tx) or, for the existing Lorenz-like chaotic attractors, according to Equations (2b),

*t = 5(y _t -x _t ), or, for the existing Lorenz-like chaotic attractors, according to Equations (2c),

Xt = 2.1(y, -x _t ),

t = 27.9x, - 2. lx _t z,, (2c-Tx) to generate a Lorenz-like chaotic masking signal x _t (t) corresponding to Equation set (2a). The transmitter can combine a message signal m(t) with the Lorenz-like chaotic masking signal x _t (t) by way of chaotic masking, and output a transmitted or communicated signal s(t) = m(t) +

In order to recover a message signal m(t) at a corresponding receiver or response system, the receiver can include a counterpart chaotic oscillator that generates a Lorenz-like chaotic signal x _r (t), which under synchronous transmitter - receiver conditions equals or approximately equals x _t (t). A receiver-side version of the message signal m(t) can be generated by way of subtracting x _r (t) from s(t), such that fn(t) = x _t (t) + m(t) - x _r (t), and therefore fn(t) = m(t) when x _r (t)≤x _t (t).

A receiver or response system can include a chaotic oscillator that operates in accordance with parameters specified by the following set of equations:

For a modified Lorenz-Like chaotic attractor :

* _r = 5.8(y _r - x _r ),

y _r = 16s(t) - (16/1.8)s(t)z _r , (2a-Rx)

* _r = s(t)y _r -1.8z _r . or, for existing Lorenz-Like chaotic attractors, according to Equations (2b), ir = 5(y _r -x _r ),

y _r = 20s(t) - 5s(t)z, (2b-Rx) * _r = s(t)y _r - 2z _r , or, for existing Lorenz-Like chaotic attractors, according to Equations (2c),

* _r = 2.1(y _r -x _r ),

r = 27.9s(t) - 2.1s(t)z _r , (2c-Rx) z _r = s(t)y _r - 0.6z _r , to generate x _r (t), which can be subtracted from a communicated signal s(t) received from the transmitter to facilitate recovery of the message signal m(t).

FIG. 4 is a schematic illustration of a representative implementation of a Lorenz-like chaotic communication system 200 that includes a transmitter 210 and a receiver 250 corresponding to Equation sets (2a-Tx) and (2a-Rx), respectively. In the communication system 200, the transmitter 200 and the receiver 250 can be implemented using essentially identical or analogous types of circuits to enable improved transmitter - receiver synchronization. In a manner analogous to that described above, the receiver 250 and the transmitter 210 differ in that at the receiver 250, a) the receiver-side oscillator loop is open; b) each of y _r and z _r incorporates the communicated signal s(t) rather than x _t (t) or x _r (t); and c) x _r (t) is generated from x _r (t). Results of a numerical simulation of the communication system 200 of FIG. 4 using as identical oscillator parameters and matching initial conditions to communicate a simplified representative message signal m(t) are provided hereafter.

FIG. 5A is a graph of a portion of a representative message signal m(t) versus time, which is input into the transmitter 210 of FIG. 4. For purpose of illustration and to aid understanding, m(t) is a simple sinusoid defined by m(t) = 0.03sin(2 ft), where the frequency f is defined to be 1 kHz.

FIG. 5B is a graph illustrating simulation results corresponding to a transmitted signal s(t) output by the transmitter 210 of FIG. 4, where s(t) equals the sum of m(t) of FIG. 5 A plus a Lorenz-like chaotic masking signal x _t (t) that is generated by the transmitter 210 in accordance with Equation set (2a-Tx).

FIG. 5C is a graph illustrating simulation results corresponding to a recovered message signal m(t) that is generated by the receiver 250. The receiver 250 operates in accordance with Equation set (2a-Rx) to generate a Lorenz-like chaotic masking signal x _r (t) that is a counterpart to the Lorenz-like chaotic masking signal x _t (t) generated by the transmitter 210. The receiver 250 implements a subtraction operation in which the receiver-side Lorenz-like chaotic masking signal x _r (t) is subtracted from the communicated or received signal s(t) to generate the recovered message signal m(t). Given that x _r (t)≡ x _t (t), it can be seen from FIG. 5C that m(t) = m(t).

FIG. 5D is a graph illustrating simulated steady state self-synchronization characteristics of the communication system 200 of FIG. 4. As indicated in FIG. 5D, variations in an original message signal m(t) result in corresponding variations in a recovered message signal m(t) in accordance with a linearly relationship, which implies that the receiver-side five-term chaotic oscillator exhibits a strong, robust, or significant tendency to self-synchronize or maintain synchronization with the transmitter-side Lorenz-like chaotic oscillator.

