Survey Report on Chaos Based Public-key Cryptosystem

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International Journal of Emerging Technology and Advanced Engineering

Website: www.ijetae.com (ISSN 2250-2459,ISO 9001:2008 Certified Journal, Volume 3, Issue 12, December 2013)

449

Survey Report on Chaos Based Public-key Cryptosystem

Adel A. El-Zoghabi

1

, Amr H. Yassin

2

, Hany H. Hussien

3

1_{Head, Department of Information Technology, Institute of Graduate Studies and Research, Alexandria University} 2_{Lecturer, Electronics & Communication, Engineering Department, Alexandria Higher Institute of Engineering and Technology}

3PHD students, Department of Information Technology, Institute of Graduate Studies and Research, Alexandria University Abstract—In recent years, encryption technology has been

developed rapidly the chaos based cryptographic algorithms. Chaos owns certain critical properties such as sensitive dependence on initial condition, random-like behavior, and continuous broadband power spectrum, which match the confusion, diffusion, and key sensitivity requirements for cryptography. Chaos based cryptographic offer sundry features over the traditional encryption algorithms such as high security, speed, and sensible computational overheads and power. This paper presents a survey of public key encryption methods based on chaos system.

Keywords— Chaos-based cryptography, Chaos System, Chaotic Map, Decryption, Encryption, Public key cryptography.

I. INTRODUCTION

Over the past decade, there has been formidable interest in studying the behaviour of chaotic systems for it characterizes such as sensitive dependence on initial conditions and control parameters, similarity to random behaviour, continuous broad-band power spectrum, ergodicity, and quasi-randomness. The nature of chaos has initiated a lot of interests in different engineering disciplines, where cryptography must be one of the most potential applications. Unlike the classical cryptographic algorithms which are primarily based on discrete mathematics, chaos-based cryptography mostly is based on the complex dynamics of nonlinear systems or maps which are deterministic. Therefore, it can provide fast and secure mechanisms for data protection, which is conclusive for multimedia data transmission over fast communication channels, such as the broadband internet communication [1].

Chaos theory is a field of study in mathematics, with applications in various disciplines including meteorology, physics, engineering, economics and biology. Chaos theory studies the behaviour of dynamical systems that are highly sensitive to initial conditions, which means that a small change in the initial state can be lead to a big different action in the final state.

Which can be observe in the Butterfly effect's shown in (Figure 1) allows the possibility that even a tiny disturbance of a butterfly flapping its wings can significantly affect whether sunny or cloudy skies will predominate days later. The chaos system have input parameters, which mean the system are not random, and have rules for every state must be taken. The irregular behaviour shown by chaotic systems is owing to the system’s intrinsic non-linearity rather than the noise [2].

FIGURE 1:CHAOTIC TRAJECTORIES [3]

The public key is cryptography algorithms which have two independent keys, one is a public key Ke is used to encrypt the message and the other is a private key Kd which is used to decrypt the message. Even though the key Ke which was used to encrypt the message is the public and is supposed to be known to all, decoding the message using that key or recovering the private decoding key Kd out of the public key Ke can’t be happen [4].

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International Journal of Emerging Technology and Advanced Engineering

450 It is computationally easy for a user to generate their public and private key-pair and to use them for encryption and decryption. The strength lies in the fact that it is "impossible" (computationally infeasible) for a properly generated private key to be determined from its corresponding public key. Thus the public key may be published without compromising security, whereas the private key must not be revealed to anyone not authorized to read messages or perform digital signatures. Public key algorithms, unlike symmetric key algorithms, do not require a secure initial exchange of one (or more) secret keys between the parties [4].

The major advantage of using the public key encryption is that they avoid the need of sharing any shared key between the transmitter and the receiver via a secure channel. In a shared key encryption scheme the transmission of a private key via a secure channel can cause a completely danger to the encryption scheme. For certain applications the transmission of a shared key via a secure channel is theoretical [4].

Chaotic cryptography depicts the use of chaos theory to execute different cryptographic tasks in a cryptographic system, which obtained in Table 1, which summarizes the connection between chaos and cryptography [5].

