We prove an $Omega(d lg n/ (lglg n)^2)$ lower bound on the dynamic cell-probe complexity of statistically $mathit{oblivious}$ approximate-near-neighbor search ($mathsf{ANN}$) over the $d$-dimensional Hamming cube. For the natural setting of $d = Theta(log n)$, our result implies an $tilde{Omega}(lg^2 n)$ lower bound, which is a quadratic improvement over the highest (non-oblivious) cell-probe lower bound for $mathsf{ANN}$. This is the first super-logarithmic $mathit{unconditional}$ lower bound for $mathsf{ANN}$ against general (non black-box) data structures. We also show that any oblivious $mathit{static}$ data structure for decomposable search problems (like $mathsf{ANN}$) can be obliviously dynamized with $O(log n)$ overhead in update and query time, strengthening a classic result of Bentley and Saxe (Algorithmica, 1980).
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