Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 468567, 9 pages

http://dx.doi.org/10.1155/2015/468567

## On the Cryptanalysis of Two Cryptographic Algorithms That Utilize Chaotic Neural Networks

^{1}School of Computer Science and Engineering, University of Electronic Science and Technology of China, Chengdu 611731, China^{2}School of Computer Science, Carleton University, 1125 Colonel By Drive, Ottawa, ON, Canada K1S 5B6

Received 14 October 2014; Revised 30 January 2015; Accepted 3 February 2015

Academic Editor: Joao B. R. Do Val

Copyright © 2015 Ke Qin and B. John Oommen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This paper deals with the security and efficiency issues of two cipher algorithms which utilize the principles of Chaotic Neural Networks (CNNs). The two algorithms that we consider are (1) the CNN-Hash, which is a one-way hash function based on the Piece-Wise Linear Chaotic Map (PWLCM) and the One-Way Coupled Map Lattice (OCML), and (2) the Delayed CNN-Based Encryption (DCBE), which is an encryption algorithm based on the delayed CNN. Although both of these cipher algorithms have their own salient characteristics, our analysis shows that, unfortunately, the CNN-Hash is not secure because it is neither Second-Preimage resistant nor collision resistant. Indeed, one can find a collision with relative ease, demonstrating that its potential as a hash function is flawed. Similarly, we show that the DCBE is also not secure since it is not capable of resisting known plaintext, chosen plaintext, and chosen ciphertext attacks. Furthermore, unfortunately, both schemes are not efficient either, because of the large number of iteration steps involved in their respective implementations.

#### 1. Introduction

Over the last few decades, the phenomenon of chaos has been widely investigated and applied in a variety of domains including social networks, control systems, and prediction. A chaotic system is characterized by salient phenomena such as its sensitivity to initial values, its pseudorandomness, and ergodicity, rendering it to be quite similar to a cryptographic system. The characteristics that render chaotic systems to be akin to cryptographic algorithms are listed below.

*(1) Chaotic Maps versus Encryption/Decryption Algorithms*. The form of a chaotic system is usually iterative, when the system is discrete, or it involves differential equations when it is continuous. As opposed to this, an encryption/decryption algorithm is usually a nonlinear mapping from the plaintext space to the ciphertext space, and this mapping is, often, not complex. The similarity between the two is that both of them can yield, as their outputs, results that appear to be random, by virtue of the underlying algorithm repeating certain steps.

*(2) Iterations versus Rounds*. For a chaotic system, each of the steps mentioned above that are “repeated” constitute a so-called “*iteration.*” As opposed to this, a cryptographic system involves a sequence of “*rounds.*” Only long-term chaotic iterations can yield sequences that appear to be random [1].

*(3) Controlling Parameters versus Keys*. If a chaotic system starts from a given initial value, different control parameters can yield different output sequences at each iteration. This, in turn, is analogous to the role of keys in a cryptographic system. The similarity between the two lies in the fact that it is computationally infeasible to deduce the initial input without knowing the controlling parameters or the keys, respectively.

*(4) Sensitivity to Initial Values versus Diffusion and Confusion*. When it concerns a chaotic system, a slightly different initial value may result in a significant difference in the output generated after a sufficiently large number of iterations. Analogously, in a cryptographic system, the change of even a single bit (whether it is in the key or the plaintext) should affect most of the ciphertext bits. Furthermore, the statistics relating the plaintext and the key should be “as complicated as possible.” Thus, if we regard the plaintext or the key as the initial value, the ciphertext should be highly sensitive to these.

*(5) Pseudorandom and **Ergodic*. The sequence of outputs generated by a chaotic system should be able to fill the entire range in a random-like manner. Analogously, a good encryption algorithm requires that the ciphertexts be randomly distributed in the cipher space.

*Brief Survey of the Field*. As a result of the above observations, chaos has also been widely applied in the field of information security since Matthews proposed the first chaotic encryption algorithm [2] in 1984. Later, Baptista and Alvarez reported two cryptographic algorithms based on the phenomenon of chaotic searching in [3–5], respectively. While Erdmann and Murphy described a stream cipher based on the so-called Henon maps [6], Kanso and his coauthors illustrated a novel hash function [7] and showed how one could achieve digital image encryption based on chaotic maps [8]. Kocarev and Tasev presented a public-key encryption [9] and random number generators [10] based on chaotic maps. A detailed list of articles that advocate the use of chaotic principles in cryptographic systems can also be found in [11, 12], and systematic reviews about chaos-based ciphers are found in [13, 14].

Now that chaotic* maps* have been proven to be useful in encryption; researchers have attempted to use Chaotic Neural Networks (CNNs), which are characterized by much more complicated dynamics than chaotic maps, to develop cryptosystems. The authors of [15–17] proposed different one-way hash functions based on different CNNs. Similarly, Yu and Cao proposed an encryption algorithm based on delayed CNNs [18]. Our present paper concerns some of these results.

*Motivation of This Paper*. Although the latter above-mentioned authors have affirmed that their schemes are secure and efficient, in this paper, we shall demonstrate that the security levels guaranteed by them are weak and that they are inefficient. For example, most chaos-based ciphers require an excessive number of iterations, without which the ciphertexts are not sensitive to plaintexts. As opposed to these, traditional ciphers, for example, the AES, only require a 10-round calculation if one utilizes a key of 128 bits. Further, since chaotic equations are typically specified on the set of real numbers, the associated accuracy of implementing these schemes using digital computations is also problematic. Indeed, when we implement the associated computations numerically, we observe that some of the significant digits will be automatically truncated, and the consequence of this is that the original system which was chaotic within the domain of “real” numbers is no longer chaotic [13]! Also, the improvement brought about by increasing the accuracy using higher-precision software entails a larger computational cost.

In this paper, we analyze two typical CNN-based cipher systems, the first of which is a one-way hash function and the second is an encryption method. However, we believe that our analysis is also valid for other CNN-based schemes.

#### 2. The CNN-Based Hash Function

##### 2.1. The Description of the CNN-Based Hash Function

The authors of [15] proposed a novel one-way hash function based on a special CNN. The structure of the network (more details about PWLCM’s dynamics and analysis can be found in [19] and omitted here to avoid repetition.) is shown in Figure 1.