A Niho power function is locally-APN if and only if its Walsh spectrum is four-valued in {-p^m, 0, p^m, 2p^m}.
On the feistel counterpart of the boomerang connectivity table: introduction and analysis of the FBCT,
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Walsh Spectrum and Boomerang Properties of Locally-APN Niho Functions
A Niho power function is locally-APN if and only if its Walsh spectrum is four-valued in {-p^m, 0, p^m, 2p^m}.