# Parity of the order of a element

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 $$G$$?