Aspects of Systems Involving Two-Fold Masking and Static-Dynamic Message Recovery Embodiments of a secure or enhanced security communication system in accordance with the present disclosure can include multiple chaotic oscillators that generate multiple chaotic masking signals that are combined with a message signal m(t) in a manner that securely obscures or conceals the message signal m(t). More particularly, in several embodiments a secure or enhanced security communication system includes a chaotic oscillator that implements a modified five-term chaotic attractor corresponding to Equation sets (la), (la-Tx) and (la-Rx) above, or the existing five-term chaotic attractors corresponding to Equation sets (lb) - (le), (lc-Tx) - (le-Tx), and (lc-Rx) - (le-Rx), above, as well as a chaotic oscillator that implements a modified Lorenz-like chaotic attractor corresponding to Equation sets (2a), (2a-Tx) and (2a-Rx) above, or the existing Lorenz-like chaotic attractor corresponding to Equation sets (2b) - (2c), (2b-Tx) - (2c-Tx), and (2b-Rx) - (2c-Rx), above.

Embodiments of a secure or enhanced security communication system in accordance with the present disclosure can further utilize both static as well as dynamic or time varying keys to facilitate the recovery or extraction of the message signal m(t) from a communicated signal s(t) that includes more than one chaotic masking signal. To enhance communication security, static and dynamic keys can be communicated from a transmitter to a receiver by way of separate communication channels. FIG. 6 is a schematic illustration of a secure communication system 300 according to an embodiment of the disclosure, based upon two-fold masking of different chaotic oscillators with static-dynamic secret keys. In general, the system 300 includes a transmitter 310 and a receiver 410 that are configured for the transfer of chaotically masked message signals as well as the transfer of dynamic or time varying message signal recovery keys. Such transfer can involve a set of wireless communication channels 312, 314, as further described below. The transmitter 310 is configured to receive a message signal m(t), and the receiver 410 is configured to output a recovered message signal m(t).

In an embodiment, the transmitter 310 includes a first and a second drive system 320, 350, and the receiver 410 correspondingly includes a first and a second response system 420, 450. The first drive system 320 and the first response system 420 include counterpart synchronizable chaotic oscillators that operate in accordance or association with a first set of equations defining a first chaotic attractor. Similarly, the second drive system 350 and the second response system 450 include counterpart synchronizable chaotic oscillators that operate in accordance or association with a second set of equations defining a second chaotic attractor.

The transmitter 310 is configured to chaotically mask the message signal m(t) using each of a first chaotic masking signal x _tl (t) generated by the first drive system 320 and a second chaotic masking signal xa(t) generated by the second drive system 350. Thus, the transmitter 310 combines the message signal m(t), the first chaotic masking signal Xu(t), and the second chaotic masking signal Χβ(ί) by way of a summing operation to produce a chaotically masked message signal si(t) = m(t) + x _tl (t) + xa(t) to thereby conceal or hide the message signal m(t). In accordance with such an embodiment, the chaotically masked message signal s^t) can be defined as a type of two-fold chaotically masked message signal due to the use of two distinct chaotic masking signals x _t i(t) and x (t) to render the message signal m(t) indistinguishable from the chaotically masked message signal si(t). The transmitter 310 transmits or communicates the chaotically masked message signal sj(t) over a first communication channel 312. In various embodiments, the first communication channel 312 can be a public or known communication channel.

The first drive system 320 and the first response system 420 are configured to synchronize their operation based a first group of static secret keys that facilitate successful recovery of the message signal m(t). Similarly, the second drive system 350 and the second response system 450 are configured to synchronize their operation based upon a second group of static secret keys that further facilitate successful message signal recovery. The first static secret key group corresponds to chaotic oscillator parameters and initial conditions for a first drive system - response system pair defined by the first drive system 320 and the first response system 420. Similarly, the second static secret key group corresponds to chaotic oscillator parameters and initial conditions for a second drive system - response system pair defined by the second drive system 350 and the second response system 450. Aspects of the first and second static secret key groups are described in greater detail below.

To further facilitate secure communication, the transmitter 310 is additionally configured to generate a set of dynamic or time varying secret keys s ₂ (t) that the receiver 410 requires in addition to the first and second static secret key groups to successfully recover the message signal m(t). In an embodiment, the set of dynamic secret keys s ₂ (t) can be a time series that is based upon or derived from a particular chaotic masking signal, such as the first chaotic masking signal xu(t). In such an embodiment, the chaotically masked message signal s^t) itself includes or incorporates the set of dynamic secret keys, in addition to including or incorporating the second chaotic masking signal χα(ΐ). The transmitter 310 can transmit the set of dynamic secret keys s ₂ (t) over a second communication channel 314 that is distinct from the first communication channel 312. In a number of embodiments, the second communication channel 314 can be a non-public, confidential, or secret communication channel.

