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A Secure Approach for Embedding Message Text on an Elliptic Curve Defined Over Prime Fields, and Building 'EC-RSA-Elgamal' Cryptographic System

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A Secure Approach for Embedding Message Text on an Elliptic Curve Defined Over Prime Fields, and Building 'EC-RSA-Elgamal' Cryptographic System 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 curve defined over prime fields, and building 'EC-RSA-ELGamal' Cryptographic System Ahmad Steef M. N. Shamma Department of Mathematics, 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 Logarithm approach to embedding message text on an elliptic curve, Problem(DLP) [1,2]. by using the concept of the RSA Algorithm and its There are special kinds of algebraic curves 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 algorithms, 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 Discrete Logarithm 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 field; Elliptic not been found any 'subexponential-time Algorithm' for Curve Cryptography(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: Integer same time. Factorization Problem(IFP), and the security of ElGamal 2 . Preliminaries and background 1 https://sites.google.com/site/ijcsis/ ISSN 1947-5500 International Journal of Computer Science and Information Security (IJCSIS), Vol. 15, No. 6, June 2017 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 group 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 equation (1): curve E over Fp , and some point 2 3 y= x + ax + b(mod p) (1) PEF∈ ()p which generates some large subgroup 3 2 from EF() with order 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 'point at infinity'; 'O '. We can define • Step 2: Alice selects number a randomly; 1 <a < n and EC over prime field as form: computes Q= aP (mod p) , and sends this to Bob. 2 3 E( Fp )={ ( x , y ) ∈ Fp × F p ; y = x+ ax + b(mod p); • Step 3: Bob selects number b randomly; 1 <b < n , and 3 2 T computes bP (mod p), (M+bQ)(mod p) , and a,b∈Fp and 4a + 27b 0mod() p} ∪ {O } sends the cipher C=( bP , M + bQ )(mod p) to The set EF()p with a special binary operation ' + ' forms an Alice. algebraic group 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 2 so from now when we mention the word 'message' , we mean 3x1 + a (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 2 https://sites.google.com/site/ijcsis/ ISSN 1947-5500 International Journal of Computer Science and Information Security (IJCSIS), Vol. 15, No. 6, June 2017 'Integer Factorization Problem(IFP)'. This system described as Let elliptic curve; E; y2= x 3 + ax + b(mod p) , and following: suppose MK, are positive integers 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<e < φ ( n ), and gcd( e ,φ ( n ))= 1 x={ KM + j ; j=1,2,3....k-1} until finding 5. Compute d such that: e. d = 1(mod φ(n )) . some value for x satisfies the equation of E (,)e n is the public – key and d is the private –key. above(i.e: x3 + ax + b becomes quadratic residue To encrypt Message ' M ', and get cipher ' C ' by this system modulo p ), and then we stop and getting the point we use equation: 2 3 (x , y )∈ E ( Fp ); y = x + ax + b(mod p) , CM= e (mod n) (6) and thus the number M is being mapped to the point (,)x y . To do decryption, and coming back to message ' M ' we use • To coming back to the number from the point equation: M (,)x y x x C d (mod n)=M , (7) just we can note that M = ; is the K K The figure (2) [1] illustrates the RSA System and how it is x used between two sides; 'Alice and Bob' to encrypt and biggest integer number less than or equals to decrypt the message ' M '. K Theorem 1 If p be an odd prime, then there are exactly p −1 p −1 quadratic residues and exactly quadratic 2 2 nonresidues modulo p [1].
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