# Why does knowing the number of points on a curve help solve ECCDLP?

Perhaps, this is a really obvious question, but I am still having trouble understanding how this all fits together. Why is knowing the number of points on an Elliptic Curve helpful in cracking it?

Because it can help factor $n$ (as in “mod $n$”) if it's composite? Is it because knowing the order of different points is helpful, and the order divides the number of points total?

Why is so much effort put into finding an efficient way to count points?

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If the number of points on the curve is $n$, then we know how to solve the ECCDLP problem in $\sqrt{n}$ time. However, if we know a factorization $n = pq$, then we can use that factorization to solve the ECCDLP problem in $\sqrt{p} + \sqrt{q}$ time. Note that this is distinctly smaller; if $n$ is 256 bits, then $\sqrt{n}$ is 128 bits; we believe that no one has the computational resources to do $2^{128}$ work. On the other hand, if $p$ and $q$ is 128 bits, then $\sqrt{p} + \sqrt{q}$ is about 65 bits; we know that performing $2^{65}$ work is achievable in practice.