The receiver 410 is configured to receive the transmitted or communicated signal s^t) by way of the first communication channel 312, and is additionally configured to receive the transmitted set of dynamic secret keys si(t) by way of the second communication channel 314. In an embodiment in which the set of dynamic secret keys corresponds to a time series of the first chaotic masking signal x _tl (t) produced by the first drive system 320, the receiver 410 utilizes the first response system 420 to generate a receiver-side version of the first chaotic masking signal x _rl (t), and removes the set of dynamic secret keys from the communicated signal si(t) by subtracting x _r i(t) from Si(t). In order for this dynamic secret key removal to be successful, the first response system 420 and the first drive system 320 must be synchronized by way of corresponding sets of static secret keys within the first static key group. As a result of subtracting x _r i(t) from si(t), the two-fold chaotically masked message signal si(t) is transformed into a segregated chaotically masked message signal s ₃ (t) that omits or excludes Xri(t), and which corresponds to or is correlated with each of the message signal m(t) and the second chaotic masking signal χ (ί).

The receiver 410 utilizes the second response system 450 to generate a receiver-side version of the second chaotic masking signal x _r2 (t). In order for x _r2 (t) to be equal or approximately equal to the transmitter-side second chaotic masking signal xa(t), the second response system 450 and the second drive system 350 must be synchronized by way of corresponding sets of static secret keys within the second static secret key group. The receiver 450 subtracts the receiver- side second chaotic masking signal x _r 2(t) from the segregated chaotically masked message signal s ₃ (t) to generate a recovered message signal m(t).

In summary, in a number of embodiments the structure and function of the transmitter 310 and receiver 410 can be correlated with the following set of mathematical equations: s ₁ (t) = m(t) + xti(t) +x _a (t)

s ₂ (t) = x,i(t)

s ₃ (t) = m(t) + x,i(t) + _Xt2 (t) - x _rl (t) (3)

= m(t) + Xt2(t) when x _rl (t)≡ x _tl (t)

m(t) = m(t) + Xt2(t) - Xr2(t)≡ m(t) when x _r2 (t)≤ χ _β (ί)

In Equation set (3) and in a manner identical or analogous to that described above, si(t) corresponds to a two-fold chaotically masked message signal that is produced, transmitted, or communicated by the transmitter 310 over a first communication channel 312; s ₂ (t) corresponds to a set of dynamic secret keys that is communicated by the transmitter 310 over a second communication channel 314; s ₃ (t) corresponds to a segregated chaotically masked message signal produced by the receiver 410 as part of a dynamic secret key removal process; and m(t) corresponds to a recovered or extracted message signal that is produced or output by the receiver 410.

In several embodiments, the transmitter 310 includes a first drive system 320 that is configured to operate as a modified five-term chaotic oscillator in a manner corresponding or analogous to Equation sets (la), (la-Tx) and (la-Rx) above, or the existing five-term chaotic attractors corresponding to Equation sets (lb) - (le), (lc-Tx) - (le-Tx), and (lc-Rx) - (le- Rx), above; and a second drive system 350 that is configured to operate as a modified Lorenz- like chaotic oscillator in a manner corresponding or analogous to Equation sets (2a), (2a-Tx) and (2a-Rx)above, or the existing Lorenz-like chaotic attractor corresponding to Equation sets (2b) - (2c), (2b-Tx) - (2c-Tx), and (2b-Rx) - (2c-Rx), above. Thus, an embodiment of the first drive system 320 can include a modified five-term chaotic oscillator that operates in accordance with the following set of equations:

*ti = 5(y _t i - x _tl ),

y ti = - tizti, (4a)

*ti = -90 + x,iy _t i

Correspondingly, an embodiment of the second drive system 350 can include a modified Lorenz-like chaotic oscillator that operates in accordance with the following set of equations:

*e = 5.8φ _β - XQ),

YQ = 16x _t2 - (d)x _t2 z _t2 , (4b)

In a manner identical or analogous that previously described, in certain embodiments, the factor d in the set of Equations (4b) can be an attractor scale factor that can be defined (e.g., in a predetermined, selectable, or programmable manner) to take on multiple values. Aspects of manners in which particular values of the attractor scale factor d can affect communication system performance are described in greater detail below.

