"A few observations on APN and AB functions".

Abstract:
There remain many difficult open problems on general S-boxes (vectorial Boolean functions). In particular little is known on the best possible nonlinearity of vectorial functions from $F_2^n$ to $F_2^m$ when $n$ is odd and $m\neq n$ or when $n$ is even and $m>n/2$; few Perfect Nonlinear (PN / bent) functions are known; important questions remain unanswered about Almost Perfect Nonlinear (APN) and Almost Bent (AB) functions. Progress is necessary in these domains for supporting the design of further block ciphers. We shall recall what are these open questions and make observations about them.
Echternach-carlet.pdf

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