Cubic power functions with optimal second-order differential uniformity
Connor O'Reilly, Ana Sălăgean
TL;DR
The paper investigates the second-order differential uniformity of vectorial Boolean functions over $\mathbb{F}_{2^n}$, with a focus on cubic monomials. It proves that the monomial $f(x)=x^{d}$ with $d=2^{2k}+2^{k}+1$ and $\gcd(k,n)=1$ attains the optimal value $\delta^2(f)=4$, while the same form with $\gcd(k,n)>1$ yields $\delta^2(f)=2^n$. It also derives necessary conditions on exponents $i,j$ for general cubic monomials $f(x)=x^{2^j+2^i+1}$ to be optimal, including affine-equivalence reductions to the cubic form when certain modular relations hold. Computational results for $4\le n\le 20$ support a conjecture that, up to affine equivalence, the cubic optimal exponents are exactly those of the form $d=2^{2k}+2^{k}+1$ with $\gcd(k,n)=1$, with rare degree-4 exceptions.
Abstract
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.