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Given an element $g$ in a cyclic group $G$ of known order $m$ its easy to test if $m$ has even or odd order. In other words $\textrm{ord}(g) \pmod 2$ can be computed easily.

In some cases where the order of the group is unknown it is also easy to test if $g$ has even order for a large subset of $g$ in $G$.

For example, if $g$ is an element in ${\mathbb{Z}_n}^*$ and the Jacobi symbol $\left(\frac{g}{n}\right)$ is $-1$ then $g$ has even order. (If it is $1$ then we can't say anything.)

What other examples are there of cryptographically interesting groups $G$ where it is easy to test if $g$ has even order (or odd order) for a large subset of $g$ in $G$?

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