Difficult: I do not quite understand the idea of encrypting messages with an elliptic curve mod p. So, you pick your message and then attempt to build your curve based on using your message as a point? Or do you have the curve first and then try to guess a point on the curve based on your message by adding things to your message? If it is the second method, how would the person you are sending the message to know which part is junk? Or even if you did not have to add any junk?
Reflective: The elliptic curve mod p doesn't seem that much more difficult than the elliptic curve without modular arithmetic. The number of points on the curve being so unpredictable is interesting, but I suppose it is just another "finding square roots mod n" problem. If using a prime p, do we need to ensure that p=3 (mod 4) as we did before?
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