In this contribution an encryption method using a chaotic oscillator, excited by “

Since Pecora and Carroll presented their work about chaos synchronization [

In recent years, encryption schemes are being studied and increasing demand exists to develop a safer encryption system for transmitting data in real time via the Internet, wireless networks, and other devices ([

The traditional standard encryption algorithm for images and data (DES) has a disadvantage when handling large amounts of data [

The online encryption in a dynamic system has the advantage of processing the signal in real time, so the analog signal (message) is encrypted while being sent.

In recent years some works were presented for the synchronization of master-slave structure ([

In this work and algorithm is proposed in order to encrypt the message using a nonlinear chaotic system; this message is combined with “

These signals excite the chaotic oscillator (the master) in order to encrypt the message more safely in the sense that the message signal is combined with sinusoidal signals; this combination increases the harmonics produced by the chaotic system and then the spectrum of the sending signal has more frequencies in the bandwidth of the signal.

In the proposed scheme the receiver (the slave) knows the frequencies to be used and estimates the sinusoidal signals which perturb the message and retrieve the message in exacta away.

In many other works (see [

The paper is organized as follows: in Section

In this paper a nonlinear system is proposed, which is excited by a signal given by the following equation:

It is observed that the signal perturbation

Various types of chaotic systems can be treated under the impulse of a sinusoidal input signal. In general the chaotic system to the encryption of the message can be seen in (

Having described the chaotic system model in general, we proceed to select the outputs of the system, which are given by

Some chaotic oscillators are studied in the following paragraphs and will be adapted to realize the desired encryption and take the form (

It is proposed that the outputs given by (

Estimation of disturbance

If an intruder is capable of obtaining the transmission signal and can eliminate the chaos, find that the message will be disturbed by a sinusoidal signal and its harmonics as this was injected into the chaotic system, so the message is not decoded and the information will be preserved.

In what follows some chaotic systems and how they are processed to take a proper structure are presented.

The Van der Pol equation provides an example of a nonlinear oscillator system. The system can be written as

Duffing equation is introduced in 1918 as a nonlinear oscillator model. The equation is defined as

This system is known as a simplified model of multiple physical systems ([

If we take

For this class of systems some assumptions are considered.

The constants

The

The message and the perturbation do not destroy the chaos.

With this assumption outputs (

Now in order to recover the transmitted message, a state estimator for system (

Consider the next theorem.

Take the outputs signals given by (

Consider the error system between (

From (

Then using control theory [

For the dynamics of

Then

Consider the Lorenz oscillator given by equations

Applying (

Estimation of disturbance on chaotic system.

Figure

Lorenz attractor with sinusoidal signal and audio signal.

Figure

(a) the original audio signal to be encrypted. ((b), (c)) the outputs

In Figure

Original audio signal and recovered signal from the decoder implementation.

Figure

Error signal between the original message and the message retrieved.

It is observed how the message and the perturbation are retrieved in exact manner, it is, the synchronization is exact and does not exist an error as other algorithms.

The method can also be applied to encrypt and to transmit digital images; for encryption, the image is modified to be sent in vector form. In this case, the receiver knows the number of rows and columns.

In this example we use the Duffing chaotic system with the perturbation given by

Figure

Sinusoidal signal retrieved and original signal.

Figure

Duffing attractor with sinusoidal signal and digital image (

Figure

Duffing attractor with sinusoidal signal and digital image (

The differences between Figures

Two examples are shown in Figures

(a) Comparison between the original information and the recovered image. (b) Chaotic signals

(a) Comparison between the original information and the recovered image. (b) Chaotic signals

In this contribution we presented a methodology to encrypt and transmit audio and image signals using chaos and regulation theory. The algorithm consisted in perturbing a chaotic system with a signal composed of

The authors declare that there is no conflict of interests regarding the publication of this paper.