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The Element Distinctness problem is to decide whether each character of aninput string is unique. The quantum query complexity of Element Distinctness isknown to be $\Theta(N^{2/3})$; the polynomial method gives a tight lower boundfor any input alphabet, while a tight adversary construction was only known foralphabets of size $\Omega(N^2)$.  We construct a tight $\Omega(N^{2/3})$ adversary lower bound for ElementDistinctness with minimal non-trivial alphabet size, which equals the length ofthe input. This result may help to improve lower bounds for other related queryproblems.

Adversary Lower Bound for Element Distinctness with Small Range