The aim of this coursework assignment is to show evidence of understanding of the Elliptic curve Diffie-Hellman (ECDH) key exchange protocol, which allows two parties, each having an elliptic curve public-private key pair, to establish a shared secret over an insecure channel. This shared secret may be directly used as a key, or to derive another key which can then be used to encrypt subsequent communications using a symmetric key cipher. It is a variant of the Diffie-Hellman protocol using elliptic curve cryptography.

Protocol

The following example will illustrate how a key establishment is made. Suppose Alice wants to establish a shared key with Bob, but the only channel available for them may be eavesdropped by a third party. Initially, the domain parameters, which are k, a, b, and G, must be agreed upon;

a and b determine the curve;

k restricts the curve to a prime field Fk;

G is a generator, a point in the field.

Also, each party must have a key pair suitable for elliptic curve cryptography, consisting of a private key d (a randomly selected integer in the interval (1, n-1], where n is the order of the field) and a public key Q (where Q = dG, that is, the result of adding G together d times). Let Alice's key pair be (d_A, Q_A) and Bob's key pair be (dB, QB). Each party must know the other party's public key prior to execution of the protocol.  

Alice computes (x_k, y_k) = d_AQ_B. Bob computes (x_k, y_k) = d_BQ_A. The shared secret is x_k (the x coordinate of the point).

Report - Please write a report using the following skeleton:

1. Show in detail all the steps of the key exchange protocol, with the calculation expanded using the numbers you were given. For the private keys (d_A and d_B) you may choose any number in the [11, n-1], where n is the order of the field.

2. How do Alice and Bob arrive at the same shared secret?

3. If Carol is intercepting the communication and captures Q_A and Q_B, can she compute Alice's and Bob's private keys?

4. A more sophisticated attack by Carol involves generating (dC, QC) for use as a reset value, using the same values of a, b, k and G that Alice and Bob are using. Explain how this would work.

5. Write a brief discussion (two paragraphs) on the comparison of ECC (Elliptic Curve Cryptography) and RSA, focusing on the advantages and disadvantages of each.

6. Include key snippets of your code. NOTE: as you are doing modulo multiplications with figures greater than 10 on an elliptic curve, your work will most probably involve some programming. You may choose a programming language of your liking, whatever you are most comfortable with. The snippet should be the fragment dealing with modular multiplication on the elliptic curve.

Attachment:- Assignment File.rar