International Journal of Computer Science and Information Security (IJCSIS), Vol. 15, No. 6, June 2017

A secure approach for embedding message text on an elliptic defined over prime fields, and building 'EC-RSA-ELGamal' Cryptographic System

Ahmad Steef M. N. Shamma Department of , Arab International University(AIU), Syria & Al-Baath University, Homs, Syria Basic Sciences Department, the Mechanical and e-mail: [email protected] Electrical Engineering, Damascus University, Syria e-mail: [email protected] A. Alkhatib Department of mathematics, Al-Baath University, Homs, Syria e-mail: [email protected]

Abstract— This paper presents a new probabilistic Cryptographic System depends on Discrete approach to embedding message text on an elliptic curve, Problem(DLP) [1,2]. by using the concept of the RSA and its There are special kinds of algebraic called Elliptic security, and such approach allows us discovering the Curves. They have been using with many of the famous message from the point, only according to the security of Asymmetric Cryptographic , and the first time was the RSA Algorithm. By mixing between the concept of this in 1985, by Neal Koblitz and Victor Miller[7,6]. That using approach and the concept of EC-ELGamal Cryptographic generates a special kind of cryptographic systems which called System, we have built a cryptographic System and we elliptic curve cryptographic systems(ECC). The security of named it 'EC-RSA-ELGamal' those systems depends basically, on Elliptic Curve Problem(ECDLP), and this problem is more Keywords RSA Cryptographic System; EC-ELGamal difficult than others like IFP and DLP, because so far there has Cryptographic System; Elliptic Curve over finite ; Elliptic not been found any 'subexponential-time Algorithm' for Curve (ECC); Embedding message text on an elliptic solving ECDLP, and there are just 'exponential-time curve. Algorithm', while there are 'subexponential-time Algorithms' 1. Introduction for solving problems like IFP and DLP. And thus the 'ECC' systems are the most secure ones in these days. For that The Cryptography is the most important science have being reason, many researchers work on offering several approaches used to secure data while transmitting in networks, so, and by using the concepts of elliptic curves with Asymmetric because of the increasing of progress of technology, the algorithms to reaching to the highest degrees of security and researchers are working to offer the best techniques and protection of information. scientific approaches for achieving the most secure steps, and There are various kinds of 'ECC Systems', and for all those depending on the concepts that related of many instance we mention the most famous and applicable ones; topics in advanced mathematics. EC-Diffie Helman, and EC-ElGamal Cryptographic Systems. Cryptography, basically, is divided into two basic types; To do encryption and decryption message texts by ECC symmetric cryptography(the encryption and decryption systems, we need using some approaches to embedding the operations use one secure key), and Asymmetric message on an elliptic curve firstly, and then applying the cryptography(the encryption and decryption operations use principles which related of encrypting and decrypting two keys; one called private and the other is public). The operations by those systems. cryptographic system which uses the symmetric approach is In this paper we presented a new and secure approach for called Symmetric Cryptographic System, and the other is embedding message text on an elliptic curve (i.e: mapping called Asymmetric Cryptographic System(or public message text onto a point on an elliptic curve) which defined Cryptographic Systems). The Asymmetric Cryptographic over prime fields, and then we built some cryptographic Systems are considered more secure than symmetric. The system. This system is a mix between the concept of EC- security of those systems depends on some open problems in ElGamal system and the concept of RSA system, and thus this both mathematics and computer science, and for example; the system depends on the two problems IFP and ECDLP at the security of RSA Cryptographic System depends on: same time. Problem(IFP), and the security of ElGamal 2 . Preliminaries and background

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2.1 Elliptic Curve over prime fields Depending on [1], we can illustrate the technique of this system and how using it for encryption and decryption According to [3,4], EC over prime field FZp= p and message text between two sides; 'Bob' is the sender of the law are defined as the following: message ' M ', and the receiver 'Alice', as the following steps:

EC over prime field Fp is a special kind of algebraic curves • Step 1: In the first, Alice and Bob should agree to the elliptic given by the following (1): curve E over F , and some point p 2 3 y= x + ax + b(mod p) (1) PEF∈ ()p which generates some large

3 2 from EF() with n and so, the information with the condition 4a+ 27bT 0( mod p) . This kind of p curves consists of all points which satisfy that equation (1) (,,)E P p is public for Alice and Bob. with some point called ''; 'O '. We can define • Step 2: Alice selects number a randomly; 1 and the identity is 'O '. The operation ' + ' • Step 4: Alice decrypts the message by applying the satisfies the properties ' group law ': following: (M+ bQ − a ( bP ))(mod p) ≡ M

• O+ P = P, ∀ p ∈ EF()p The Figure (1) [1] illustrates all steps above.

• The inverse of the point P( x , y )∈ E ( Fp ) is: −P =( x , − y ), and P + ( − P ) =O • Point addition:

Let P(, x1 y 1 ),(, Q x 2 y 2 )∈ E ( Fp ); P ≠ ± Q then;

P+ Q = ( x3 , y 3 );

2 y− y  2 1 (2) x 3 =  −x1 − x 2 , x2− x 1 

y2− y 1  y 3 =  ()x1− x 3 − y 1 , (3) x2− x 1 

• Point doubling: Figure 1 Description of 'EC-ALGamal Cryptographic System'

Let P( x1 , y 1 )∈ E ( Fp ); P ≠ − P then;

