If the order of elliptic group is prime then every point is a generator of that group. I tested the above statement on some elliptic curves and found it true. Does that really work on all curves?

Is there any lemma or theorem which states that?


This is true of any group of prime order, over elliptic curves or not. This is due to Lagrange's Theorem which states that the order of a subgroup $H$ of group $G$ divides the order of $G$.

Since orders are elements of the ring of integers and since this is a principal ideal domain, unique factorization exists and primes make sense. Or put another way, primes behave how you've been taught they do. Therefore the only possible subgroups of a group of order $p$ are the trivial group of order $1$ and the group of order $p$, i.e. the group itself.

This has a consequence, usually presented as a lemma of Lagrange's theorem - no element can generate a subgroup, so it must generate the full group.

For elliptic curve groups, there is a valid proper subgroup, the trivial subgroup consisting only of the identity, in this case $\mathcal{O}$.


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