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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 and defined over the ring where . After showing isomorphism between and , we define a composition operation (in the form of a mapping) on their union set. Then we have discussed our proposed cryptographic schemes based on the elliptic curve . We also illustrate the coding of points over E, secret key exchange and encryption/decryption methods based on above said elliptic curve. Since our proposed schemes are based on elliptic curve of the particular type, therefore the proposed schemes provides a highest strength-per-bit of any cryptosystem known today with smaller key size resulting in faster computations, lower power assumption and memory. Another advantage is that authentication protocols based on ECC are secure enough even if a small key size is used.

Elliptic curve cryptography has been an active area of research since 1985 when Koblitz (Ref. [

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. [

An elliptic curve

The Addition operation is defined over

If

If

Let

where

Now we discuss the auxiliary result of this section. For a prime number p, let

Lemma 2.1. (Ref. [

Proof. Let

which implies

In (1) take the conjugate

Multiply (1) and (2), we get

We deduce

Lemma 2.2. (Ref. [

Proof. Assume that

invertible. By Lemma 2.1, we have

We deduce that

Theorem 2.3. For two isomorphic abelian groups

such that

where f is the isomorphism between

identity element e and all elements in E are invertible.

Proof. It is clear that

To show that e is the identity element with respect to binary operation

Let x in E. If

because

Else

because

We have

Let

If

If

If

Let

where O is the point at infinity.

Corollary 3.1. If

Proof. Let

This implies that

which is a contradiction.

Hence

Theorem 4.1. Let f be a mapping from

Then f is a bijection.

Proof. First we show that f is well defined.

Let

Hence f is well defined.

f is one-one. Let

This implies that

So,

Hence, f is one-one.

f is onto. Let

This implies that

Thus, f is onto.

f is homomorphism. Let

Case I. When

As we know that addition of two different points

where

So we have

where

Again

where

It is obvious that

Therefore

Case II. When

where

Again

where

It is evident that

Therefore,

Case III. When

We have

and

Thus

Therefore, in either case f is an homomorphism. Hence f is a bijection.

Corollary 4.2. For two isomorphic abelian groups

such that

where f is the isomorphism between

Proof. Keeping in view the result of theorem-2.3, corollary-2.4, and theorem-3.1, it is evident that

Corollary 4.3. If

Proof. Since

Now,

This implies that

Therefore,

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.

It is described with the help of illustration 5.1 and illustration 5.2.

Illustration 5.1. For

Since,

Therefore

and

Coding of element

Let

Illustration 5.2. For

Let

The above scheme helps us to encrypt and decrypt any message of any length.

1) For a publically integer p, and an elliptic curve

2) P generates a subgroup say

Now, key exchange between Alice and Bob can be described as follows

3) Alice chooses a random number

4) Bob chooses a random number

5) Alice computes

6) Bob computes

7) Alice and Bob are agree with a point

Remark. With the secret key

Illustration 5.3. Let

are two elliptic curve defined over the same field

1) Alice chooses a random number

2) Bob chooses a random number

3) Alice computes

4) Bob computes

5) Alice and Bob are agree with a point

Now, exchange of secret key involves the following steps:

1) Encode the message m on the point

2) Choose a random number k, compute

3) Public key is

4) Private key is

To encrypt

Decryption of the message

This operation is shown in

Illustration 5.4. The

are two elliptic curves defined over the same field

Alice’s message is the point

Bob has chosen his secret random number

and calculated

Bob publishes the point. Alice chooses the random number

and

Alice sends (7966,6354) and (5011,2629) to Bob, who multiplies the first of these point by

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.

This research work is supported by University Grant commission (UGC) New Delhi, India under the Junior Research Fellowship student scheme..

ManojKumar,PratikGupta, (2016) Cryptographic Schemes Based on Elliptic Curves over the Ring Zp[i]. Applied Mathematics,07,304-312. doi: 10.4236/am.2016.73027