More infinite classes of APN-like Power Functions
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In the literature, there are many APN-like functions that generalize the APN properties or are similar to APN functions, e.g. locally-APN functions, 0-APN functions or those with boomerang uniformity 2. In this paper, we study the problem of constructing infinite classes of APN-like but not APN power functions. For one thing, we find two infinite classes of locally-APN but not APN power functions over _2^2m with m even, i.e., ℱ_1(x)=x^j(2^m-1) with (j,2^m+1)=1 and ℱ_2(x)=x^j(2^m-1)+1 with j = 2^m+2/3. As far as the authors know, our infinite classes of locally-APN but not APN functions are the only two discovered in the last eleven years. Moreover, we also prove that this infinite class ℱ_1 is not only with the optimal boomerang uniformity 2, but also has an interesting property that its differential uniformity is strictly greater than its boomerang uniformity. For another thing, using the multivariate method, including the above infinite class ℱ_1, we construct seven new infinite classes of 0-APN but not APN power functions.
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