In a number of embodiments, the receiver 410 includes a first response system 420 that is configured as a structural and/or functional synchronizable counterpart or analogue to the five- term chaotic oscillator corresponding to Equation set (4a) in accordance with the following set of equations:

Zri = - 0 + S ₂ (t)y-i

The receiver 410 additionally includes a second response system 450 that is configured as a structural and/or functional synchronizable counterpart or analogue to the Lorenz-like chaotic oscillator corresponding to Equation set (4b) in accordance with the following set of equations: r ₂ = 5.8(y _r2 - x _r2 ),

y _r2 = 16S ₃ (t) - (d)S ₃ (t)z _r2 , (5b)

¾ = S ₃ (t) y _r2 - 1.8z _r2

FIG. 7 is a schematic illustration of a representative implementation of a multiple chaotic oscillator static - dynamic key secure communication system 300a according to an embodiment of the disclosure. In an embodiment, the system includes a transmitter 310a having a first driving system 320a that includes a five-term chaotic oscillator configured to produce a first chaotic masking signal x _t i(t) in accordance with Equation set (4a) above; and a second driving system 350a that includes Lorenz-like chaotic oscillator 350a configured to produce a second chaotic masking signal Χβ(ΐ) in accordance with Equation set (4b) above. The transmitter 310a additionally includes a masking unit 380 that is configured to receive and combine an input or original message signal m(t), the first chaotic masking signal xu(f), and the second chaotic masking signal Χβ(ΐ) to produce a two-fold chaotically masked message signal s^t). In various embodiments, such combination is performed by way of a set of addition operations.

The transmitter 310a is configured to provide, output, or transmit the two-fold chaotically masked message signal s^t) to a first communication channel 312a, as well as a dynamic secret key corresponding to a time series of the first chaotic masking signal xu(t) to a second communication channel 314a. The first communication channel 312a can be a public or known channel, while the second communication channel 314a can be channel that is distinguishable or distinct from the first communication channel, such as a secret or private channel.

The system 300a also includes a receiver 410a having a first response system 420a and a second response system 450a. The first response system 420a includes a five-term chaotic oscillator that is a counterpart to the transmitter's five-term chaotic oscillator, and which is synchronizable therewith in accordance with a first group of static secret keys. The receiver's five-term chaotic oscillator is configured to receive a set of dynamic secret keys that the transmitter 310a has communicated or output by way of the second communication channel 314a. The receiver's five-term chaotic oscillator is further configured to generate or output a receiver-side five-element chaotic masking signal x _rl (t) in accordance with Equation set (5a) above that is derived or generated using the set of dynamic secret keys.

The first response system 420a additionally includes a segregation unit 480 that is configured to receive each of the receiver-side five element chaotic masking signal x _r i(t) and the communicated two-fold chaotically masked message signal si(t). The segregation unit 480 subtracts the receiver-side five element chaotic masking signal x _r i(t) from the communicated signal Si(t) to generate a segregated chaotically masked message signal s ₃ (t) that corresponds to each of the message signal m(t) and the second chaotic masking signal x^O).

The second response system 450a includes a Lorenz-like chaotic oscillator 450a that is a counterpart to the transmitter's Lorenz-like chaotic oscillator 350a, and which is synchronizable with the transmitter-side Lorenz-like chaotic attractor 350a in accordance with a second group of static secret keys. The second response system's Lorenz-like chaotic oscillator 450a is configured to receive the segregated chaotically masked message signal s ₃ (t), and generate a receiver-side version of the second chaotic masking signal x _r2 (t) therefrom in accordance with Equation set (5b) above.

The second response system 450a additionally includes a message recovery unit 490 that is configured to receive each of the receiver-side second chaotic masking signal x _r2 (t) and the segregated chaotically masked message signal s ₃ (t). The message recovery unit 490 subtracts x _r2 (t) from s ₃ (t) to produce or provide a recovered message signal m(t) at a receiver output.

Aspects of Static Secret Keys

Any given set of static secret keys specifies parameters and initial conditions for a particular chaotic oscillator. In various embodiments, a static secret key group includes a transmitter- side set of static secret keys that specifies parameters and initial conditions for a particular transmitter-side chaotic oscillator, as well as a receiver-side set of static secret keys that specifies parameters and initial conditions for a receiver-side chaotic oscillator that is a synchronizable counterpart to the transmitter-side chaotic oscillator.

The first drive system 320a and the first response system 420a are configured to synchronize their operation based a first group of static secret keys that includes a first set of drive static secret keys and a first set of response static secret keys. More particularly, for the representative implementation of the secure communication system 300a of FIG. 7, the first set of drive static secret keys can specify parameters and initial conditions for the transmitter- side five-term chaotic oscillator, and the first set of response static secret keys can specify parameters and initial conditions for the counterpart receiver-side five-term chaotic oscillator.

