Walsh Spectrum and Boomerang Properties of Locally-APN Niho Functions

Recently, the Walsh spectrum and boomerang properties of special power functions have aroused widespread research interest, owing to their important applications in cryptography and information security. In particular, locally-APN functions may offer superior resistance against differential cryptanalysis compared to other functions of equivalent differential uniformity. Up till now only a small number of locally-APN functions have been studied. In this paper, we show that a Niho type power function is locally-APN if and only if its Walsh spectrum takes four values in $\{-p^m,0,p^m,2p^m\}$. Equivalently, the associated cyclic codes have four nonzero weights: $p^{m-1}(p-1)(p^m+k)$ for $k=0,1,-1,-2$. Moreover, we also study properties of Niho type locally-APN power functions, including their differential spectrum, Walsh spectrum, Feistel Boomerang Connectivity Table(FBCT for short) and second-order zero differential spectra.