Cryptography – Communication system using cryptography – Symmetric key cryptography
Reexamination Certificate
1999-08-25
2004-06-01
Barrón, Gilberto (Department: 2132)
Cryptography
Communication system using cryptography
Symmetric key cryptography
C380S046000, C380S274000
Reexamination Certificate
active
06744893
ABSTRACT:
The present invention relates generally to chaotic communications systems and more particularly to a chaotic communication system utilizing a chaotic receiver estimation engine. This estimation engine both synchronizes and recovers data by mapping probability calculation results onto the chaotic dynamics via a strange attractor geometrical approximation. For synchronization, the system does not require either a stable/unstable subsystem separation or a chaotic system inversion. The techniques employed can be implemented with any chaotic system for which a suitable geometric model of the attractor can be found.
BACKGROUND
Chaos is an area of science and mathematics that is characterized by processes that are nonlinear, such as equations that have a squared or higher order term. Chaotic processes are iterative, in that they perform the same operations over and over. By taking the result of the process (equations) and performing the process again on the result, a new result is generated. Chaos is often called deterministically random motion. Chaotic processes are deterministic because they can be described by equations and because knowledge of the equations and a set of initial values allows all future values to be determined. Chaotic processes also appear random since a sequence of numbers generated by chaotic equations has the appearance of randomness. One unique aspect of chaos compared to an arbitrarily chosen aperiodic nonlinear process is that the chaotic process can be iterated an infinite number of times, with a result that continues to exist in the same range of values or the same region of space.
A chaotic system exists in a region of space called phase space. Points within the phase space fly away from each other with iterations of the chaotic process. Their trajetories are stretched apart but their trajectories are then folded back onto themselves into other local parts of the phase space but still occupy a confined region of phase space A geometrical shape, called the strange attractor, results from this stretching and folding process. One type of strange attractor for a chaotic process called the Rossler system is depicted in
FIG. 1
, and illustrates the stretching and folding process. The chaotic attracter exists in perpetuity in the region of phase space for which the chaotic system is stable.
In an unstable system, two points in phase space that are initially close together become separated by the stretching and the folding, which causes the points to be placed correspondingly farther apart in the phase space than they originally were. Repeating the process (iterating) accentuates the situation. The points not only diverge from each other, but both trajectories move toward infinity and, therefore, away from the stable region of perpetual chaotic existence.
The sequences of points that result from two closely spaced initial conditions become very different very quickly in chaotic processes. The Henon chaotic system, for example, has been shown to start with two initial conditions differring by one (1) digit in the 15
th
decimal place. The result was that within 70 iterates the trajectories were so different that subtracting them resulted in a signal that was as large as the original trajectories' signals themselves. Therefore, the stretching and folding causes the chaotic process to exhibit sensitive dependence on initial conditions. A receiver without perfect information about a transmitter (a “key”) will be unable to lock to the transmitter and recover the message. Even if a very close lock were achieved at one point in time, it is lost extremely quickly because the time sequences in the transmitter and receiver diverge from each other within a few iterations of the chaotic process.
Chaotic nonlinear dynamics may be utilized in telecommunications systems. There is interest in utilizing and exploiting the nonlinearities to realize secure communications, while achieving reductions in complexity, size, cost, and power requirements over the current communications techniques. Chaotic processes are inherently spread in frequency, secure in that they possess a low probability of detection and a low probability of intercept, and are immune to most of the conventional detection, intercept, and disruption methods used against current ‘secure’ communications systems based on linear pseudorandom noise sequence generators. Chaotic time sequences theoretically never repeat, making them important for such applications as cryptographic methods and direct sequence spread spectrum spreading codes. In addition, chaotic behavior has been found to occur naturally in semiconductors, feedback circuits, lasers, and devices operating in compression outside their linear region.
Unfortunately, many of these characteristics also complicate the task of recovering the message in the chaotic transmission. A fundamental problem permeating chaotic communications research is the need for synchronization of chaotic systems and/or probabilistic estimation of chaotic state, without which there can be no transfer of information. Without synchronization, meaning alignment of a local receiver chaotic signal or sequence with that of the transmitter, the characteristic of sensitive dependence on initial conditions causes free-running chaotic oscillators or maps to quickly diverge from each other, preventing the transfer of information. Probability-based estimates of chaotic state are typically made via correlation or autocorrelation calculations, which are time consuming. The number of iterates per bit can be reasonable in conjunction with synchronization, but values of several thousand iterates per bit are typically seen without synchronization.
A general communications system employing chaos usually consists of a message m(t) injected into a chaotic transmitter, resulting in a chaotic, encoded transmit signal y(t). This signal is altered as it passes through the channel, becoming the received signal r(t). The receiver implements some mix of chaotic and/or statistical methods to generate an estimate of the message m
e
(t).
The recovery of a message from transmitted data depends on the ability of the receiver to perform either asynchronous or synchronous detection. There are three fundamental approaches to chaotic synchronization delineated in the literature of the prior art: (1) decomposition into subsystems, (2) inverse system approach, and (3) linear feedback.
If the approach is a decomposition into subsystems, the chaotic system is divided into two or more parts: an unstable subsystem usually containing a nonlinearity and one or more stable subsystems. The stable subsystem may contain a nonlinearity if its Lyapunov exponents remain negative. The Lyapunov exponents of all subsystems must be computed to evaluate the stability or instability of each one, where negative Lyapunov exponents indicate stability and positive Lyapunov exponents indicate instability. The arbitrary division into stable and unstable systems is accomplished by trial and error until successful. The driver or master system is the complete chaotic system, and the driven or slave system(s) consists of the stable subsystem(s). In general, synchronization will depend on circuit component choices and initial conditions.
The problem with using the decomposition approach is threefold. First, the decomposition into subsystems is arbitrary, with success being defined via the results of Lyapunov exponent calculations. There is presently no known method by which to synthesize successful subsystem decomposition to satisfy a set of requirements. Second, the evaluation of Lyapunov exponents can become extremely involved, complicated, and messy. The time and effort spent evaluating a trial, arbitrary decomposition can be extensive, with no guarantee of success. Third, there can be several possible situations for the subsystem decompositions for a given chaotic process. A chaotic process could have single acceptable decomposition, multiple acceptable decompositions, or no acceptable decompositions. Since the method is by trial-and-error an
Barrón Gilberto
Southwest Research Institute
Taylor Russell & Russell P.C.
Zand Kambiz
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