The second drive system 350a and the second response system 450a are configured to synchronize their operation based upon a second group of static secret keys that includes a second set of drive static secret keys and a second set of response static secret keys. For the representative implementation of a secure communication system 300a of FIG. 7, the second set of drive static secret keys can specify parameters and initial conditions for the transmitter- side Lorenz-like chaotic oscillator, and the second set of response static secret keys can specify parameters and initial conditions for the corresponding receiver-side Lorenz-like chaotic oscillator.

Because the transmitter-side and receiver-side five-term chaotic oscillators exhibit strong or reasonably strong self-synchronization characteristics, and the transmitter-side and receiver- side Lorenz-like chaotic oscillators also exhibit strong, robust, or reasonably strong self- synchronization characteristics, particular initial conditions for counterpart transmitter-side and receiver-side chaotic oscillators can differ from each other to a certain extent. For instance, for the representative implementation of the communication system 300a of FIG. 7, the initial conditions corresponding to the transmitter-side and the receiver-side first chaotic masking signals Xti(t), x _r i(t) can differ somewhat from each other; and/or the initial conditions corresponding to the transmitter-side and the receiver side second chaotic masking signals X(2(t), Xr2(t) can differ somewhat from each other.

Successful synchronization can be achieved as follows :

Case 1 : Masking of the modified Lorenz-like and the existing Lorenz-like chaotic attractors

The initial conditions of (x _t , y _t , z _t ) and (x _r , y _r , z _r ) can be different, significantly different, or vastly different values. The more the values of the initial conditions differ, the more transient time is needed prior to successful synchronization.

Case 2 : Masking of the modified five-term and the existing five-term chaotic attractors (a) The initial conditions of (y _t , z _t ) and (y _r , z _r ) must be the same because (y _r , z _r ) generates x _r at the response system.

(b) The initial conditions of (x _t ) and (x _r ) can be different, significantly different, or vastly different values. The more the values of the initial values differ, the more transient time is needed prior to successful synchronization.

Case 3 : Two-Fold Masking of the modified five-term (and the existing five-term) and the modified Lorenz-like (and the existing Lorenz-like) chaotic attractors

(a) The initial conditions of (y _tl , z _t j) and (y _rl , z _r i) must be the same because (y _r i, z _rl ) generates x _rl at the response system.

(b) The initial conditions of (x _t i) and (x _r i) can be different, significantly different, or vastly different values. The more the values of the initial values differ, the more transient time is needed prior to successful synchronization.

(c) The initial conditions of , Ya, za) and (x _r2 , y _r2 , z _r2 ) can be different, significantly different, or vastly different values. The more the values of the initial values differ, the more transient time is needed prior to successful synchronization.

In view of the foregoing, embodiments of communication systems according to the present disclosure can exhibit robust message signal recovery in the presence of sources of communication channel disruption, such as noise or signal reflections.

Table 1 provides a representative example illustrating a manner in which particular initial conditions within a first static secret key group can differ, and also a manner in which particular initial conditions within a second static secret key group can differ. More particularly, in Table 1 a first static secret key group includes a first set of drive static secret keys corresponding to a transmitter-side five-term chaotic oscillator, and a first set of response static secret keys corresponding to a receiver-side five-term chaotic oscillator. The first set of drive static secret keys has initial conditions (x _t i _, y _t i, z _t i) given by (5, 5, 5), while the first set of receive static secret keys has initial conditions (x _r i _, y _r i _, z _r i) given by (4.8, 5, 5).

Furthermore, in Table 1 a second static secret key group includes a second set of drive static secret keys corresponding to a transmitter-side Lorenz-like chaotic oscillator, and a second set of response static secret keys corresponding to a receiver-side Lorenz-like chaotic oscillator. The second set of drive static secret keys has initial conditions (XQ, y , za) given by (1, 1, 1), while the second set of receive static secret keys has initial conditions (x _r2) y _R 2, z _r2 ) given by (0.8, 1, 1).

Table 1: Representative First and Second Static Secret Key (SSK) Groups and Initial

Condition Differences Therewith™

As indicated in Table 1, a parameter b corresponding to each of a transmitter-side and a receiver-side five-term chaotic oscillator is defined to be equal to 90; and a parameter d corresponding to each of a transmitter-side and a receiver-side Lorenz-like chaotic oscillator is defined to be equal to (16/180). In a number of embodiments, particular chaotic oscillator parameters can be specified or defined (e.g., in a predetermined, selectable, or programmable manner) to exhibit different values in view of secure communication system operating conditions or operating environment. That is, a parameter configuration for an embodiment of a secure communication system can be established or configured and/or modified or reconfigured in view of a secure communication system operating situation or context. For instance, as further detailed below, a Lorenz-like chaotic oscillator parameter d may be selected, altered, or programmed to facilitate chaotic attractor scaling in view of chaotic oscillator power conditions. Table 2 illustrates a representative manner in which a Lorenz-like chaotic oscillator parameter d can be adjusted, altered, or reconfigured to differ from that shown in Table 1. 1 ^st SSK Group

