In 1998, M.S. Baptista proposed a chaotic cryptosystem using the ergodicity property of the simple lowdimensional and chaotic logistic equation. Since then, many cryptosystems based on Baptista’s work have been proposed. However, over the years research has shown that this cryptosystem is predictable and vulnerable to attacks and is widely discussed. Among the weaknesses are the non-uniform distribution of ciphertexts and succumbing to the one-time pad attack (a type of chosen plaintext attack). In this chapter the authors give a mathematical treatment to the phenomenon such that the cryptosystem would no longer succumb to the one-time pad attack and give an example that satisfies it.
Top1.0 Introduction
The relationship between chaos and cryptography makes it natural to employ chaotic systems to design new cryptosystems. It is based on the facts that chaotic signals are usually noise-like and chaotic systems are very sensitive to initial conditions. Their sensitivity to initial conditions and their spreading out of trajectories over the whole interval seems to be a model that satisfies the classic Shannon requirements of confusion and diffusion (Shannon, 1949). From 1989 onwards, many different chaotic encryption systems have been proposed. The most celebrated cryptosystems based on the ergodicity property of chaotic maps is presented by Baptista (1998) and has received more and more attentions in the past literature (Grassi & Mascolo, 1998; Alvarez, et.al., 1999; Chu & Chang 1999; Alvarez, et.al., 2000; Jakimoski, & Kocarev, 2001; Li, et.al., 2001; Wong, et.al., 2001; Garcia & Jimenez, 2002; Wong, 2002; Palacios & Juarez, 2002; Alvarez, et.al., 2003; Pareek, et.al., 2003; Li, et.al., 2003; Wong, 2003; Wong, et.al., 2003; Alvarez, et.al., 2004; Li, et.al., 2004; Alvarez & Li 2006). Researchers in this field have also constructed chaotic cryptosystems without using chaotic synchronization (most are designed for implementation on digital circuits or computers (Jakimoski, & Kocarev, 2001; Alvarez, et.al., 2003)) and secure communications based on chaotic synchronization of analog circuits (Baptista, 1998; Alvarez, et.al., 1999; Alvarez, et.al., 2000).