Paper 2024/1838

Pushing the QAM method for finding APN functions further

Abstract

APN functions offer optimal resistance to differential attacks and are instrumental in the design of block ciphers in cryptography. While finding APN functions is very difficult in general, a promising way to construct APN functions is through symmetric matrices called Quadratic APN matrices (QAM). It is known that the search space for the QAM method can be reduced by means of orbit partitions induced by linear equivalences. This paper builds upon and improves these approaches in the case of homogeneous quadratic functions over $\mathbb{F}_{2^n}$ with coefficients in the subfield $\mathbb{F}_{2^m}$. We propose an innovative approach for computing orbit partitions for cases where it is infeasible due to the large search space, resulting in the applications for the dimensions $(n,m)=(8,4)$, and $(n,m)=(9,3)$. We find and classify, up to CCZ-equivalence, all quadratic APN functions for the cases of $(n,m)=(8,2),$ and $(n,m)=(10,1)$, discovering a new APN function in dimension $8$. Also, we show that an exhaustive search for $(n,m) = (10,2)$ is infeasible for the QAM method using currently available means, following partial searches for this case.

Note: The paper was submitted to the special issue of Cryptography and Communications https://link.springer.com/journal/12095