Security and secrecy are some of the important concerns in the communications world. In the last years, several encryption techniques have been proposed in order to improve the secrecy of the information transmitted. Chaos-based encryption techniques are being widely studied as part of the problem because of the highly unpredictable and random-look nature of the chaotic signals. In this paper we propose a digital-based communication system that uses the logistic map which is a mathematically simple model that is chaotic under certain conditions. The input message signal is modulated using a simple Delta modulator and encrypted using a logistic map. The key signal is also encrypted using the same logistic map with different initial conditions. In the receiver side, the binary-coded message is decrypted using the encrypted key signal that is sent through one of the communication channels. The proposed scheme is experimentally tested using Arduino shields which are simple yet powerful development kits that allows for the implementation of the communication system for testing purposes.
Security and secrecy in communications are some of the most important concerns in societies nowadays. With the advent of worldwide networks and digital communication techniques, the cryptographic techniques that once were restricted to military and state affairs are now covering several domains such as banks, private companies, medical organizations, and so forth. This has led to a very active research field oriented to finding optimal solutions to the problem of communications security [
There are basically two main approaches to designing secure communication systems based on chaotic dynamics: analog and digital. Analog communication systems based on chaos are possible because of the possibility of synchronization [
Several examples of chaos-based communication systems can be found in the literature. For instance, Zapateiro et al. [
Numerous works can be found in the literature that use the logistic map for improving security in communications. The logistic map is a nonlinear discrete map originally used for modeling population growth of different species as well as economic and political phenomena [
In this paper we present a digital chaos communication system in which the logistic map is used to encrypt the message and key of the transmission. A simple Delta modulator is used along with one of the chaotic maps to encrypt the message. The Delta modulation technique is one of the most simple and robust methods of analog-to-digital (ADC) schemes requiring serial digital communications of analog signals [
This paper is organized as follows. Section
The objective of this paper is to design and implement a communication system to transmit a message
Block diagram of the communication system.
The details of these blocks will be outlined in the following sections of this chapter.
The logistic map has its origins in the works by the Belgian mathematician Pierre-François Verhulst in the first half of the 18th century [
The logistic map, the discrete-time version of Verhulst’s logistic model, is chaotic under certain conditions. Its equation is
Logistic map bifurcation diagram.
As can be seen in the bifurcation diagram, there are different regions that depend on the value of
Logistic map Lyapunov exponent.
In the next sections, we will use a logistic map as part of an encryption/decryption scheme for transmitting information. We will explain the details of the prototype of this communication system which is implemented on two Arduino Uno boards.
The communication system implemented in this work consists of a transmitter and a receiver whose cores are the Arduino Uno R3 microcontroller boards [
The flow diagram of the programs executed by each Arduino is shown in Figures
Flow diagram of the transmitter Arduino codes.
Flow diagram of the receiver Arduino codes.
The communication begins when a message
The next step is the Delta modulation. This kind of modulation can be viewed as an 1-bit ADC conversion scheme since it generates one output bit per input sample. The scheme of the Delta modulation is shown in Figure
Diagram of the simple Delta modulator.
Delta modulation example of a sine input signal. (a) Input signal and reconstructed signal comparison. (b) Modulated output.
After one bit from the Delta modulator is obtained, the next step is the message encryption. In order to do so, two logistic maps are called to generate two values
Part 1 (1) (2) (3) (4) (5) (6) (7) Part 2 (8) (9) (10) (11) (12) (13) (14) Part 3 (15)
In order to increase the security of the system, the key,
The key is then finally encrypted by applying the XOR function to the variables
The signals
In the receiver, the signals
Truth table for
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0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 1 |
1 | 1 | 0 |
Truth table for
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0 | 0 | 0 |
0 | 1 | 1 |
1 | 0 | 0 |
1 | 1 | 1 |
The Karnaugh maps technique [
Once
Part 1 (1) (2) (3) (4) (5) (6) Part 2 (7) (8) (9) (10) (11)
The signal
As shown in Figure
Circuit diagram of the analog electronics in the receiver.
For the experiments, the logistic maps were implemented with
Numbers generated by the logistic map with
Figures
125 Hz sine wave message. Blue: sent message. Yellow: retrieved message.
125 Hz sine wave message. Blue: sent message. Yellow: encrypted message signal.
125 Hz sine wave message. Blue: sent message. Yellow: encrypted key.
125 Hz sine wave message. Blue: sent message. Yellow: auxiliary signal
In subsequent experiments, different frequencies and waveforms were tested. Figure
125 Hz triangular wave message. Blue: sent message. Yellow: retrieved message.
70 Hz sine wave message. Blue: sent message. Yellow: retrieved message.
Random wave message. Blue: sent message. Yellow: retrieved message.
In this paper we presented a communication system based on chaotic logistic maps and an experimental realization of it. The proposed communication system uses a simple Delta modulator to modulate the message signal and a logistic map for encryption. A key signal is also generated and encrypted in order to retrieve the message in the receiver side without the need for synchronization. The whole system was implemented with Arduino Uno microcontroller boards that run the encryption and decryption algorithms in the transmitter and receiver, respectively. The experiment results showed the feasibility of using the Arduino microprocessors for the task proposed. With the proposed scheme, it is possible to transmit signals whose bandwidth is 500 Hz approximately.
The authors declare that there is no conflict of interests regarding the publication of this paper.
First author is supported by the fellowship from CAPES/Programa Nacional de Pós-Doutorado from Brazil. This work was funded by the European Union (European Regional Development Fund) and the Spanish Ministry of Economy and Competitiveness through the research projects DPI2012-32375/FEDER, DPI2011-28033-C03-01, and DPI2014-58427-C2-1-R and by the Government of Catalonia (Spain) through 2014SGR859.