Cubic power functions with optimal second-order differential uniformity

We discuss the second-order differential uniformity of vectorial Boolean functions, a relevant cryptographic property due to indication of resistance to the boomerang attack. First, we discuss connections with the second-order zero differential uniformity and its recent literature. We then prove the optimality of monomial functions with univariate form $x^d$ where $d=2^{2k}+2^k+1$ and $\gcd(k,n)=1$, and begin work towards generalising such conditions to all monomial functions of algebraic degree 3. Finally, we discuss further questions arising from computational results.