Design and performance evaluation of parallel elliptic curve cryptosystem with GF(P) projective coordinates

Elliptic Curves Cryptosystem (ECC) has been introduced as a secure and efficient public key algorithm. A number of elliptic curves representations have been presented, such as Standard (Weierstrass), Edwards, Binary Edwards, Montgomery curves, and others. ECC’s computations suffer the long time inversion operation when applied using the usual affine coordinates, the use of serial design also increases the time delay, which affects the performance of ECC. The efficient selection of appropriate coordinates system to be applied with particular curve is one of the main concerns when designing efficient and high-speed ECC architecture. Moreover, other factors that play a crucial role in designing efficient ECC for different applications have not been intensively addressed in the majority of present ECC designs. These factors include area, system utilization, resources consumption,area*time (AT), and area*time2 (AT2) cost factors. The variation in elliptic curves and security applications in recent years, calls for finding several design solutions (choices) of ECC that fit the different security applications according to the requirements of particular application and the available resources. It is worth mentioning that relatively few research works were conducted on the prime field (GF(p)). The approach adopted in this thesis uses several projective coordinates to apply ECC computations over GF (p), in order to eliminate inversion operation. In addition to the current projective coordinates, projection form (X/Z2, Y/Z2) was proposed to be used for Edwards ECC. To improve performance even further, this work proposed using parallel hardware designs by utilizing the inherent parallelism in ECC computations. Our proposed designs were supported by mathematical analytical study and solutions for different ECCs presented. The proposed designs were also implemented using VHDL, and then the Xilinx tool was used to synthesize the designs. A number of comparisons were conducted to highlight enhancements achieved using presented ECC designs. The designs proposed improved the performance of the Binary Edwards ECC considerably. The best performance level was achieved using homogeneous coordinates. This projection also showed the highest performance for both Montgomery and Standard curves when applied using four and five parallel multipliers (PM) respectively. Furthermore, the performance of Edwards ECC using projection (X/Z2, Y/Z2) overcame other known projective coordinates systems. This thesis proposed several design solutions for the aforementioned curves byvarying the degree of parallelism for ECC designs. The proposed designs provided an attractive trade-off between mentioned factors, which improved these factors. Furthermore, this research determined the most efficient coordinates to be applied with particular parallelization level for ECC. Such findings and others presented in this work lead to the building of efficient ECCs that satisfies different applications.

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