Enhanced Scalar Multiplication Algorithm over Prime Field Using Elliptic Net

Abstract

Scalar multiplication in elliptic curve cryptography is the most expensive and time-consuming operation. The elliptic curve cryptography attracted interest due to the development of modern technology since it could offer the equivalent high level of security with a reduced length of key. Therefore, improving elliptic curve scalar multiplication performance has always been the primary goal of cryptography. In this paper, a novel scalar multiplication algorithm based on the modified double and double add via elliptic net with Karatsuba method was proposed in order to enhance the efficiency of scalar multiplication. In the experimental results, the elliptic net equivalence sequence was applied to the Twisted Edwards curve together with safe curves of numsp384t1 and numsp512t1. At the point operational level, the proposed method reduced the cost of multiplication by 46.15% and 42.30% for double and double add, respectively, when compared to elliptic net using eight blocks method. The proposed double lowered the multiplication cost by 12.5% and the squaring cost by 20% when compared to elliptic net using ten temporary variables method. Following this, proposed double add cost reductions of 6.25% and 20% were obtained to multiplication and squaring. At the field operational level, in comparison to the binary method, the eight-block elliptic net method, and the elliptic net method with ten temporary variables for the 384 bits scenario, the developed scalar multiplication algorithm obtained cost reductions of 57.6%, 31.3%, and 13.2%, respectively. On 512 bits with similar comparison, the designed algorithm exhibited better performance by averages 59.2%, 31.0% and 13.2%. The results signified that the designed algorithm over prime field performed better at the point and field operational levels with larger scalar bit size.