We have a cyclic field Fp where p is a prime number, a generator g, and an order n. A generator is an element such that $g^n=1$. A random number x has been chosen as the private key, selected from the interval from 1 to n−1. Then the public key will be $y=g^x$. The discrete logarithm problem is to determine x given y and g. This problem is hard (or hard only on elliptic curves?). Why do we need elliptic curves instead of just using a cyclic field and a generator?
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