TABLE 1

COMPARISONS OF CHAOTIC AND CRYPTOGRAPHIC PROPERTIES

The rest of the paper is organized as follows: Section 2 describes the main chaotic maps existed in the studied literature. A literature review on public key cryptosystem based chaotic system is found in section 3. The summary point is introduced in section 4 .Section 5 includes the conclusion.

II. TYPES OF CHAOTIC MAPS

After analysing the aforementioned articles, various chaotic maps were found. The most frequent ones are described below:

A.Chebyshev polynomials

Are sequences of orthogonal polynomials which are related to de Moivre's formula and which can be defined recursively. The chebyshev map is a chaotic map which is defined as follows:

Tn(X)=2Tn-1(X)-Tn-2(X)

Where T0(x) = 1 and T1(x) = x.

The Chebyshev polynomials Tn or Un are polynomials of

degree n and the sequence of Chebyshev polynomials of either kind composes a polynomial sequence. Chebyshev polynomials are polynomials with the largest possible leading coefficient, but subject to the condition that their absolute value is bounded on the interval by 1.

B.Coupled map lattice (CML)

Is a dynamical system that models the behavior of non-linear systems (especially partial differential equations). They are predominantly used to qualitatively study the chaotic dynamics of spatially extended systems. This includes the dynamics of spatiotemporal chaos where the number of effective degrees of freedom diverges as the size of the system increases. Features of the CML are discrete time dynamics, discrete underlying spaces (lattices or networks), and real (number or vector), local, continuous state variables. Studied systems include populations, chemical reactions, convection, fluid flow and biological networks. More recently, CMLs have been applied to computational networks identifying detrimental attack methods and cascading failures.

C.Logistic maps

Logistic map is general form of the chaotic map. It is a nonlinear polynomial of second degrees and can be expressed by using the following equation:

Xn+1=rXn(1-Xn)

Chaotic Characteristic

Cryptographic property

Description

Ergodicity

Mixing property

Auto-similarity

Confusion

The output of the system seems similar for any input

Sensitivity to initial

conditions and

control parameters

Diffusion

A small difference in the input produces avery different output

Deterministic Deterministic

Pseudo-randomness

A deterministic procedure that produces pseudo-randomness

Complexity Algorithmic

Complexity

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International Journal of Emerging Technology and Advanced Engineering

451 Where r is a system parameter lies between 0 to 4, Xn is

map variable lies between 0 to 1, Xn is the initial condition

of the logistic map and n is number of iteration used for generating the iterative values. By varying the system parameter r, following behaviours are observed and shown in (Figure 2):

 When the value of r lies between 0 to 1, the iterative values ultimately die, which are sovereign of initial condition.

 When the value of r lies between 2 to 3, the iterative values first oscillate around some value and then finally stabilize on the same value.

 When the value of r lies between 3 and 3.45 (approximately), the iterative values oscillate between two values forever, which are dependent on r.

 When the value of r lies between 3.45 and 3.56 (approximately), the iterative values oscillate between four values.

 As the value of becomes greater than or equal to 3.57, this logistic map is converted into chaotic map, because a slight variations in the initial condition produces dramatically different iterative values over time, exhibit chaotic behaviour and trajectory of these iterative values is called chaotic attractor.

FIGURE 2:LOGISTIC MAP BIFURCATION DIAGRAM [6]

III. LITERATURE REVIEW

Over the past two decades, many researchers were utilized a chaotic system to design public key algorithms in order to provide high security.

A recent survey of the literature indicates that there has been an increasing interest in the application of different classes of chaotic maps to problems related to cryptography in the past few years. Recent works have examined the use of chaos in public key cryptosystems [7-23].

Public-key encryption with chaos, 2004

Ljupco Kocarev proposed public-key encryption algorithms based on iteration of one-dimensional Chebyshev chaotic maps and two-dimensional of torus automorphisms chaotic map. The public-key encryption algorithm used the following map:

Y=Tp(X)(Mod N)

Where X€{0,1, . . . ,N−1},Where N and p are integers,

and Tp is chebyshev map of order p. The chebyshev maps

can be commuted from the known results of the periodic structure of torus automorphisms. The model generalized the traditional algorithms by replacing powers with matrix powers, choosing the matrix to be a matrix which defined a two-torus automorphism. The torus automorphism map can be implemented by 2 x 2 matrixes M with integer entities. The requirement that the matrix M has integer entities ensures that M maps torus into itself. Experimental result shows that the proposed encryption schemes are both secure and practical, and can be used also for digital signature [7].

