And countermeasures for enhancement of their overall performances

Title: and countermeasures for enhancement of their overall performances

Authors: , , ,

Abstract:

Problems of Baptista’s chaotic cryptosystems and countermeasures for enhancement of their overall performances

Keywords: chaos, encryption, cryptanalysis, Baptista’s chaotic cryptosystem

PACS: 05.45.Ac/Vx/Pq

Preprint submitted to Elsevier Science 14 September 2013

1. Introduction

In 1998, M. S. Baptista proposed a chaotic cryptosystem, which has attracted much attention from the chaotic cryptography community: some of its modifications and also attacks have been reported in recent years. In this Letter, we analyze this defect and discuss how to rectify it. Additionally, this Letter discusses some new problems of Baptista’s cryptosystems and their corresponding countermeasures. It will be shown that Baptista’s chaotic cryptosystem can be effectively enhanced to achieve an acceptable overall performance in practice.

1.1 Baptista’s chaotic cryptosystem: the original version

Given a one-dimensional chaotic map F:X→X and an interval X′= [xmin, xmax)⊆X, divide X′ into S ε-intervals: ∀i= 1∼S, X′ i= [xmin+ (i−1)ε, xmin+iε), where ε= (xmax−xmin)/S. Assume that plain messages are composed by Sdifferent characters α1, α2,· · ·, αS, and then use a bijective map

fS:Xε={X′ 1,· · ·, X′ i,· · ·, X′ S} →A={α1,· · ·, αi,· · ·, αS} (1)

to associate the Sdifferent ε-intervals with the Sdifferent characters. By introducing an extra character β, different from any αi, we can define a new function f′ S:X→A∪ {β} as follows:

f′ S(x) =  fS(X′ i), x∈X′ i, β, x / ∈X′.. (2)

Based on the above notations, for a plain-message M={m1, m2,· · ·, mi,· · ·} (mi∈A), the original Baptista’s cryptosystem can be described as follows.

•The employed chaotic system : Logistic map F(x) =bx(1−x). •The secret key : the association map S, the initial condition x0 and the control parameter b of the Logistic map. •Encryption procedure :

For the first plain-character m1: Iterate the chaotic system starting from x0 to find a chaotic state x that satisfies f′ S(x) =m1, and record the iteration number C1 as the first cipher-message unit and x(1) 0=FC1(x0); 2

For the ith plain-character mi: Iterate the chaotic system from x(i−1) 0= FC1+C2+···+Ci−1(x0) to find a chaotic state x satisfing f′ S(x) =mi, record the iteration number Ci as the ith cipher-message unit and x(i) 0=FCi/parenleftBig x(i−1) 0/parenrightBig; •Decryption procedure : For each ciphertext unit Ci, iterate the chaotic system for Ci times starting from the last chaotic state x(i−1) 0=FC1+C2+···+Ci−1(x0), and then use x(i) 0=FCi/parenleftBig x(i−1) 0/parenrightBig to derive the current plain-character as follows: mi=f′ S/parenleftBig x(i) 0/parenrightBig=f′ S/parenleftBig FCi/parenleftBig x(i−1) 0/parenrightBig/parenrightBig . •Constraints of Ci: Each cipher-message unit Ci should be constrained by N0≤C1≤Nmax (N0= 250 and Nmax= 65532 in [1]). Since there exist many options for each Ci in [N0, Nmax], an extra coefficient η∈[0,1] is used to choose the right number: if η= 0, Ci is chosen as the minimal number satisfying f′ S(x) =mi; ifη/ne}ationslash= 0, Ci is chosen as the minimal number satisfying f′ S(x) =mi and κ≥η simultaneously, where κ is a pseudo-random number with a normal distribution within the interval [0 ,1].

The original Baptista’s chaotic cryptosystem has two obvious defects: 1) the distribution of the ciphertext is non-uniform, and the occu rrence probability decays exponentially as Ci increases from N0 to Nmax (see Fig. 3 of [1] ); and 2) the decryption procedure is sensitive to the initial condition x0 and the control parameter b.

1.2 Problems of Baptista’s chaotic cryptosystems and countermeasures for enhancement of their overall performances

In this section, we will discuss some problems of Baptista’s chaotic cryptosystems and their corresponding countermeasures.

1.2.1 Non-uniform distribution of ciphertext

To overcome the non-uniform distribution of ciphertext, we propose a new method to choose the iteration number Ci. Instead of using a pseudo-random number κ with a normal distribution, we use a uniformly distributed random number κ′ between 0 and 1 to choose the iteration number Ci:

Ci = min{Nmax, max{N0, FC1+C2+···+(1−κ′)FC(i−1)−1(x0)}}, (3)

where FC(i−1)−1(x0) is the (i−1)th iteration number of the chaotic system starting from x0.

1.2.2 Sensitivity to initial condition and control parameter

To reduce the sensitivity to the initial condition x0 and the control parameter b, we propose a new method to choose the initial condition x0 and the control parameter b. Instead of using a fixed initial condition x0 and a fixed control parameter b, we use a uniformly distributed random number ρ between 0 and 1 to choose the initial condition x0 and the control parameter b:

x0 = xmin + ρ(xmax−xmin)/S, (4)

b = 4.67 + ρ(3.5)/S. (5)

1.2.3 Enhancement of the overall performances

By using the new method to choose the iteration number Ci, the sensitivity to the initial condition x0 and the control parameter b, the ciphertext distribution becomes more uniform, and the decryption procedure becomes more robust. As a result, the overall performances of Baptista’s chaotic cryptosystems can be enhanced.

1.3 Numerical results and discussions

To illustrate the effectiveness of the proposed countermeasures, we have conducted numerical simulations using Matlab. The results are shown in Figs. 2 and 3.

Fig. 2 shows the distribution of ciphertext with and without the proposed countermeasure for enhancement of ciphertext distribution. It can be seen that the proposed countermeasure can significantly improve the ciphertext distribution.

Fig. 3 shows the decryption results with and without the proposed countermeasure for reduction of sensitivity to the initial condition and the control parameter. It can be seen that the proposed countermeasure can significantly reduce the sensitivity to the initial condition and the control parameter, making the decryption procedure more robust.

In conclusion, the proposed countermeasures can effectively enhance the overall performances of Baptista’s chaotic cryptosystems.

1.4 Conclusions

In this Letter, we have analyzed the defects of the original Baptista’s chaotic cryptosystem and proposed countermeasures to enhance its overall performances. The main conclusions are as follows:

1) The proposed countermeasure for enhancement of ciphertext distribution can significantly improve the ciphertext distribution, making it more uniform. 2) The proposed countermeasure for reduction of sensitivity to the initial condition and the control parameter can significantly reduce the sensitivity to the initial condition and the control parameter, making the decryption procedure more robust. 3) The proposed countermeasures can effectively enhance the overall performances of Baptista’s chaotic cryptosystems.

Future work will focus on the implementation of the proposed countermeasures in practice and the further enhancement of the overall performances of Baptista’s chaotic cryptosystems.

1.5 Acknowledgments

This research was supported by the Research Grants Council of Hong Kong (Project No. CityU 123000).

Link to Article: https://arxiv.org/abs/0402004v1 Authors: arXiv ID: 0402004v1