Advances on APN Functions

Speaker: Dr. Eimear Byrne , School of Mathematical Sciences, University College Dublin, Ireland


A function f defined on a finite field L is called almost perfect nonlinear ( APN) if there are at most two solutions to the equation f(x+a)-f(x) = b for each a,b in L with a nonzero. APN functions arise in coding theory, cryptography and sequences, especially for fields of characteristic 2. Monomial APN functions co rrespond to m-sequences and cyclic codes of minimum distance 5. Many APN functio ns are useful as substitution boxes of block-ciphers, having optimal resistance to a differential attack (by definition) and to a linear attack when defined on a field of odd degree over GF(2). For some time, it was conjectured that any APN function was equivalent to one of a short list of monomials and much work has b een done towards a full classification. However, since 2006, a number of new fam ilies of APN functions have been discovered, inequivalent to any of the known po wer functions. In this talk we discuss these new results and related open proble ms.