How should I define order according to domain parameters in elliptic curve pairing groups?
I think you should probably specify more details about the current curve, or give more insight about the parameters you are choosing, eg: is this a curve over $GF(p)$ or is this a curve over $GF(2^m)$. What is your $r$ value? Without such detail I can't give a very accurate answer but the order of the subgroup generated by a given point $P$, $&lt;P&gt;$ can be calculated from a combination of bruteforcing and checking the divisors of the number of elements inside the original curve (Since these are cyclic groups you really only need to apply lagrange theorem and check if the current divisor $d$ of $n$ satisfies $P^d = O$). There are polynomial time algorithms to retrieve the order of the original curve group (check Schoof's algorithm).