A matrix approach for constructing quadratic APN functions

Abstract

A one to one correspondence is given between quadratic homogeneous APN functions and a special kind of matrices which we call as QAM’s. By modifying the elements of a known QAM, new quadratic APN functions can be constructed. Based on the nice mathematical structures of the QAM’s, an efficient algorithm for constructing quadratic APN functions is proposed. On $$\mathbb {F}_{2^7}$$ , we have found 471 new CCZ-inequivalent quadratic APN functions, which is 20 times more than the number of the previously known ones. Before this paper, It is only found 23 classes of CCZ-inequivalent APN functions on $$\mathbb {F}_{2^8}$$ . With the method of this paper, we have found 2,252 new CCZ-inequivalent quadratic APN functions, and this number is still increasing.