Additive Mixing Modulation for Public Key Encryption Based on Distributed Dynamics, 2005

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International Journal of Emerging Technology and Advanced Engineering

452

Public-key encryption based on generalized synchronization of coupled map lattices, 2005

Xingang Wang proposed a new technique for constructing public-key cryptosystems by generalized synchronization of the one-way coupled map lattices with the method of ―Merkle’s puzzles‖. The public key algorithm can be implemented via many steps; the transmitter is composed two one-way coupled map lattices (OCML (K) via generalized synchronization in coupled map chaotic system which act as pseudorandom number (PN) generator for each lattice system and generated the plaintext message for the following parameters OCML (A) that act as symmetric encryption public key. Then, the receiver generated decryption key OCML (B) that act as private key. The security of the system is depend on the number of session have been transmitted. The model has many features such as the flexibility in security update, convenient in key deducing, and feasible for real applications. While in this scheme the key tracing is only operated at the receiver side according to Merkle’s method [9].

A New Public Key Cryptography Algorithm Using Chaotic Systems and Hyperelliptic Curves, 2006.

Rodrigo Abarz´ua proposed a new assistance between chaotic system and hyper-elliptic curves for producing a new strong public key cryptography. The model basic idea is that the partners need a common secret key to exchange a secret message. So, a Diffie-Hellman algorithm based on (hyper-elliptic curves C) is used to generate a private key (β). That operates as initial condition parameters for the chaotic system. After that synchronization is done between the two partners through a common external signal for achieve (γ) parameter which adds to the private key to produced a new key called (β,γ). The proposed algorithm is solving the discreet mathematical problem that has in the traditional public key system. The model is needs more statistical analysis to be sure about its efficiency and security [10].

A Cryptosystem Based on Iterations of Chaotic Map, 2007

Shuichi Aono presented a new cryptosystem by using iterations of an expansion chaotic map. The expansion map is modified logistic map as follow.

The proposed cryptosystem is a symmetric-key cryptography that has characteristic of public-key cryptography, that used three kinds of keys, a public key, a private key and a common private key. A common private key means shared secret key between the decryptor and the encryptor. And, the keys are decided by the decryptor, which is initial point (X0) and a parameter (α). The number

of iterations (A), that acts a private key of the decryptor. The decryptor achieves X0, XA as a public key. The

security of the cryptosystem becomes more safety by setting the different key at the encryption. The encryption process using the private key (B), while the decryption process using the common private key (α) and the private key (A). The security of the system is effective against some of the well-know attacks, but its need to do more analysis on the security of the system [11].

A Chaos Based Public Key Cryptosystem, 2009

M.R.K. Ariffin proposed an asymmetric encryption system based on chaotic beta-transformation mapping called ―AAβ‖ . The model consist of many steps, firstly the sender generate a random private key dS, then generate a public key through the AAβ function that used the following equitation:

Then the public key (eS) transmitted to the receiver. The

receiver generated his private key dB, and generated an encrypt key via the above AAβ function equitation called

(eSR )based on the sender public key (eS), then generated his

public key (eB), finally the receiver encrypted the message

based on (eSR ) key and transmitted it back to sender.

Which generated the encryption key (eRS ) to decrypted the

message . Experimental result show that the model is more secure and more efficient and the private key can be extended to 128 bit to achieve more security [12].

A Chaos Public-Key Cryptosystem Based on Semi-Group Features, 2009

Bi Dayuan proposed a public-key cryptosystems based on chebyshev polynomials, and a defined feature of semi group of chebyshev polynomials. The public key algorithm used the following map:

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International Journal of Emerging Technology and Advanced Engineering

453 The modular computation is introduced and the definitive range of chebyshev polynomials is extended from [-1,1] to real field. A new one-way drop door function, which is based on extended Chebyshev polynomials, is established. In the end, it is proven the proposed algorithms are practical and feasible [13].