1 ^st Set of Drive SSKs 1 ^st Set of Response SSKs

Tx (x _t1 , y _t1 , z _t1 ) b Rx (x _r1 , y _r1 , z _r1 ) b

(5, 5, 5) 90 (4.8, 5, 5) 90

2 ^nd SSK Group

2 ^nd Set of Drive SSKs 2 ^nd Set of Response SSKs

Tx (Xt2. yt2 . ^z t2) d ^Rx ( ^x r2- yr2. ^z r2> d

(1 , 1 , 1) 16/1.8 (0.8, 1 , 1) 16/1.8

Table 2: Representative First and Second Static Secret Key (SSK) Groups Having a Lorenz- Like Chaotic Oscillator Parameter d that Differs from Table 1

FIG. 8 is a graph showing portions of a representative message signal m(t) versus time, which is defined as a simple sinusoid given by m(t) = 0.3sin^ft), where the frequency f is specified to be 1 kHz. The description that follows provides representative examples illustrating particular types of conditions under which successful and unsuccessful message signal recovery can occur. These representative examples correspond to secure communication systems 300a configured to include a five-term chaotic oscillator and a Lorenz-like chaotic oscillator in a manner identical or analogous to that shown in FIG. 7. Such communication systems 300a generate a two-fold chaotically masked message signal sj(t) by combining the representative message signal m(t) of FIG. 8, a first chaotic masking signal x _tl (t) corresponding to a five-term chaotic attractor, and a second chaotic masking signal Xa(t) corresponding to a Lorenz-like chaotic attractor. Additionally, the first chaotic masking signal x _t i(t) is utilized in the representative examples to generate a set of dynamic secret keys s ₂ (t).

Representative Example 1 : Successful Message Recovery (Two Substantially Similar-Power-

Level Chaotic Atractors ^" )

In a first representative Example corresponding to successful message signal recovery, chaotic oscillators within a transmitter 310a and a receiver 410a are defined to operate under equal, approximately equal, or similar power conditions. Chaotic oscillator parameters and initial conditions corresponding to a first and a second static secret key group suitable for the first representative example are provided in Table 1. In particular, a Lorenz-like chaotic oscillator parameter d can be defined as (16/180). Graphs illustrating simulation results corresponding to particular signals or waveforms generated by portions of a communication system 300a operating in accordance with this representative Example are described hereafter with reference to FIGs. 9 - 14.

FIG. 9 is a graph showing a portions of a second chaotic masking signal xc(t) generated by the communication system's Lorenz-like chaotic oscillator. FIG. 10 is a graph showing a set of dynamic secret keys s ₂ (t) generated by a five-term chaotic oscillator, and which is output or transmitted to a secret communication channel 314a. The set of dynamic secret keys s ₂ (t) is identical or analogous to the first chaotic masking signal x»(t) produced by the communication system's five-term chaotic oscillator. As indicated by FIGs. 9 and 10, the relative power levels of the first and second chaotic masking signals xu(t), xc(t) are approximately equal or substantially similar.

FIG. 11 is a graph showing portions of a two-fold chaotically masked message signal s^t) output to a public communication channel 312a. It is evident from a comparison of FIGs. 10 and 11 that the waveform characteristics, patterns, or properties of Si(t) and s ₂ (t) are different. Such differences facilitate enhanced communication security.

FIG. 12 is a graph showing portions of a successfully recovered message signal m(t) versus time, and FIG. 13 is a graph showing the transient behavior of the successfully recovered message signal m(t), using a frequency of 15 Hz for purpose of illustration. FIG. 14 is a graph showing steady state self-synchronization characteristics of the communication system 300a of representative Example 1. As indicated by FIGs. 12 - 14, the communication system 300a exhibits strong or robust self-synchronization characteristics, and hence strong or robust message signal recovery behavior. Representative Example 2: Successful Message Recovery (Two Substantially Different- Power-Level Chaotic Atractors)

In a second representative example corresponding to successful message signal recovery, chaotic oscillators within a transmitter 310a and a receiver 410a are defined to operate under different power conditions. Chaotic oscillator parameters and initial conditions corresponding to a first and a second static secret key group suitable for the second representative example are provided in Table 2, in which a Lorenz-like chaotic oscillator parameter d is defined as (16/1.8). Graphs illustrating simulation results corresponding to particular signals or waveforms generated by portions of a communication system 300a operating in accordance with this representative Example are described hereafter with reference to FIGs. 15 - 17.

