I am interested in the elliptic curve
$$
y^2 = x^3 + 7
$$
where both $x$ and $y$ are in the finite residue class field $F\_p$ with $p=2^{256}-2^{32}-2^9-2^8-2^7 -2^6-2^4 -1$. Those parameters are used in the secp256k1 standard. 


There are $N$ tuples $(x,y)$, $N=115'...'337$. I assume this number is computed using the ideas by Schoof? What's the strategy to compute a point (x,y) on this curve such that the cyclic group generated by this point is matching $N$. 



Elliptic curve over Galois Field, Blockchain
Yes, Schoof's point counting algorithm (1985) is generally used for this purpose. See R. Schoof: *Elliptic Curves over Finite Fields and the Computation of Square Roots mod p.* Math. Comp., 44(170):483–494, 1985.


It was the first deterministic polynomial time (in the size of the elliptic curve group, which is a prime here) algorithm for counting points on elliptic curves. 


Schoof's algorithm was improved by Elkies and Atkin. 


**Disclaimer:** I am not an expert in this area, but the answer is common knowledge if one works/teaches cryptography. The paper G. Musiker: *Schoof's Algorithm for Counting Points on* $E({\mathbb {F}}\_{{q}})$ available online [here](http://www.math.umn.edu/~musiker/schoof.pdf) may help as well.


