FPGA Implementation of a Chaotic Map with No Fixed Point

Abstract

The employment of chaotic maps in a variety of applications such as cryptosecurity, image encryption schemes, communication schemes, and secure communication has been made possible thanks to their properties of high levels of complexity, ergodicity, and high sensitivity to the initial conditions, mainly. Of considerable interest is the implementation of these dynamical systems in electronic devices such as field programmable gate arrays (FPGAs) with the intention of experimentally reproducing their dynamics, leading to exploiting their chaotic properties in real phenomena. In this work, the implementation of a one-dimensional chaotic map that has no fixed points is performed on an FPGA device with the objective of being able to reproduce its chaotic behavior as well as possible. The chaotic behavior of the introduced system is determined by estimating the Lyapunov exponents and its chaotic behavior is also analyzed using bifurcation diagrams. Simulations of the system are realized via Matlab, as well as in C and the very high-speed integrated circuit (VHSIC) hardware description language (VHDL). Experimental results on FPGA show that they are like those obtained in the simulations; therefore, this chaotic dynamical system could be used as an element in some encryption schemes such as in the generation of cryptographically secure pseudorandom numbers.