Distribution of elliptic twins over fixed finite fields: Numerical results
This addresses a specific issue in cryptography regarding the randomness of standardized elliptic curves, but it is incremental as it builds on prior theoretical work without confirming it.
The study tackled the problem of verifying the distribution of elliptic twin curves over NIST-standardized prime fields, finding numerical results that suggest with ~99% probability that the P-384 curve was not selected from a uniform distribution of prime-order curves.
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.