On the Niho Type Locally-APN Power Functions and Their Boomerang Spectrum
Xi Xie, Sihem Mesnager, Nian Li, Debiao He, Xiangyong Zeng · IEEE Transactions on Information Theory · 2022
This article focuses on the so-called locally-APN power functions introduced by Blondeau, Canteaut and Charpin, which generalize the well-known notion of APN functions and possibly more suitable candidates against differential attacks. Specifically, given two coprime positive integers$m$and$k$such that$\gcd (2^{m}+1,2^{k}+1)=1$, we investigate the locally-APN-ness property of the Niho type power function$F(x)=x^{s(2^{m}-1)+1}$over the finite field$\mathbb {F}_{2^{2m}}$for$s=(2^{k}+1)^{-1}$, where$(2^{k}+1)^{-1}$denotes the multiplicative inverse modulo$2^{m}+1$. By employing finer studies of the number of solutions of certain equations over finite fields, we prove that$F(x)$is locally-APN and determine its differential spectrum. We emphasize that computer experiments show that this class of locally-APN power functions covers all Niho type locally-APN power functions for$2\leq m\leq 10$. In addition, we also determine the boomerang spectrum of$F(x)$by using its differential spectrum, which particularly generalizes a recent result by Yan, Zhang and Li.