Python programming for an modified elliptic curve cryptography

Project Title: An Efficient RFID Authentication Protocol to Enhance Patient Medication Safety Using Elliptic Curve Cryptography There are two phases in protocol, setup yhase and authentication phase. Before giving details, I defined some notations used in the protocol. q A large prime number. G A additive group with order q consisting of points on an elliptic curve. P A generator of G Xi The identifier of the ith tag which is a random point in G. y The private key of the server. Y The public key of the server where Y=yP. r,k Two random numbers. h() A secure hash function. Setup Phase The server generates his private key and public key in the phase. He also generates the identifier of each tag. 1). The server generates a random number y ∈ Zq as his private key and computes his public key Y=yP. 2). The server chooses a random point Xi in G as the ith tag’s identifier. Then the server stores the ith tag’s identifier and related information in its database. The server also stores [Xi,Y,P] into the ith tag’s memory. Authentication Phase When interrogating a tag, the server starts the phase to authenticate each other. As show in the attached file, the details of the phase are presented as follows. 1). The server generates a random number r∈Zq, com- puters C0=rY and broads the message {C0} to the ith tag. 2). On receiving {C0}, the i th tag generates a random number k∈Zq, computes K=kP, C1=kC0, R=K+K,C2=Xi+R and C3=h(Xi,K). Then the i th tag sends the message {C1,C2,C3} to the server. 3). Upon receiving {C1,C2,C3}, the server computes K=y−1r−1C1, R=K+K and Xi=C2−R. Then the server checks whether C3 and h(Xi,K) are equal. If they are not equal, the server rejects the session; otherwise, the server searches its database for Xi. If succeeds, the server confirm that the tag is a legal one; otherwise, the server computes C4 = h(Xi, 3K) and the message {C4} to the ith tag. 4). Upon receiving {C4}, the ith tag checks whether C4 and h(Xi, 3K) are equal. If they are not equal, the ith tag rejects the session; otherwise, the ith tag confirms that the server is a legal one.