A Degree Bound for the c-Boomerang Uniformity
Matthias Johann Steiner
令 𝔽_q 为有限域，F ∈ 𝔽_q[X] 为次数 d = deg(F) 的多项式，且满足 (d, q) = 1。本文证明了当 c ≠ 0 时，F 的 c-Boomerang 均匀性具有以下上界：当 c² ≠ 1 时不超过 d²，当 c = -1 时不超过 d·(d-1)，当 c = 1 时不超过 d·(d-2)。对于所有 c 的情况，我们给出了 F ∈ 𝔽_q[X] 的紧示例。此外，在证明 c = 1 的情况时，我们建立了以下结论：对于特征为 p 的域 k 和 a ∈ k∖{0}，二元多项式 F(x) - F(y) + a ∈ k[x,y] 在 p ∤ deg(F) 时是绝对不可约的。
Let 𝔽_q be a finite field, and let F ∈𝔽_q [X] be a polynomial with d = deg( F ) such that ( d, q ) = 1. In this paper we prove that the c-Boomerang uniformity, c ≠ 0, of F is bounded by - d^2 if c^2 ≠ 1, - d · (d - 1) if c = -1, - d · (d - 2) if c = 1. For all cases of c, we present tight examples for F ∈𝔽_q [X]. Additionally, for the proof of c = 1 we establish that the bivariate polynomial F (x) - F (y) + a ∈ k [x, y], where k is a field of characteristic p and a ∈ k ∖{ 0 }, is absolutely i...
