Cryptographic Schemes Based on Elliptic Curves over the Ring

Elliptic Curve Cryptography recently gained a lot of attention in industry. The principal attraction of ECC compared to RSA is that it offers equal security for a smaller key size. The present paper includes the study of two elliptic curve


Introduction
Elliptic curve cryptography has been an active area of research since 1985 when Koblitz (Ref. [1]) and Miller (Ref. [2]) independently suggested using elliptic curves for public-key cryptography. A lot of work has been done on elliptic curve cryptography (Ref. [3]- [7]). Because elliptic curve cryptography offers the same level of security as compared to RSA with considerably shorter keys, it has replaced traditional public key cryptosystems, especially, in environments where short keys are important. Public-key cryptosystems are computationally demanding and, hence, the fact that elliptic curve cryptography has been shown to be faster than traditional pub-lic-key cryptosystems is of great importance. Elliptic Curve Cryptographic (ECC) schemes are public-key mechanisms that provide the same functionality as RSA schemes. However, their security is based on the hardness of a different problem, namely the Elliptic Curve Discrete Logarithmic Problem (ECDLP). Most of the products and standards that use public-key cryptography for encryption and digital signatures use RSA schemes. The competing system to RSA is an elliptic curve cryptography. The principal attraction of elliptic curve cryptography compared to RSA is that it offers equal security for a smaller key-size.

Auxiliary Result
In this section first we discuss some essential arithmetic of elliptic curves, and then we mention some auxiliary results which are necessary to prove the main result. Although a lot of literature exist on arithmetic of elliptic curves (Ref. [8]- [11]), a simple and easier arithmetic of elliptic curves are given by the following (Ref. [ Multiply (1) and (2), we get a ib a ib c id c id  We deduce that k c i + is not invertible. This completes the proof of the result.
where f is the isomorphism between 1 G and 2 G . Then ⊕ is an internal composition law, commutative with identity element e and all elements in E are invertible.
Proof. It is clear that ⊕ is an internal composition law over E.
To show that e is the identity element with respect to binary operation ⊕ .
and e is the unit element of ( ) = and e is unit element of ( )

Main Result
x y E ∈ then 2 3 y x ax b = + + , then 2 3 and there is three cases arise: As we know that addition of two different points ( ) where f is the isomorphism between

Cryptographic Applications
In this section we shall illustrate our proposed methods for coding of points on Elliptic Curve, then exchange of secret key and finally use them for encryption/decryption.

Coding of Element on Elliptic Curve
It is described with the help of illustration 5.  x y Z ∈ for j = 0 or 1 and z = 0 or 1. Then coding method is given by 0 1 0 1 x x y y z which produces the following codes The above scheme helps us to encrypt and decrypt any message of any length. , choose the binary code of point S as a private key, which transformed on the decimal code 30760000265000001 .

ECC Key Generation Phase
Now, exchange of secret key involves the following steps:

ECC Encryption Phase
To encrypt m P , a user choose an integer r at random and sends the point ( ) , m b r Q P r P ⋅ + ⋅ . This operation is shown in Figure 1.

ECC Decryption Phase
Decryption of the message ( ) m is done by multiplying the first component ( )  Bob then subtracts the result from the last point that Alice sends him. Note that he subtracts by adding the point with the second coordinate negated: Bob has therefore received Alice's message.