The time complexity of some algorithms for generating the spectra of finite simple groups

The spectrum $\omega(G)$ is the set of orders of elements of $G$. We consider the problem of generating the spectrum of a finite nonabelian simple group $G$ given by the degree of $G$ if $G$ is an alternating group, or the Lie type, Lie rank and order of the underlying field if $G$ is a group of Lie type.