On Quadratic Almost Perfect Nonlinear Functions and Their Related Algebraic Object

It is well known that almost perfect nonlinear (APN) functions achieve the lowest possible differential uniformity for functions defined on fields with even characteristic, and hence, from this point of view, they are the most ideal choices for S-boxes in block and stream ciphers. They are also interesting as the link to many other areas, for instance topics in coding theory and combinatorics. In this paper, we present a characterization of quadratic APN functions by a certain kind of algebraic object, which is called an APN algebra. By this characterization and with the help of a computer, we discovered 285 new (up to CCZ equivalence) quadratic APN functions on F27 , which is a remarkable contrast to the currently known 17 such functions. Furthermore, 10 new quadratic APN functions on F28 are found. We propose some problems and conjectures based on the computational results.

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