Boomerang connectivity table is a new tool to characterize the vulnerability of cryptographic functions against boomerang attacks. Consequently, a cryptographic function is desired to have boomerang uniformity as low as its differential uniformity. Based on generalized butterfly structures recently introduced by Canteaut, Duval and Perrin, this paper presents infinite families of permutations of \({\mathbb {F}}_{2^{2n}}\) for a positive odd integer n, which have the best known nonlinearity and boomerang uniformity 4. Both open and closed butterfly structures are considered. The open butterflies, according to experimental results, appear not to produce permutations with boomerang uniformity 4. On the other hand, from the closed butterflies we derive a condition on coefficients \(\alpha , \beta \in {\mathbb {F}}_{2^n}\) such that the functions

where \(R_i(x,y)=(x+\alpha y)^{2^i+1}+\beta y^{2^i+1}\) and \(\gcd (i,n)=1\), permute \({{\mathbb {F}}}_{2^n}^2\) and have boomerang uniformity 4. In addition, experimental results for \(n=3, 5\) indicate that the proposed condition seems to cover all such permutations \(V_i(x,y)\) with boomerang uniformity 4.

回旋镖连通性表是一种新工具，用于表征加密功能针对回旋镖攻击的脆弱性。因此，期望密码功能具有回旋镖均匀性低至其微分均匀性。基于最近Canteaut，杜瓦尔和佩兰引入广义蝶形结构，提出了无限的排列的家庭\（{\ mathbb {F}} _ {2 ^ {2N}} \）对于正奇整数Ñ，其具有最著名的非线性和回旋镖均匀性4.考虑开放和封闭蝶形结构。根据实验结果，开放的蝴蝶似乎不会产生具有飞旋镖均匀性的排列4。另一方面，从封闭的蝴蝶中，我们得出了系数的条件\（\ alpha，\ beta \ in {\ mathbb {F}} _ {2 ^ n} \）这样的函数

其中\（R_i（x，y）=（x + \ alpha y）^ {2 ^ i + 1} + \ beta y ^ {2 ^ i + 1} \）和\（\ gcd（i，n）= 1 \），置换\（{{\ mathbb {F}}} _ {2 ^ n} ^ 2 \）并具有回旋镖均匀性4。此外，\（n = 3，5 \）的实验结果表明，建议的条件似乎以回旋镖均匀性覆盖了所有此类置换\（V_i（x，y）\） 4。