Biblio

Group exponentiation is an important operation used in many public-key cryptosystems and, more generally, cryptographic protocols. To expand the applicability of these solutions to computationally weaker devices, it has been advocated that this operation is outsourced from a computationally weaker client to a computationally stronger server, possibly implemented in a cloud-based architecture. While preliminary solutions to this problem considered mostly honest servers, or multiple separated servers, some of which honest, solving this problem in the case of a single (logical), possibly malicious, server, has remained open since a formal cryptographic model was introduced in [20]. Several later attempts either failed to achieve privacy or only bounded by a constant the (security) probability that a cheating server convinces a client of an incorrect result. In this paper we solve this problem for a large class of cyclic groups, thus making our solutions applicable to many cryptosystems in the literature that are based on the hardness of the discrete logarithm problem or on related assumptions. Our main protocol satisfies natural correctness, security, privacy and efficiency requirements, where the security probability is exponentially small. In our main protocol, with very limited offline computation and server computation, the client can delegate an exponentiation to an exponent of the same length as a group element by performing an exponentiation to an exponent of short length (i.e., the length of a statistical parameter). We also show an extension protocol that further reduces client computation by a constant factor, while increasing offline computation and server computation by about the same factor.

Security of the information is the main problem in network communications nowadays. There is no algorithm which ensures the one hundred percent reliability of the transmissions. The current society uses the Internet, to exchange information such as from private images to financial data. The cryptographic systems are the mechanisms developed to protect and hide the information from intruders. However, advancing technology is also used by intruders to breach the security of the systems. Hence, every time cryptosystems developed based on complex Mathematics. Elliptic curve cryptography(ECC) is one of the technique in such kind of cryptosystems. Security of the elliptic curves lies in hardness of solving the discrete logarithms problems. In this research, a new cryptographic system is built by using the elliptic curve cryptography based on square matrices to achieve a secure communication between two parties. First, an invertible matrix is chosen arbitrarily in the the field used in the system. Then, by using the Cayley Hamilton theorem, private key matrices are generated for both parties. Next, public key vectors of the both parties are generated by using the private keys of them and arbitrary points of the given elliptic curve. Diffie Hellman protocol is used to authenticate the key exchange. ElGamal plus Menezes Qu Vanstone encryption protocols are used to encrypt the messages. MATLAB R2015a is used to implement and test the proper functioning of the built cryptosystem.