“Semifields: theory and elementary constructions”

Finite semifields, also known as finite non-associative division algebras, generalize finite fields. Associative semifields are (skew)fields. A slightly more general notion is a presemifield (where the existence of a multiplicative unit is no longer required). Each finite semifield has p elements for some r (p a prime). We can think of addition coinciding with the addition in the field GF (p). A geometric representation of a presemifield is the corresponding projective plane. Those semifield planes are characterized as being translation planes and also dual translation planes. The basic equivalence relation of isotopy relies on a group of motions GL(r, p) × GL(r, p) × GL(r, p). This is a natural notion as presemifields are isotopic if and only if the corresponding semifield planes are isomorphic. We want to discuss some basic problems and elementary constructions. One basic problem concerns isotopy invariants and substructures. In fact, the naive notion of a sub(pre)semifield is not satisfactory as it is not stable under isotopy. The corresponding isotopy-invariant notion are substructures. Further basic problems include a canonical passage from a given presemifield to an isotopic semifield, criteria when a presemifield is essentially commutative (meaning: isotopic to commutative), proofs of non-isotopy, containment of essentially non-commutative substructures in essentially commutative presemifields and the question if the theory of finite semifields is in some way essentially equivalent to the theory of finite commutative semifields. The constructions to be discussed include the trans-characteristic construction and the projection construction. In odd characteristic (p odd) the theory of commutative semifields may be equivalently expressed in terms of PN functions (PN=perfectly nonlinear). If x∗y is the presemifield multiplication, then the corresponding PN function is Q(x) = x∗x. The relation between semifield multiplication and PN function is formally identical to the relation between a symmetric bilinear form and the corresponding quadratic form in odd characteristic, via the polarisation formula. This leads to the phenomenon that the theory of commutative semifields in odd characteristic has two different natural analogues in characteristic 2, commutative semifields in characteristic 2 and the theory of APN functions (APN=almost perfectly nonlinear), a basic notion in the cryptographic theory of S-boxes with applications also to coding theory. The trans-characteristic construction arose from a construction of a family of APN-functions by lifting the proof from characteristic 2 to arbitrary characteristic. Those PN functions are binomials, linear combinations of two monomials. The projection construction is much more general. It applies in all characteristics and is not limited to the commutative case.