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

Cryptography, basically, is divided into two basic types; symmetric cryptography(the encryption and decryption operations use one secure key), and Asymmetric cryptography(the encryption and decryption operations use two keys; one called private and the other is public). The cryptographic system which uses the symmetric approach is called Symmetric Cryptographic System, and the other is called Asymmetric Cryptographic System(or public Cryptographic Systems). The Asymmetric Cryptographic Systems are considered more secure than symmetric. The security of those systems depends on some open problems in both mathematics and computer science, and for example; the security of RSA Cryptographic System depends on: Integer Factorization Problem(IFP), and the security of ElGamal Cryptographic System depends on Discrete Logarithm Problem(DLP) [1,2]. There are special kinds of algebraic curves called Elliptic Curves. They have been using with many of the famous Asymmetric Cryptographic algorithms, and the first time was in 1985, by Neal Koblitz and Victor Miller [7,6]. That using generates a special kind of cryptographic systems which called elliptic curve cryptographic systems(ECC). The security of those systems depends basically, on Elliptic Curve Discrete Logarithm Problem(ECDLP), and this problem is more difficult than others like IFP and DLP, because so far there has not been found any ‘subexponential-time Algorithm’ for solving ECDLP, and there are just ’exponential-time Algorithm’, while there are ‘subexponential-time Algorithms’ for solving problems like IFP and DLP. And thus the ‘ECC’ systems are the most secure ones in these days. For that reason, many researchers work on offering several approaches by using the concepts of elliptic curves with Asymmetric algorithms to reaching to the highest degrees of security and protection of information.

There are various kinds of ‘ECC Systems’, and for instance we mention the most famous and applicable ones; EC-Diffie Helman, and EC-ElGamal Cryptographic Systems.

To do encryption and decryption message texts by ECC systems, we need using some approaches to embedding the message on an elliptic curve firstly, and then applying the principles which related of encrypting and decrypting operations by those systems.

In this paper we presented a new and secure approach for embedding message text on an elliptic curve (i.e: mapping message text onto a point on an elliptic curve) which defined over prime fields, and then we built some cryptographic system. This system is a mix between the concept of EC-ElGamal system and the concept of RSA system, and thus this system depends on the two problems IFP and ECDLP at the same time.

According to [3,4] ( , ), ( , ) ( );

• Point doubling:

Note 1 all calculations above are computed module p .

Depending on [1], we can illustrate the technique of this system and how using it for encryption and decryption message text between two sides; ‘Bob’ is the sender of message ’ M ‘, and the receiver ‘Alice’, as the following steps: Note 2 Any message text can be represented as number or sets of numbers according to the encoding system which we use it, so from now when we mention the word ‘message’ , we mean some number.

RSA Cryptographic System is the most common applications of Public-Key Cryptography(Asymmetric systems). It was published by Rivest, Shamir and Adelman 1978. It uses two distinct keys, public-key which possible to be known to everyone and the other is private-key which is secured and not allowed to exchange between the sender and receiver. The security of this system depends on some open problem in both mathematics and computer science which is ‘Integer Factorization Problem(IFP)’. This system described as following:

1. Choose two distinct large random prime numbers p and q . ( , ) e n is the public -key and d is the private -key. To encrypt Message ’ M ‘, and get cipher ’ C ’ by this system we use equation:

To do decryption, and coming back to message ’ M ’ we use equation:

The figure (2) [1] illustrates the RSA System and how it is used between two sides; ‘Alice and Bob’ to encrypt and decrypt the message ’ M ‘. 3. Proposed approach for mapping message text onto a point on an elliptic curve defined over prime field

The mapping (or embedding) message (some number) onto a point on an elliptic curve is an important approach to do encryption and decryption message texts in every ECC system.

The references [1,3], presented the most practical approach known for mapping message text onto a point on an elliptic curves which defined over finite fields