FIG. 15 is a graph showing a portions of a second chaotic masking signal Xt2(t) generated by the communication system's Lorenz-like chaotic oscillator. FIG. 16 is a graph showing a set of dynamic secret keys s ₂ (t) generated by a five-term chaotic oscillator, and which is output or transmitted to a secret communication channel 314a. The set of dynamic secret keys s ₂ (t) equals or corresponds to the first chaotic masking signal Xti(t) produced by the communication system's five-term chaotic oscillator. As can be seen from, the relative power levels of the first and second chaotic masking signals x _tl (t), xa(t) are significantly different. Such a power level difference indicates that the five-term and the Lorenz-like chaotic oscillators of the secure communication system 300a of representative Example 2 can operate under significantly or very different power conditions.

FIG. 17 is a graph showing portions of a two-fold chaotically masked message signal Sj(t) output to a public communication channel 312a. A successfully recovered message signal m(t) versus time corresponding to representative Example 2 is identical or essentially identical to that shown in FIG. 12 for representative Example 1.

A comparison of FIGs. 16 and 17 reveals that the waveform characteristics, patterns, or properties of st(t) and s ₂ (t) bear similarities or resemblance to each other. Such resemblance may weaken communication security; however, an acceptable level of communication security can still be expected in various communication situations because chaotic oscillator behavior exhibits sensitive dependence upon initial conditions. In other words, very small differences in transmitter-side initial conditions give rise to very large differences in receiver-side message recovery results due to the so-called "butterfly effect."

Representative Example 3: Unsuccessful Message Recovery (Faulty Static Secret Keys ^' )

A third representative Example corresponding to unsuccessful message recovery considers a situation in which a receiver-side set of static secret keys are mismatched or inappropriate with respect to a transmitter-side set of static secret keys. In the third representative Example, chaotic oscillator parameters and initial conditions corresponding to a first and a second static secret key group suitable for the first representative example are provided in Table 3. As shown in Table 3, a Lorenz-like chaotic oscillator parameter d is defined to be (16/1.8), which can correspond to a reduced communication security situation resulting from chaotic oscillators operating under substantially different power conditions.

Table 3: Representative Mismatch Between Transmitter-Side and Receiver-Side Static Secret

Keys (SSKs) Resulting in Unsuccessful Message Recovery

FIG. 18 is a graph showing portions of an unsuccessfully recovered message signal rn(t) versus time as a result of the transmitter-side and receiver-side static secret key mismatch of representative Example 3. As indicated by FIG. 18, even in view of a reduced communication security situation, inappropriate or significantly mismatched static secret keys results in unsuccessful message signal recovery. Representative Example 4: Unsuccessful Message Recovery (Faulty Dynamic Secret Keys ^' )

Even when transmitter-side and receiver-side static secret keys are suitably or appropriately matched in a manner that can enable successful message recovery, unsuccessful message recovery will occur in situations in which dynamic secret keys are improperly generated at the receiver-side.

FIG. 19 is a schematic illustration of a communication system 300b having a receiver 410b that is (mis)configured to generate dynamic secret keys using an incorrect or inappropriate communication channel. In the fourth representative Example, the communication system 300b is configured in accordance with the chaotic oscillator parameters and initial conditions specified in Table 4. As shown in Table 4, the communication system 300b is configured to have matching transmitter-side and receiver-side static secret keys. Additionally, a Lorenz- like chaotic oscillator parameter d is defined to be equal to (16/1.8), which can correspond to a reduced communication security situation resulting from chaotic oscillators that operate under significantly different power conditions.

Table 4: Matched Transmitter-Side and Receiver-Side Static Secret Keys (SSKs) for

Representative Example 4

As shown in FIG. 19, the communication system 300b is (mis)configured to generate a set of receiver-side dynamic secret keys using a first or public communication channel 312b over which the two-fold chaotically masked message signal Si(t) is transmitted, rather than a second or secret communication channel 314a such as that illustrated in FIG. 7. More specifically, the receiver 410b includes a first response system 420b having a five-term chaotic oscillator that is configured to receive signals transmitted by way of the public communication channel 312b, i.e., the first response system 420b is configured to receive or respond to a two-fold chaotically masked message signal s^t), and attempt to generate a receiver-side first chaotic masking signal x _rl (t) and a corresponding set of receiver-side dynamic secret keys therefrom.

Because signals transmitted by way of the public communication channel 312b include signal components generated in accordance with the chaotic oscillator parameters and initial conditions of the transmitter-side Lorenz-like chaotic oscillator, the first response system's five-term chaotic oscillator cannot successfully synchronize with signals transmitted over the public communication channel, and message signal recovery will be unsuccessful.

