Scalar Multiplication on Elliptic Curves in Cryptography: Choosing an Efficient Generator

One of the major challenges in the current world of cryptography is the efficient exchange of data, especially with regard to the initial peer-to-peer key transfer. Establishing a cost effective way of sending back and forth this initial code has not yet proven fruitful, and this research hopes to establish efficient generators (a generator is a number or point on a coordinate plane that serves as the basis for the initial exchange of data, as expounded upon by the paper by Diffie and Hellman). Using Python software, different functions were executed to better ascertain the efficiency of different generators across elliptic curves. The thing that was quantified is the Hamming weight, which is the “heaviness” or amount of ones in a string of binary code. Although at this stage the maximum and minimum values only differed by a value of three, it will be exacerbated when the small test values are scaled up to industry standards, somewhere in the range of 2²⁰⁰ and 2⁴⁰⁰, although anywhere in this range is perfectly secure. In the future, this research will be extended to this range of values and more conclusive data will be enumerated and expounded upon. A second facet of the investigation is synthesizing a specific table of values that will allow for more succinct ways to calculate generators; the challenge is how to minimize the tradeoff between storage, speed, and security.