# How to Generate Low-Order Generator Points on Elliptic Curves

How can one generate a 'Generator Point' on an elliptic curve that has an extremely low order.

Take this Elliptic Curve from HTB Cyber Apocalypse 2024. The order of G is 11. How can one replicate this but for different custom curves.

Curve in question:

p = 6811640204116707417092117962115673978365477767365408659433165386030330695774965849821512765233994033921595018695941912899856987893397852151975650548637533
F = GF(p)
E = EllipticCurve(F, [726, 42])
G=E(926644437000604217447316655857202297402572559368538978912888106419470011487878351667380679323664062362524967242819810112524880301882054682462685841995367, 4856802955780604241403155772782614224057462426619061437325274365157616489963087648882578621484232159439344263863246191729458550632500259702851115715803253)

print(G.order()) #### Output is 11

• This question is similar to: Invalid curve attack: finding low order points. If you believe it’s different, please edit the question, make it clear how it’s different and/or how the answers on that question are not helpful for your problem. Commented Jul 12 at 6:51
• Commented Jul 12 at 14:52

1. Find the order $$n$$ of the “custom” elliptic curve (using Schoof's algorithm or some variant of it);
2. Hope that $$n$$ has some small prime factor $$\ell$$;
3. If you take a random point $$P$$ on the curve, $$G = [n/\ell] P$$ will be of order $$\ell$$ with high probability (and if it isn't, you can pick a different $$P$$ and try again).
For the different problem of constructing an elliptic curve with a point of order $$\ell$$, there are better approaches, however (involving the modular curve $$X_1(\ell)$$).