We discuss the second-order differential uniformity of vectorial Boolean functions. The closely related notion of second-order zero differential uniformity has recently been studied in connection to resistance to the boomerang attack. We prove that monomial functions with univariate form $x^d$ where $d=2^{2k}+2^k+1$ and $\gcd(k,n)=1$ have optimal second-order differential uniformity. Computational results suggest that, up to affine equivalence, these might be the only optimal cubic power functions. We begin work towards generalising such conditions to all monomial functions of algebraic degree 3. We also discuss further questions arising from computational results.

翻译：本文讨论向量布尔函数的二阶差分均匀度。与二阶零差分均匀度密切相关的概念最近被研究用于抵抗回旋攻击。我们证明，具有单变量形式$x^d$的单项函数，其中$d=2^{2k}+2^k+1$且$\gcd(k,n)=1$，具有最优的二阶差分均匀度。计算结果表明，在仿射等价意义下，这些函数可能是唯一最优的三次幂函数。我们开始致力于将此类条件推广到所有代数次数为3的单项函数。我们还讨论了由计算结果引发的进一步问题。