Subgroups, Generators, Orders, And Cofactors

Core analogy: Treat the finite curve group as a circular route.

• Curve order $\#E$: total number of stops in the complete group.
• Point order $\operatorname{ord}(P)$: lap length—the first positive $n$ with $nP=\mathcal{O}$.
• Generated subgroup $\langle P\rangle$: all stops reachable by repeatedly taking the $P$-step.
• Generator: a point whose multiples reach every element of the group being discussed.
• Base point $G$: the public generator selected by a parameter set.
• Subgroup order $n$: $n=\operatorname{ord}(G)$.
• Cofactor $h$: number of subgroup-sized cosets in the full group: $$\#E=hn, \qquad h=\frac{\#E}{n}.$$

Lagrange's theorem guarantees that a point's order divides the curve order.

Running example

$P=(1,5)$ takes a 14-stop lap and generates the whole curve. Choosing $G=2P=(3,3)$ skips every other stop, producing a prime-order subgroup with

$$\#E=14, \qquad n=7, \qquad h=2.$$

Protocol bridge: A private scalar $k$ produces the public point $Q=kG$; production systems make $n$ enormous.