cc-differential uniformity, (almost) perfect cc-nonlinearity, and equivalences. (arXiv:2301.09325v1 [cs.IT])

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