Elliptic curves, current status?

["James A. Donald" <jamesd@echeque.com>]

I have read:

That public key cryptography using elliptic curves is much 
faster than using integers, for comparable security.

That public key cryptography using elliptic curves is much 
slower than using integers, for comparable security.

That the numbers required have many more bits.

That the numbers required have somewhat less bits.

That elliptic curves are about the same, but the programs 
are a lot nastier.

Can someone tell me the true story?


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