FIG. 20 is a graph showing portions of an unsuccessfully recovered message signal m(t) versus time as a result of the receiver-side misconfiguration with respect to the generation of dynamic secret keys of representative Example 4.

Aspects of Representative Enhanced Security Communication Processes

FIG. 21 A is a flow diagram of a representative process 500 for transmitter-side, encoder-side, or masking-side enhanced security communication in accordance with the present disclosure. In an embodiment, the process 500 includes a first process portion 502 that involves defining or specifying a set or group of transmitter-side, encoder-side, or masking-side static secret keys. Such static secret keys can include a first set of transmitter-side, encoder-side, or masking-side static secret keys that corresponds to a first chaotic attractor; and a second set of transmitter-side, encoder-side, or masking-side static secret keys that corresponds to a second chaotic attractor distinct from the first chaotic attractor. The first and second chaotic attractors can respectively correspond to a five-term chaotic attractor and a Lorenz-like chaotic attractor, e.g., in a manner identical, analogous, or generally analogous to that described above.

The process 500 further includes a second process portion 504 that involves generating a transmitter-side, encoder-side, or masking-side first chaotic masking signal, and a third process portion 506 that involves generating a transmitter-side, encoder-side, or masking-side second chaotic masking signal. The first and second chaotic masking signals can be generated using a (modified) transmitter-side five-term chaotic oscillator and a transmitter-side Lorenz- like chaotic oscillator, respectively, in a manner identical or analogous to that described above.

The process 500 additionally includes a fourth process portion 508 that involves generating a set of dynamic secret keys, for instance, using the first chaotic masking signal. The process 500 also includes a fifth process portion 510 in which a received or retrieved message signal is combined with the first and second chaotic masking signals to produce a two-fold chaotically masked message signal.

Finally, the process 500 includes a sixth process portion 512 that involves outputting, transmitting, or transferring the two-fold chaotically masked message signal, and a seventh process portion 514 that involves outputting, transmitting, or transferring the set of dynamic secret keys.

FIG. 21B is a flow diagram of a representative process 550 for receiver-side, decoder-side, or unmasking-side enhanced security communication in accordance with the present disclosure. In an embodiment, the process 550 includes a first process portion 552 that involves defining or specifying a set or group of receiver-side, decoder-side, or unmasking-side static secret keys. Such static secret keys can include a first set of receiver-side, decoder-side, or unmasking-side static secret keys that corresponds to the first chaotic attractor; and a second set of receiver-side, decoder-side, or unmasking-side static secret keys that corresponds to the second chaotic attractor.

A second process portion 554 and a third process portion 556 include receiving or retrieving the two-fold chaotically masked message signal and the set of dynamic secret keys, respectively. A fourth process portion 558 includes generating a receiver-side, decoder-side, or unmasking-side first chaotic masking signal that is synchronized or linked to or correlated with the set of dynamic secret keys, and a fifth process portion 560 includes generating a receiver-side, decoder-side, or unmasking-side second chaotic masking signal.

A sixth process portion 562 includes segregating the set of dynamic secret keys from the twofold chaotically masked message signal (e.g., using the receiver-side first chaotic masking signal) to generate a segregated chaotically masked message signal, e.g., in a manner identical, analogous, or generally analogous to that described above. Finally, a seventh process portion 564 includes recovering or extracting a message signal from the segregated chaotically masked message signal using the receiver-side second chaotic masking signal.

Depending upon embodiment details, particular aspects of one or more process portions described above with respect to the transmitter-side, encoder-side, or masking-side enhanced security communication process 500 and/or the receiver-side, decoder-side, or unmasking-side enhanced security communication process 550 can be performed using hardware (which can include dedicated and/or programmable or reconfigurable hardware) and/or software (e.g., program instructions that are stored in a memory, and which are executable by a processing unit or state machine). Additionally, particular process portions can be performed simultaneously or in a sequenced or delayed manner, e.g., in accordance with implementation details and/or a type of enhanced security communication situation under consideration.

Aspects of particular embodiments of the disclosure as described above can address one or more shortcomings or disadvantages of prior systems, devices, apparatuses, and/or processes. While advantages associated with certain embodiments have been described within the context of those embodiments, other embodiments may also exhibit such advantages, and not all embodiments need necessarily exhibit such advantages to fall within the scope of the disclosure. It will be appreciated that several of the above-disclosed and other structures, features and functions, or alternatives thereof, may be desirably combined into other different devices, systems, or applications. The above-disclosed structures, features and functions, or alternatives thereof, as well as various presently unforeseen or unanticipated alternatives, modifications, variations or improvements thereto that may be subsequently made by one of ordinary skill in the art, are encompassed by the following claims.