A new RSA public key encryption scheme with chaotic maps

ABSTRACT


INTRODUCTION
Cryptography is defined as the set of protocols and procedures that are necessary for secure communications in the existence of third parties. Cryptography is divided into two basic types: private key encryption, and public key encryption. In the former, a specific key (i.e., private key) has to be known by the sender and receiver to be able to encrypt and decrypt messages. This means that a secure channel in private key encryption is required to share the key. In reality, it is not easy to attain such secure channel. Diffie and Hellman [1] introduced Public Key Cryptography (PKC), which solves the drawback of private key cryptography; A single number theoretic cryptographic assumptions, on which many public key encryption protocols are based on (i.e., discrete logarithm, or factoring a large composite number) [1][2]. The security of a given protocol depends mostly on the cryptographic assumptions. If these assumptions can be hacked easily, then the cryptosystem will not be secure anymore [3]. Several cryptographic protocols try to enhance the system security by adding extra multiple hard problems that need to be solved simultaneously. Unlike protocols that depend on a single hard problem, these extra hard problems will definitely make the whole system more secure.
The first key distribution protocol, which is based on two different assumptions, was proposed in 1988 by K.S. McCurley [4]. This protocol was inefficient, because it was very hard to select module and to achieve similar difficulty for these two assumptions. To maintain acceptable efficiency L, Harn et al. [5] proposed a cryptosystem protocol that was based on two distinct cryptographic assumptions: Discrete Logarithm (DL), and Factoring (FAC). This new protocol has improved the security, while maintaining  Tahat) 1431 the implementation efficiency. Later, many other cryptosystem protocols were proposed [6][7][8][9], most of which are based on combining two problems such DL and FAC, Elliptic Curve Discrete Logarithm (ECDL), Knapsack problem, and many more. Some of these protocols achieve the optimal goal, which is an efficient secure system. In this paper, we propose a crypto-system protocol that is based on both of chaotic maps and factorization problems. The new protocol improves the overall security, and needs a lower number of operations in both of the encryption and decryption processes. Therefore, the proposed crypto-system is more practical for realistic applications. The fashion into which the rest of this paper is arranged into is as follows: In Section 2, we briefly introduce the necessary mathematical framework used in the paper. In the section 3, the new proposed encryption scheme is introduced. In Sections 4, 5 and 6, we analyze the security and efficiency of the proposed scheme. We finally conclude in Section 7.

CHAOTIC MAPS
Chaotic theory has been heavily used in designing secure communication protocols since the 1990s [10][11][12][13][14][15], while chaotic maps have been utilized in the design of symmetric encryption protocols in [16][17][18][19]. Designing a chaotic map setting is usually difficult, but generally creates secure and efficient protocols. That is because chaotic map-based protocols have low computational costs when compared with other modular exponential computing based protocols or protocols that are based on scalar multiplication on elliptic curves.

Chebyshev maps
A map of a Chebyshev polynomial, : → of degree , can be defined with the subsequent recurrent relation [20]: with 0 ( ) = 1, and 1 ( ) = , the headmost Chebyshev polynomials are, A significant property of Chebyshev polynomials is the semi-group property: An instant sequel of the above property is that Chebyshev polynomials under composition commute, i.e., ( ) = ( ). Under the action of the map : ([−1, 1]) = [−1, 1], the interval [−1, 1] is invariable. Thus, a Chebyshev polynomial confined to the interval [−1, 1] will be the prominent chaotic map for all > 1. It has a unique invariant measure ( ) = √1− 2 , which is absolutely continuous with positive Lyapunov exponent = . The Chebyshev map, for, = 2, reduces to the familiar logistic map. Two presumably intractable problems related to Chebyshev polynomials [21] are: Definition 1. Chaotic maps discrete logarithm (CMDL) problem: Given a random number ∈ ℤ * , and an element ∈ ℤ , the task of the CMDL problem is to find an integer such that = ( )( ). Definition 2. Chaotic maps Diffie-Hellman (CMDH) problem: Given a random number ∈ ℤ p * , and two elements, r ( ) and s ( ), for unknown values and , the task of the CMDH problem is to compute rs ( ).

