ELLIPTIC CURVE cryptosystems (ECC) are new generations of public key cryptosystems that have a smaller key size for the same level of security. The exponentiation on ELLIPTIC CURVE is the most important operation in ECC, so when the ECC is put into practice, the major problem is how to enhance the speed of the exponentiation. It is thus of great interest to develop algorithms for exponentiation, which allow e cient implementations of ECC. In this paper, we improve e cient algorithm for exponentiation on ELLIPTIC CURVEs de ned over Fp in terms of a ne coordinates. The algorithm computes 2n2(2n1P +Q) directly from random points P and Q on an ELLIPTIC CURVE, without computing the intermediate points. Moreover, we apply the algorithm to exponentiation on ELLIPTIC CURVEs with width{w Mutual Opposite Form (wMOF) and analyze their computational complexity. This algorithm can speed up the wMOF exponentiation of ELLIPTIC CURVEs of size 160{bit about (21. 7%) as a result of its implementation with respect to a ne coordinates.

ELLIPTIC CURVE cryptography system due to the short key length and high level of security is most important encryption system for use in electronic voting. The problem with this system is a lot of computation time due to the complexity of computational operations on ELLIPTIC CURVE is over. Multiplication of ELLIPTIC CURVE cryptography system is time consuming operations that about 85% of the time spent implementing the encryption algorithm stems. Hence, we propose an optimal method to reduce the cost of providing time of multiplication operations. The proposed method improved in two main parts, the parts of the control and computing encryption algorithm, has the good performance. The result of evaluation and comparison of the proposed method with some conserned algorithms, shows that this method compared to other algorithms, is faster and very good performance.

According to the spread of various elections and the need for safety and confidence of the voting process، electronic voting has been of considerable attention. Numerous studies show that the electronic voting system can be proved to the requirements of a voting process. One suitable method for the implementation of electronic voting can be homomorphic encryption. Homomorphic encryption method is an encryption system that is isomorphic to the operation. In this paper، using ELLIPTIC CURVE encryption، a new electronic voting protocol is provided، which can be monitored by all. The system is based on the difficulty of solving discrete logarithm problem on ELLIPTIC CURVE and provides voters and key safety. The proposed protocol was tested for 100 milion voters in a referendum in which the total number of votes was counted in the 30 seconds.

This paper presents a new and efficient implementation approach for the ELLIPTIC CURVE cryptosystem (ECC) based on a novel finite field multiplication in GF (2m) and an efficient scalar multiplication algorithm. This new finite field multiplication algorithm performs zero chain multiplication and required additions in only one clock cycle instead of several clock cycles. Using modified (limited number of shifts) Barrel shifter, the partial result is also shifted in one clock cycle instead of several clock cycles. Both the canonical recoding technique and the sliding window method are applied to the multiplier to reduce the average number of required clock cycles. In the scalar multiplication algorithm of the proposed implementation approach, the point addition and point doubling operations are computed in parallel. The sliding window method and the signed-digit representation are also used to reduce the average number of point operations. Based on our analysis, the computation cost (the average number of required clock cycles) is effectively reduced in both the proposed finite field multiplication algorithm and the proposed implementation approach of ECC in comparison with other ECC finite field multiplication algorithms and implementation approaches.

Sending and receiving information confidentially has always been very important. The cryptography Plays a major role in exchanging information securely. For this purpose, several encryption algorithms are designed and implemented. Among the designed algorithms, the old RSA and Diffie-Hellman algorithms have been superseded by the ELLIPTIC CURVE encryption algorithm, due to its unique features. The proposed method has higher speed for image encryption and decryption in comparison with the current method. For this purpose, the grouping of pixels by remainder Chinese theorem has been employed. The proposed method is applicable on other data such as text, sound and video.