P+ P =2 P = ( x3 , y 3 ); Note 2 Any message text can be represented as number or sets of numbers according to the encoding system which we use it, 2 3x2 + a  so from now when we mention the word 'message' , we mean 1 (4) some number. x 3 =  − 2x 1 , 2y 1  3.2 RSA Cryptographic System [1,2] RSA Cryptographic System is the most common 2 3x1 + a  applications of Public-Key Cryptography(Asymmetric y 3 =  ()x1− x 3 − y 1 , (5) 2y systems). It was published by Rivest, Shamir and Adelman 1  1978. It uses two distinct keys, public-key which possible to Note 1 all calculations above are computed module p . be known to everyone and the other is private-key which is secured and not allowed to exchange between the sender and 2.2 EC-ELGamal Cryptographic system receiver. The security of this system depends on some open problem in both mathematics and computer science which is

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' Problem(IFP)'. This system described as Let elliptic curve; E; y2= x 3 + ax + b(mod p) , and following: suppose MK, are positive such that 1. Choose two distinct large random prime numbers p and q . (M+ 1). K < p , so, we want to mapping the number 2. Compute n= p. q M onto some point on E , and to do that just we apply the 3. Compute Euler's function of n; φ (n)=(p-1).(q-1) . following: 4. Choose an integer e such that: • We compute a set of values of x ; 1

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• Step 4: if x satisfies the equation of E above(i.e: (354)185 (mod 989)=439=M .  3 x+ ax + b is a quadratic residue modulo p ), then move to step 5, else move to step 2. 4. Building a new cryptographic system 'EC-RSA-ELGamal' • Step 5: stop and the number M is being mapped to the Depending on the ideas showed in paragraphs (3.2) and (3) 2 3 above, we shall present a cryptographic system and how using point (x , y )∈ E ( Fp ); y = x + ax + b(mod p) . it to do encryption and decryption. To coming back from the point (,)x y to the number M , Since, this system basically, depends on the concepts of we have to know the number d; e . d = 1(mod φ (n )) , then RSA and EC-ELGamal Systems, so we named it 'EC-RSA- we apply the following: ELGamal' system. 2 3 M= x d (mod n) Let elliptic curve; E; y= x + ax + b(mod p) , and

suppose (,)()x y∈ E Fp which generates some large Note 3 In our approach above, the number d is not allowed to subgroup from and its order is . Suppose the any one does not have permission. Just whom have permission EF()p n can discover the number d from the point (,)x y . And number M we want to encrypt and decrypt it by our discovering it is dependent on the security of RSA algorithm cryptographic system. To do that, we apply the following which depends basically on the problem 'IFP'. steps: • Step 1: select two distinct prime numbers q and r such Theorem 2 if p is odd and , then the number that M< n = q. r < p , and then move to step 2 a is a quadratic residue modulo p if and only if • Step 2: find e 1

Suppose e =5 ⇒ To do decryption, and coming back from the point Q to the • Step 3: x= M e (mod n)=354 ⇒ number M , we apply the following steps: 3 • Step 1: Compute the point A(,) x y by using the • Step 4: x+71 x + 602 is a quadratic residue modulo 0 0 0 (2,3,4,5) as following: p= 1009

• Step 5: stop and the number M = 439 is being mapped to (xQQ , y )(+ − a1 . b 1 P ) (mod p)= A0 = ( x 0 , y 0 ) (9)

the point (x , y )= (354,88) ∈EF (1009 ); 2 3 • Step 2: Compute the message M by applying the equation y= x + ax + b(mod 1009) following: To coming back from the point (354,88) to the number d x 0 (mod n)=M; e . d = 1(mod φ (n )) M = 439 , we have to know the number Note 4 We can note that EC-RSA-ELGamal Cryptographic d; e . d = 1(mod φ (n )) , and then we can apply the System mentioned above offers alternative factor for security following: if the point (,)x y has been known by someone does not x d (mod n)=M have permission, especially according to the increasing in By Extended Euclidian Algorithm for instance we have progress of technology, thus if the point (,)x y is caught 'by d =185 ⇒

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some way' by some eavesdropper, there is additional factor for 5. Discussion and conclusion security depending on the security of RSA algorithm . In this paper, we showed a new approach for embedding message on an elliptic curve defined over prime fields, and 4.1 Numerical Example 2 this approach allows us to secure that message according to Let elliptic curve; the security of the RSA Algorithm. But according to the E; y2= x 3 + 71 x + 602(mod 1009) , the number of condition; ''M< n < p ,the EC-RSA-ELGamal 2 points on E is #E(F1009 ) = 1060 = 2 .5.53, and the point Cryptographic System mentioned in this paper has some P( x, y) = ( 1,237) which degree is s= 530 = 2.5.53 [1]. weakness, because the number n must be more than 1024-bit according to the security of RSA algorithm, and thus the Let M= 439, and we want to apply EC-RSA-ELGamal number p , also, must satisfy the same, but when we working Cryptographic System to encrypt and decrypt the message M on EC over prime fields we want to select the number p

small as possible as we can and still keeping on the security at The following steps illustrate that: the same time. • Step 1: Let q =23, r=43 ⇒ n=989, φ (n)=924 In this paper, we understand that the EC-RSA-ELGamal • Step 2: select 1

the point (x , y )= (354,88) ∈EF (1009 ); 2 3 1. Song Y. Yan, "COMPUTATIONAL y= x + ax + b(mod 1009) AND MODERN CRYPTOGRAPHY", 1st ed, Wiley, 2013. • Step 6: select randomly; 2. W. Stallings, " Cryptography and Network Security, Principle and practice", Sixth ed, 2014. a1 =17, a2 = 432; 1

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