Distribution of elliptic twins over fixed finite fields: Numerical results

TL;DR

This paper presents numerical results on the distribution of elliptic twin curves over specific NIST prime fields, suggesting non-uniformity in the selection of the P-384 curve for cryptographic use.

Contribution

It provides new numerical data on elliptic twin distributions over NIST primes and discusses implications for the curve selection process.

Findings

Results suggest P-384 was not chosen uniformly among prime-order curves.

Numerical evidence supports deviation from theoretical distribution models.

Plans for future experiments to further investigate elliptic curve distributions.

Abstract

We report numerical results, and describe plans for future experiments, related to the number of prime-order curves and "elliptic twin" curves over the primes P-224, P-256, and P-384 standardized by NIST for cryptographic applications. Although these results are not sufficient to confirm the formula of Shparlinski and Sutantyo 2014 over these fields, they strongly suggest (~99% probability) that the NIST curve P-384 was not chosen from a uniform distribution over prime-order curves.