On the Niho Type Locally-APN Power Functions and Their Boomerang Spectrum

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(2m + 1, 2k + 1) = 1, we investigate the locallyAPN-ness property of the Niho type power function F (x) = xs(2m- 1)+1 over the finite field F 22m for s = ( 2 k + 1)-1, where (2k + 1)-1 denotes the multiplicative inverse modulo 2m + 1. By employing finer studies of the number of solutions of certain equations over finite fields, we prove that F (x) is locallyAPN 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 <= m <= 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.