  Specific vectorial boolean functions, such as S-Boxes or APN functions have
many applications, for instance in symmetric ciphers. In cryptography they must
satisfy some criteria (balancedness, high nonlinearity, high algebraic degree,
avalanche, or transparency) to provide best possible resistance against
attacks. Functions satisfying most criteria are however difficult to find.
Indeed, random generation does not work and the S-Boxes used in the AES or
Camellia ciphers are actually variations around a single function, the inverse
function in F_2^n. Would the latter function have an unforeseen weakness (for
instance if more practical algebraic attacks are developped), it would be
desirable to have some replacement candidates. For that matter, we propose to
weaken a little bit the algebraic part of the design of S-Boxes and use finite
semifields instead of finite fields to build such S-Boxes. Since it is not even
known how many semifields there are of order 256, we propose to build S-Boxes
and APN functions via semifields pseudo-extensions of the form S_{2^4}^2, where
S_{2^4} is any semifield of order 16 . Then, we mimic in this structure the use
of functions applied on a finite fields, such as the inverse or the cube. We
report here the construction of 12781 non equivalent S-Boxes with with maximal
nonlinearity, differential invariants, degrees and bit interdependency, and
2684 APN functions.
