"Analytic Rank Upper Bound of Elliptic Curves Given the Digit Size of the Coefficients of the Planar Cubic"

"Computing high-rank elliptic curves, a fundamentally difficult problem, infiltrates many fields of mathematics and cryptography. One of the most notable examples is in Elliptic Curve Cryptography, a stronger counterpart compared to Rivest\u2013Shamir\u2013Adleman (RSA), which requires the in-depth study of the properties of these algebraic curves. As such, it is necessary to find an algorithm that can generate high-rank curves given a certain lower bound, avoiding the consequences of computing the rank by hand. In this paper, we present an algorithm that generates high-rank elliptic curves E over Q verified by Bober's analytic rank program. Using this algorithm, a set of 900 elliptic curves was generated with their corresponding analytic rank upper bound, testing the relationship between digit sizes of the elliptic curve generating cubic, F = (d - e + g -h - i)xyz + (e - h)yz^2 - (d - e + i)x^2 z +(-g+i)x^2 y + y^2 z + x^3  + i xz^2, and the analytic rank upper bound. From our data, a cubic relationship is observed. Finally, as a case study, we isolated two notable elliptic curves with analytically high rank generated from the program. "