Public key cryptosystems based on chaotic Chebyshev polynomials, 2009

K. Prasadh proposed an expansion of the public key encryption based on chebyshev polynomials with a modest hash function. The proposed model can be applied on multilevel inputs types such as images, and video. The Chebyshev map is a chaotic map which is defined as follows:

Where; T0(x) = 1 and T1(x) = x, n >=2

The algorithm used a gorgeous property called semi group property. The sender obtained the public key of the receiver, which contained a large number integer (s) and a computed (Ts(x)) based on the previous equation to get (x, Ts(x)) then encrypted the message after converted it to the interval (-1, 1). The cipher-text sent to the receiver. The decryption process done based on the receiver private key (large number integer (s)). The addition process for secure images and videos done by using multilevel scrambling in the encryption of images, such as arnold cat scrambling (which is a simple and elegant demonstration and illustration of some of the principles of chaos-namely), and phase scrambling technique (randomizes the phase of the r, g and b layers of an image), which made the cryptosystem more secured and robust making it difficult for any intruder to crack the original video. A non XOR-ing technique was used to make the hashing algorithm more secure against the chosen plain text attacks [14].

Fast algorithms of public key cryptosystem based on Chebyshev polynomials over finite field, 2011

LI Zhi-hui proposed two modified fast algorithms based on the chebyshev polynomial chaotic system to progress the efficiency of the public key cryptosystem. The two general algorithms with running time of O (lbn), which takes O (lbn) matrix multiplications to compute the An are the matrix algorithm and the characteristic polynomial algorithm, which are practical but not optimized. Through adoption of different procedures, the number of required operations in the modified algorithms is reduced, and the running time is decreased.

The new algorithm relevant with eigenvalues of matrix used in representation of chebyshev polynomial to converted the computation of chebyshev polynomial to modular exponentiation operations. The proposed scheme can be used in practice to get the best performance with a missing analysis about the security [15].

Parameter Selection in Public Key Cryptosystem based on Chebyshev Polynomials over Finite Field, 2011

Zhihui Li. introduced a new impersonation of the chebyshev polynomial chaotic map to improve the security of a public key cryptosystem. The model is based on modified some properties of the chebyshev polynomial sequence and defined some principles for parameter selection, which make the cryptosystem more secure and reduced the time cost. The security of the cryptosystem is relevant with its corresponding chebyshev polynomial sequence, whose characteristic is in turn decided by parameters x and p. It is found if parameters are not selected properly the cryptosystem could be broken easily [16].

A Rapid Cryptography Algorithm Based on Chaos and Public Key, 2012

Yun-peng Zhang presented a fast public-key algorithm using collective application of chaotic system and public key cryptosystems. A rapid public-key algorithm used a chaotic system to produce knapsack interfering vector based on logistic and chebychev chaotic mapping and using the mod theorem to hide the double sequences of the chaotic system. The model combined the plain message after converted it to binary message with a random sequence. The key model is generated by two random sequences that combine after get the mod operation for them to obtain the public key applied on the modified message. The super increasing sequence S as one of a subset of a backpack acts as private key for the model. Simulation the model is efficiency and better, controllable security [17].

Chaotic Based Key Management and Public-key Cryptosystem, 2012

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International Journal of Emerging Technology and Advanced Engineering

454 The results obtained show that proposed chaotic public-key cryptosystem enhances the performance and security [18].

Chaos Synchronization based Data Transmission with Asymmetric Encryption, 2012

Santo Banerjee proposed an asymmetric cryptographic scheme for guarantee the security of data being transmitted in the previous model AAβ-Cryptosystem which has implementation speed of O (n2) [12]. The asymmetric cryptographic scheme utilizes the factorization problem of the two large primes and is implemented only by using the multiplication operation for both the encryption and the decryption process. The proposed model is based on the chaos synchronization bidirectional coupling which encourage flexibility to the role of the communicating parties and produced a full duplex mode of communication. The developed encrypted data is much faster than traditional asymmetric encryptions. The system possesses a vast key space and is strong to the main statistical attacks. The system can be also effectively used in the real time [19].