Public-key encryption with Chebyshev polynomial
System based on chaotic theory is usually defined on real numbers. In fact, any encryption algorithm, which utilizes chaotic maps, upon its implementation on a computer (e.g., finite-state machine), it turns into a transformation onto itself from a finite set. Because floating-point has a wide dynamic rage, its implementation seems applicable for software implementation of Chebyshev polynomials. Nevertheless, floating-point cannot be used in public-key encryption for the following reasons: -There is no uniform distribution for floating-point numbers, on the real axis, over any given interval. Moreover, there is an existence of redundant number representations in floating-point arithmetic caused by normalized calculations. As the same real signal value is represented by some floating-point numbers [22]. There is a restriction on the message length because a Chebyshev polynomial is a non-invertible.
In [23], the public key encryption protocol uses Chebyshev polynomials. This algorithm can be explained as follows: Let a large integer set s be generated by Thomas, then let a number ∈ [−1, 1] be generated randomly, and let ( ) be computed. Thomas's public key is ( , ( ) ), his private key is . Bob denotes the message as number ∈ [−1, 1] , then creates a large integer and calculates ( ), ( ) = ( ( )), and = ( ). Bob relays the cipher-text = ( ( ), ) to Thomas. To recover plain-text from , Thomas utilizes the private key to compute ( ) = ( ( )), and recovers the text by calculating = ∕ ( ). Let , , be the lengths (in bits) of , and , respectively, and let -bit precision arithmetic be employed in the algorithm software implementation. Then ≤ − − [12,23].
-When floating-point representation is used to implement chaotic maps, it is hard to implement tools for the purpose of analysing the structure of the periodicity of the periodic orbits. Furthermore, there is no hope in establishing a link between the number and chaos theory.

Modified Chebyshev polynomials
The following map will be used to show an ElGamal and RSA public-key algorithms to Chebyshev maps: : Theorem 2.3.2 Let be an odd prime and let ∈ ℤ such that 0 ≤ < . Then the period of the sequence ( ) ( ) for = 01,2, …, is a divisor of 2 − 1.

THE PROPOSED PUBLIC KEY ENCRYPTION
We propose in this section our new protocol, which is based on chaotic maps and factoring problems. The new protocol comprises three parts: key generation, encryption, and decryption.

Key generation
In general, it is assumed that it is desired to join the proposed crypto-system as entity A. For key generation purposes, the creation of a public and a private key requires performing a set of processes. We describe these processes in the following steps: Steps 1: Select two large random primes and of almost same size. Steps 2: Compute = and = ( 2 − 1)( 2 − 1). Steps 3: Choose a random integer , 1 < < ( ) such that gcd ( , ( )) = 1.

Encryption
Encryption algorithms are normally involved in the cryptographic process. Many iterations that include substitutions and transformations are performed in these algorithms on original data (known as plaintext). This is done so as to make the process of identifying the data by a hacker or intruder complicated [24]. In this paper, we consider the plaintext space as ℤ n . Assume that a user ℬ wishes to send a message ∈ ℤ n to using 's public key. Then ℬ has to carry-out the following steps: Steps 1: Select ∈ ℤ * and find 1 = ( )(mod ).

Decryption
Generally, the process of decryption is reversing all operations carried-out to perform the encryption [25]. It entails transforming the encrypted data back to the original form in order to allow the receiver to understand it. In this paper, to recover the message from(s 1 , s 2 , s 3 , s 4 ), should carry-out the following: Steps 1: Compute = ( 1 )(mod ).
Note that, in RSA key generation, the two integers and are called, respectively, the encryption exponent, and the decryption exponent. While is called the modulus. It was shown in Section 3.2 that 1 ( ) ≡ (mod ). By the same argument, Lastely, since and are distinct primes, the Chinese remainder theorem may be use to show that:

EXAMPLE
To illustrate the impact of the proposed scheme, we have used artificially small parameters into a representative example as follows: -Key generation: The user choose p = 13, q = 17 and compute n = 221, φ = 43384. selects a random integer e = 317, and find the unique integer, Chooses two random integers a = 211 and b = 311 such that 0 ≤ a, b ≤ φ(n) − 1, and he also choosesα = 107, β = 179 ∈ ℤ n * and computes:

