We present a polynomial-time reduction of the discrete logarithm problem in any periodic (or torsion)
semigroup (SGDLP) to the classic DLP in a _subgroup_ of the same semigroup.
It follows that SGDLP can be solved in polynomial time by quantum computers, and that
SGDLP has subexponential complexity whenever the classic DLP in the corresponding groups has subexponential complexity. We also consider several natural constructions of nonperiodic semigroups,
and provide polynomial time solutions for the DLP in these semigroups.
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