Cubic power functions with optimal second-order differential uniformity

TL;DR

This paper investigates the second-order differential uniformity of vectorial Boolean functions, identifying specific monomial functions with optimal properties and exploring their uniqueness and generalizations.

Contribution

It proves certain cubic monomial functions have optimal second-order differential uniformity and suggests these may be unique up to affine equivalence.

Findings

Certain monomial functions with specific exponents have optimal second-order differential uniformity.

Computational evidence indicates these might be the only such optimal cubic power functions up to affine equivalence.

Work is ongoing to generalize these conditions to all degree 3 monomial functions.

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.