Computational searches for quadratic APN functions with subfield coefficients

Abstract

Almost perfect nonlinear (APN) functions are important in fields such as algebra, combinatorics, cryptography, etc. Finding new APN functions is of special importance in cryptography. This is because when used in modern block ciphers, they are optimal against differential cryptanalysis. In this thesis, we discuss how the matrix approach for constructing quadratic APN functions developed by Yu et al. can be adapted to the case of functions over $\F_{2^n}$ with coefficients in a subfield $\F_{2^k}$. This adaptation allows us to search for functions of this form and using the notion of linear equivalence, we can significantly restrict the search space. Using this method, we classify all quadratic APN functions with coefficients in $\F_{2^2}$ over $\F_{2^8}$ up to CCZ-equivalence. To the best of our knowledge, no such search has been carried out before. The classification resulted in 27 CCZ-equivalence classes covering all quadratic APN functions with coefficients in $\F_{2^2}$ over $\F_{2^8}$ of which one seems to be new.