Boomerang uniformity of a class of power maps

TL;DR

This paper investigates the boomerang uniformity of a specific class of power maps over finite fields, revealing that it is 2 or 4 depending on the field size, and that it is strictly less than the differential uniformity.

Contribution

It establishes the boomerang uniformity values for an infinite class of power maps and compares it to their differential uniformity, providing new insights into their cryptographic properties.

Findings

Boomerang uniformity is 2 when n ≡ 0 mod 4.

Boomerang uniformity is 4 when n ≡ 2 mod 4.

Differential uniformity exceeds boomerang uniformity for these maps.

Abstract

We consider the boomerang uniformity of an infinite class of (locally-APN) power maps and show that its boomerang uniformity over the finite field $\F_{2^n}$ is $2$ and $4$, when $n \equiv 0 \pmod 4$ and $n \equiv 2 \pmod 4$, respectively. As a consequence, we show that for this class of power maps, the differential uniformity is strictly greater than its boomerang uniformity.