The main goal of this project is to refine our understanding of certain combinatorial objects (mainly some specific classes of (vectorial) Boolean functions) and the design of linear codes that are characterized with some additional properties that are proved useful in certain cryptographic applications.
Good linear codes over GF(2) (that are sometimes optimal) can be derived using some special classes of vectorial Boolean functions such as APN (almost perfect nonlinear) and AB (almost bent) functions, which are simply mappings from to .
From the known design approaches, it is apparent that a certain combinatorial structure is necessarily imposed for the defining sets (e.g. support of a bent function) both for the purpose of easier analysis and better control of the code parameters.
Therefore, the underlying combinatorial structure of certain classes of (vectorial) Boolean functions (such as APN, AB, bent and plateaued functions) seems to play a crucial role in this context.
Nevertheless, not all optimal linear codes are projective which implies that the defining set is then a multiset and the problem of specifying such codes is intrinsically hard.
In this context, there are some initial observations (ongoing work) on the multiset D that generate optimal codes (as a modification of the bent support) but there is no proper understanding of this process.
Most notably, the property of being APN can also be stated in terms of its associated dual code and hence there exists a close connection between two seemingly unrelated objects.
More precisely, an APN function over GF(2)ⁿ can be alternatively specified through the parity check matrix H of size 2ⁿ – 1 x 2ⁿ whose corresponding linear code has minimum distance d = 5, which is quite interesting even though these matrices become large since their sizes are exponential in n.
Concludingly, the proposed project aims at finding further connections between certain important discrete structures which may increase our knowledge about their structural behaviour.