In this work, we study functions that can be obtained by restricting a vectorial Boolean function F:F2n→F2n\documentclass[12pt]{minimal}
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\begin{document}$$F :\mathbb {F}_{2}^n \rightarrow \mathbb {F}_{2}^n$$\end{document} to an affine hyperplane of dimension n-1\documentclass[12pt]{minimal}
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\begin{document}$$n-1$$\end{document} and then projecting the output to an n-1\documentclass[12pt]{minimal}
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\begin{document}$$n-1$$\end{document}-dimensional space. We show that a multiset of 2·(2n-1)2\documentclass[12pt]{minimal}
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\begin{document}$$2 \cdot (2^n-1)^2$$\end{document} EA-equivalence classes of such restrictions defines an EA-invariant for vectorial Boolean functions on F2n\documentclass[12pt]{minimal}
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\begin{document}$$\mathbb {F}_{2}^n$$\end{document}. Further, for all of the known quadratic APN functions in dimension n<10\documentclass[12pt]{minimal}
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\begin{document}$$n < 10$$\end{document}, we determine the restrictions that are also APN. Moreover, we construct 6368 new quadratic APN functions in dimension eight up to EA-equivalence by extending a quadratic APN function in dimension seven. A special focus of this work is on quadratic APN functions with maximum linearity. In particular, we characterize a quadratic APN function F:F2n→F2n\documentclass[12pt]{minimal}
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\begin{document}$$F :\mathbb {F}_{2}^n \rightarrow \mathbb {F}_{2}^n$$\end{document} with linearity of 2n-1\documentclass[12pt]{minimal}
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\begin{document}$$2^{n-1}$$\end{document} by a property of the ortho-derivative of its restriction to a linear hyperplane. Using the fact that all quadratic APN functions in dimension seven are classified, we are able to obtain a classification of all quadratic 8-bit APN functions with linearity 27\documentclass[12pt]{minimal}
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\begin{document}$$2^7$$\end{document} up to EA-equivalence.
