The Equivalence of Apn and Ab Functions and Their Generalizations

Vectorial Boolean functions are used in cryptography, in particular in block ciphers. An important condition on these functions is a high resistance to differential and linear cryptanalysis, which are the main attacks on block ciphers. The functions which possess the best resistance to the differential attack are called almost perfect nonlinear (APN). Almost bent (AB) functions are those mappings which oppose an optimum resistance to both linear and differential attacks. Before this work, only a few classes of APN and AB functions had been known and all these classes happened to be extended affine equivalent (EA-equivalent) to power functions. In this work we construct the first classes of APN and AB polynomials EA-inequivalent to power mappings by using the equivalence relation of functions (which we call CCZ-equivalence) introduced by Carlet, Charpin and Zinoviev (1998). Then the constructed APN and AB functions are used to solve other related problems.