(Difficult) I was left with a few unresolved questions regarding the elliptic curve Diffie-Hellman Key Exchange: First of all, if Alice and Bob Choose N(a) and N(b) such that N(a)N(b)G = "infinity", this would yield a useless key. In choosing the point G and the Elliptic curve E, would it be better to verify that the only multiples kG for which kG = "infinity" on E is when k is extremely large, or would this limitation weaken the security of the system? Secondly, is there a size limitation on the values they pick for N(a) and N(b)? For example N(a)N(b) < p, where p is the modulo of our elliptic curve.
(Reflective) After reading about many of the uses of elliptic curves in cryptography, it seems to me that elliptic curves may be even more useful than RSA or ElGamal. The only thing that has not been mentioned here is the performance of each system. That is, which method best preserves security while remaining fast and efficient. We saw with RSA that making our two primes slightly larger would greatly increase the time required to perform a brute force attack. Are elliptic curves quicker to calculate then fast modular exponentiation?
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