Deterministic algorithm returns element of order >D mod N in O(D^{1/2+o(1)}) time for D > exp(sqrt(2 log N log log N)), or factors N or reports primality.
The Mathematics of Paul Erd
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Deterministically finding an element of large order in $\mathbb{Z}_N^*$
Deterministic algorithm returns element of order >D mod N in O(D^{1/2+o(1)}) time for D > exp(sqrt(2 log N log log N)), or factors N or reports primality.