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In this article, we introduce new notions $cc$-differential uniformity,$cc$-differential spectrum, PccN functions and APccN functions, and investigatetheir properties. We also introduce $c$-CCZ equivalence, $c$-EA equivalence,and $c1$-equivalence. We show that $c$-differential uniformity is invariantunder $c1$-equivalence, and $cc$-differential uniformity and $cc$-differentialspectrum are preserved under $c$-CCZ equivalence. We characterize$cc$-differential uniformity of vectorial Boolean functions in terms of theWalsh transformation. We investigate $cc$-differential uniformity of powerfunctions $F(x)=x^d$. We also illustrate examples to prove that $c$-CCZequivalence is strictly more general than $c$-EA equivalence.

cc-differential uniformity, (almost) perfect cc-nonlinearity, and equivalences