The differential spectrum and boomerang spectrum of a class of locally-APN functions
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In this paper, we study the boomerang spectrum of the power mapping F(x)=x^k(q-1) over 𝔽_q^2, where q=p^m, p is a prime, m is a positive integer and (k,q+1)=1. We first determine the differential spectrum of F(x) and show that F(x) is locally-APN. This extends a result of [IEEE Trans. Inf. Theory 57(12):8127-8137, 2011] from (p,k)=(2,1) to general (p,k). We then determine the boomerang spectrum of F(x) by making use of its differential spectrum, which shows that the boomerang uniformity of F(x) is 4 if p=2 and m is odd and otherwise it is 2. Our results not only generalize the results in [Des. Codes Cryptogr. 89:2627-2636, 2021] and [arXiv:2203.00485, 2022] but also extend the example x^45 over 𝔽_2^8 in [Des. Codes Cryptogr. 89:2627-2636, 2021] into an infinite class of power mappings with boomerang uniformity 2.
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