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Improved ECC Performance Using NAF Algorithm for Binary Edward and Edward Elliptic Curves


Mohammad Alkhatib


Vol. 19  No. 6  pp. 98-109


Improving the performance of elliptic curve crypto-systems (ECC) to satisfy modern cryptographic applications has become a pressing need. This research presents high-speed ECCs for Binary Edward and Edward curves using Non adjacent Form (NAF) algorithm. Proposed ECCs use NAF algorithm to perform scaler multiplication, which is the main operation in encryption process. This contributed mainly in minimizing the time delay via reducing the number of point addition operations performed during scaler multiplication. Furthermore, Homogenous projective coordinates were used to avoid time consuming-modular inversion operation. Parallel design implementations were used to accelerate ECC computations and achieve the highest performance level. Experimental results show that ECC implementations using NAF algorithm overcome corresponding implementations using other methods such as binary method for all possible design choices. A variety of parallel designs in addition to sequential design implementations were examined for both Edward and Binary Edward curves. The Binary Edward ECC using NAF algorithm and 7 parallel multipliers (PM) design accomplished the shortest time delay. So, it represents an attractive choice for security applications that require high-speed crypto-processor. Moreover, ECC sequential design using NAF algorithm scored better performance results in comparison with similar designs that use the binary method. Sequential design is a preferable choice for applications where available resources are limited. Presented ECC designs are implemented using VHDL, and synthesized using the Xilinx tool with target FPGA. This article also presents performance analysis for the different design choices of the two types of elliptic curves using both NAF and binary algorithms. This helps in developing the most efficient ECC for different security applications.


Elliptic curves cryptosystem, Time-consumption, Resources, NAF algorithm, Binary method, Projective Coordinates, Security applications.