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The vectorial nonlinearity of a vector valued function is its distance fromthe set of affine functions. In 2017, Liu, Mesnager and Chen conjectured ageneral upper bound for the vectorial linearity. Recently, Carlet proved alower bound in terms of the differential uniformity. In this paper, we improveCarlet's lower bound. Our method is elementary, it relies on the fact that thelevel sets of an APN functions are Sidon sets. We give a survey on Sidon setsin elementary abelian 2-groups. We study the completeness problem of Sidon setsobtained from hyperbolas and ellipses of the finite affine plane.

Thin Sidon sets and the nonlinearity of vectorial Boolean functions