Study of Efficient Scalar Multiplication over Elliptic Curve

Elliptic Curve Scalar multiplication is the process of repeatedly adding a point on a curve to itself [1]. Many scholars working in the field of cryptography have been drawn to studies in Scalar Multiplication over Elliptic Curves (EC) over finite fields in recent years to see how elliptic curves cryptography (ECC) may be implemented and how to reduce its complexity [2]. Elliptic curve scalar multiplication utilising the point-halving algorithm [3], then the double-base (DB) chain algorithm, and finally step multi-base representation (SMBR) are the most efficient approaches used in Elliptic curve cryptography, however each technique has its own set of drawbacks. As a result, it is critical to develop a new approach that may be used to effectively deploy ECC while also decreasing its complexity. For affine coordinates, the study introduces the Treble algorithm, which is a new algorithm. We kept working with the binary concept or double and add operation with the help of the treble technique to make it more efficient, which refers to the use of all input values in producing any form of output, including how much time and energy is necessary. The results demonstrate that our contribution can improve EC scalar multiplication significantly. Elliptic Curve Scalar Multiplication is diverse aspect of Cryptography.