NIST standardized 5 elliptic curves (P-192, P-224, P-256, P-384, P-521) for prime fields. When I looked into openssl, these curves are named as prime192v1, secp224r1, prime256v1, secp384r1, secp521r1. Is there any reason why the name convention was different for different curves?
I tried to collect some benchmarks on these curves. I calculated the time required to
(1) sample one group element,
(2) multiply 2 random group elements and
(3) exponentiate a group element with a random value in range [1, order of group].
Here are the results. The benchmarks have many anamolies that I could not explain. For P-224 curve, it took unreasonably long time for sampling random element. For P-224 curve, if it takes 2.27microseconds for multiplication, then it should take at least 224*2.27microseconds (roughly 0.5ms) for exponentiation. But exponentiation is much faster. Exponentiation is unreasonably faster even for P-256 and P-521 curves as well. Can anyone please explain why this is the case?