On the Niho type locally-APN power functions and their boomerang spectrum
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In this article, we focus on the concept of locally-APN-ness (“APN" is the abbreviation of the well-known notion of Almost Perfect Nonlinear) introduced by Blondeau, Canteaut, and Charpin, which makes the corpus of S-boxes somehow larger regarding their differential uniformity and, therefore, possibly, more suitable candidates against the differential attack (or their variants). Specifically, given two coprime positive integers m and k such that (2^m+1,2^k+1)=1, we investigate the locally-APN-ness property of an infinite family of Niho type power functions in the form F(x)=x^s(2^m-1)+1 over the finite field 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 (with even characteristic) as well as some subtle manipulations of solving some equations, we prove that F(x) is locally APN and determine its differential spectrum. It is worth noting 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.
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