How to compare the order of elements in cyclic groups?

In a cyclic group with randomly looking behavior like the one used in secp256k1, is there any known efficient algorithm to compare the order of two randomly given elements $$P_1$$ and $$P_2$$ and find out which one will appear first starting from a known generator point of the cycle? If such algorithm exists, would that be considered a weakness?!

Consider that we have two known integers $$a$$ and $$b$$ so that $$a>b$$. Taking $$P_1=a\times G$$ and $$P_2=b\times G$$ we can say $$P_2$$ has a greater order in the cycle, meaning that starting from the generator point it will appear before than $$P_1$$.

By this definition is there any known algorithm based which one can compare the order of two random elements of the group $$P_1$$ and $$P_2$$?

• You really shouldn't use the word "order" here. "Order" in group theory has a precise meaning, and using it to mean anything else creates confusion. – fkraiem Jan 6 '19 at 13:04
• What word do you suggest to use instead of "order"? – PouJa Jan 6 '19 at 16:15
• The integer $a$ such that $P_1 = a\cdot G$ is called the discrete logarithm of $P_1$ (relative to $G$), and similarly for $P_2$, so you could say that you wish to show that the discrete logarithm of $P_2$ is larger than that of $P_1$. – fkraiem Jan 6 '19 at 16:21

By this definition is there any known algorithm based which one can compare the order of two random elements of the group $$P_1$$ and $$P_2$$?
To see this fix the reference generator $$G$$ to the standard one for the curve with the generator having order $$q$$. Furthermore let $$P=xG$$ be the challenge point for which we want to recover $$x$$. Now define $$a=1$$ and $$b=q-1$$.
1. Compute $$m=\frac{a+b}{2}$$ and $$M=mG$$.
2. Check whether $$M=P$$ and if so, return $$m$$.
3. Check whether $$M using the conjectured algorithm. If this is true, set $$a\gets m$$, otherwise set $$b\gets m$$.
This is a standard binary search as you may have recognized and it will find the discrete logarithm in about $$\log_2 q$$ scalar point multiplications and algorithm evaluations which is quite efficient.