  S-boxes are an important primitive that help cryptographic algorithms to be
resilient against various attacks. The resilience against specific attacks can
be connected with a certain property of an S-box, and the better the property
value, the more secure the algorithm. One example of such a property is called
boomerang uniformity, which helps to be resilient against boomerang attacks.
How to construct S-boxes with good boomerang uniformity is not always clear.
There are algebraic techniques that can result in good boomerang uniformity,
but the results are still rare. In this work, we explore the evolution of
S-boxes with good values of boomerang uniformity. We consider three different
encodings and five S-box sizes. For sizes $4\times 4$ and $5\times 5$, we
manage to obtain optimal solutions. For $6\times 6$, we obtain optimal
boomerang uniformity for the non-APN function. For larger sizes, the results
indicate the problem to be very difficult (even more difficult than evolving
differential uniformity, which can be considered a well-researched problem).