Implementation of AES and RSA Using Chaos System, 2013

Bhavana Agrawal proposed two cryptographic algorithms AES and RSA Using Chaos. In RSA the plaintext mixed with Chaos sequence and then applied for encryption and decryption process. It’s observed that RSA with chaos take less time to execute, and secured as compose to RSA. The results of the proposed chaos based AES and RSA gives a significant improvement for the accepted sequences probability over a wide range of chaos initial conditions. Chaos based AES and RSA makes the system more complex and fast as compare to the Conventional AES and RSA [20].

Two-Dimension Chaotic-Multivariate Signature System, 2013

Xiaoyan Sun proposed a high secure crossbred public key cryptosystem based on two chaotic maps which are, the central map, and the chaotic system map. The proposed model is resistant to quantum algorithm (which is a famous algorithm attack all cryptosystem, and very efficient with polynomial time). The model firstly transformed the plaintexts via an affine transformation and then encrypted by the central map in multivariable cryptosystem. The first two elements of central map outputs act as initial values in the two-dimension chaos system and are transformed by another affine transformation. Finally, the cipher texts are derived by adding the outputs of chaos system and the second affine transformation.

Due to the chaos system, the weakness of the traditional multivariate cryptosystems is substitute and therefore the security is enhanced, but the computing efficiency of proposed algorithm is probably less than the traditional Multivariate Public Key Cryptology (MPKC) [21].

An Improved Public Key Encryption Algorithm Based on Chebyshev Polynomials, 2013

Jinhui Sun proposed an improved public key encryption algorithm based on modified semi-group property of the chebyshev polynomials chaotic map. The model imported the alternative multiply coefficient Ki to forge the cipher text sensitively which can make the cipher text-only attack out of work. The chosen of Ki is decided by the value of the defined equation as follow:

Tr(Ts(X)Mod N

Where the number of Ki can be chosen as required. All of these measures make the system not only can resist chosen-cipher text attack and tamper attack, but also has the function of identity authentication. Experimental results and performance analyses show that the improved algorithm has much higher security and practical value [22].

Multi Message Signcryption Based On Chaos With Public Verifiability, 2013

Aditya Kumar proposed a new multi message signcryption schemes. The purposed schemes achieved the public verifiability without the knowledge of sender’s or receiver’s private key. The model generated a chaotic key generator for the encryption and decryption process. The signcryption scheme consists of four algorithms which are key generation algorithm, dynamic chaotic key generator, signcryption algorithm and un-signcryption algorithm. The key generation algorithm generated two key the public key and the private key form a random sequence. The dynamic chaotic key generator generated the key based on the sender public key which used in the encryption process. Experimental results show that the scheme is secure and can be more secure by using a large degree of polynomials, which may affect the efficiency of the system in some cases in comparison to non-chaotic signcryption schemes [23].

IV. SUMMARY

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International Journal of Emerging Technology and Advanced Engineering

455 1.One-dimensional chebyshev polynomials chaotic map

and two- dimensional of torus autmorphisms are iterations to produce public key.

2.The chaotic distributed nonlinear dynamic system was used between the transmitter (public signal) and the receiver (private signal) via communication channel (which called coupled bi-direction).

3.Synchronization of one-way coupled map lattices was used as a trap-door function for conclude the private key from the public key.

4.Expansion map, which is modified logistic map was used for create the public and private key.

5.One-way chaotic beta-transformation map was used for key exchange and logistic map was used as private key generation.

6.The central map was used in the encryption process as initial values in the two-dimension logistic chaos system .

V. CONCLUSION

In this paper, we summarize some recent researches about the application of chaos in the field of public cryptography. The chaos based public key cryptosystem is a good idea of building very complicated asymmetric cryptosystem, which has been faster and security than traditional asymmetric encryption schemes.

Lastly, according to the findings of this work, the trend for the years to come regarding the use of chaos for public key cryptography tasks will be focused mainly on Chebyshev polynomials map, and Logistic map.

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