I refer to elliptic curve groups over prime fields and their application in cryptography.
If the order of a group is prime, it follows (am I wrong?) that:
- the group is cyclic and
- every element beside the zero element is a generator.
To my knowledge the first statement is well known and last statement is a consequence of the theorem:
If G is a cyclic group of order n and $\phi$ is Euler'sPhi-Function, then G has exactly $\phi(n)$ generators.
Now, if n=p is prime, then $\phi(p) = p-1$. Therefore it follows (2).
But: The order of P-256 (n=115792089210356248762697446949407573529996955224135760342422259061068512044369) is prime. So every point in this group beside the point at infinity should be a generater of the whole group. Why is a particular base point given in all the literature:
when every point is an equally good possible base point? is it just a convention to use that particular point and for what reason?