SECURITY
The proposed crypto-system' security is found on factoring and chaotic map. To depict the heuristic security at our scheme, a collection of common attacks were considered in the following: Attack 1: Assume that an attacker desires to recover all secret values ( , , , , ), utilizing all accessible system information. In this scenario, the attacker has to conduct factoring and chaotic maps solutions. S/he needs to find the primes of for factoring, which can usually be solved using the number field sieve method [9]. Nevertheless, the size of modulus influences this method, and computationally cannot factor an integer of size 1024-bit and above. If the two prime numbers p and q are chosen well, it will definitely increase the resistance of the scheme to attack by the special-purpose factorization algorithms. For chaotic maps to find and from 1 = 2 ( )(mod ) and 2 = 2 ( )(mod ), and if the same level of security is used over primes, then the attacker has to solve integer factorization problem and chaotic map. Also, the integers and must be large to prevent exhaustive search attack. One obvious encryption practice is to use different parameters , and for different messages, because if a sender used the same parameters for encryption of two message say 1 and 2 , then s/he would obtain 4 = ( 1 ) ( 2 ) ( + 1) (mod ) and ′ 4 = ( 1 ) ( 2 ) ( + 1) (mod ). So, from the relation 2 = ′ 4 4 1 , an attacker who knows the message 1 can recover 2 . Note, the new proposed algorithm is randomized, parameters , and are randomly chosen by the sender. Also, it can be proved that an attacker cannot find the cipher text of 1 2 even if he knows the corresponding ciphertext of messages 1 and 2 . Attack 2: If the attacker manages to factor the modulus , then, he can use and to calculate the value = ( 1 )(mod ) and = 4 −1 ( + 1)(mod ) = ( 1 ) ( 2 )(mod ). To recover the message from ( 1 ) ( 2 )( mod ), he has to find and . And that is the computationally infeasible assumption of the chaotic maps. Attack 3: Assume that the attacker is able to solve the chaotic maps problem, and thus obtain the integers 2 and 2 .Then, he will know 2 ( 2 ) mod = 2 ( ) = ( 1 )(mod ) and 2 ( 3 ) mod = 2 ( ) = ( 2 ), which is not enough to recover the message. The attacker still has to compute = ( 1 )(mod ) to find = 4 −1 ( + 1)(mod ), and since the factorization of is not known, it is infeasible to computationally compute . Attack 4: Now, let us assume that an oracle which can break the proposed scheme exists (i.e., the corresponding cipher-text is obtained through from the message). Now, we can show the security of the proposed scheme by the following the theorem. Theorem: If there exists an oracle that is able to break the suggested scheme, then it is also able to break the DRSA and CM.

Int J Elec & Comp Eng
ISSN: 2088-8708  Proof: If = 0 = , then 1 = 2 ( ) = 1 = 2 ( ) and so to be a particular case of the proposed scheme is satisfied by the dependent RSA crypto-system. Therefore, if an oracle exists such that it is capable of breaking the proposed scheme, then it is capable also of breaking the dependent RSA scheme. Assume that there is an oracle that is capable of breaking the proposed scheme. We will show that can also break CM. Given that ( , , ) is the public key and assume that is the private key of the CM, with = ( )(mod ) Assume that a cipher text, ( , ) was captured by an attacker, which is encrypted by the CM scheme, and s/he desires to recover the original message . So, there is a ∈ {0, … , − 2} such that = ( ) (mod ) and = ( )(mod ). First, s/he selects a prime such that ∤ and finds = . Secondly, s/he selects integers , 1 , 1 , 1 ∈ {1, … , − 1} such that: Since, ( ) = (mod ) and ( ) = 1 (mod ), then ( ) = 1 (mod ). Similarly, ( ) = 1 (mod ).

PERFORMANCE EVALUATION
In this section, evaluation of the new proposed scheme performance in terms of computational complexity and communication costs is carried-out. The notations which are used in this paper are listed and defined in Table 1. Table 2 shows taht the total computational complexity that is required by the proposed scheme is 10 ℎ + 6 + 3 , which is equivalent to merely 1.8s. It shows that it is much faster than other schemes. From the obtained results in Table 2, it is clear that the proposed scheme based on chaotic maps and factoring problems has beaten the trivial DRSA and QER schemes in series. It is also more efficient than the trivial use of the DRSA and ELGamal schemes in series.

CONCLUSION
In conclusion, this paper proposed a new crypto-system based on integer factorization and chaotic maps discrete logarithm (CMDL) problems. The new crypto-system has enhanced the overall security when compared with other major public key crypto-systems algorithms. The suggested scheme needs minimum number of operations performed in the encryption and decryption algorithms, which makes it very efficient. We have proved that the new proposed scheme demands a much lower computational cost than other schemes. We have proved that our scheme is robust against several attacks. Hence, our proposed scheme is as secure as RSA algorithm.