Improved scalar multiplication algorithm in affine coordinate system using elliptic net

In elliptic curve encryption, scalar multiplication (SM) is the most expensive and time-consuming operation. The elliptic curve cryptography attracts interest since it offers the same high security with a lower key length, owing to the advancement of modern technologies. Thus, this study designed a new scalar multiplication algorithm using six blocks of the elliptic net in a double and double-add method that cost 12M+6S in each block. This study also proposed a new formula for double block via the elliptic net method that saves four multiplications and four squaring from the prior double step. Experimental results over prime field p were conducted using safe curves namely numsp384t1 and numsp512t1, with equivalence sequences that satisfied gcd(p-1, 3). In the case of the 384-bits, results indicate that the developed scalar multiplication algorithm accelerates the running time by 65.96 % compared to the binary method, 44.81 % compared to the elliptic net without equivalent sequences, 30.28 % compared to the elliptic net with temporary variables, and 19.71 % compared to the seven blocks of the elliptic net with Karatsuba method. In a similar comparison for the 512-bits case, the proposed algorithm attained are 67.23 %, 44.65 %, 30.37 %, and 22.64 % faster, respectively.

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