Topological Invariants for Linear Codes and APN Functions

Abstract

In this paper, we try to apply methods from topological data analysis (TDA) to study geometric properties invariant under code equivalence transformation, especially for the linear codes associated with almost perfect nonlinear (APN) functions which offer optimal resistance to differential attacks and are very important in the design of block ciphers in cryptography. By employing persistent homology from TDA and tools from graph theory, we present new CCZ-invariants for APN functions. Some of them are computationally efficient and sufficient to distinguish many known APN functions, including $x^{3}$ and $x^{9}$ (resp. $x^{33}$ ) over $\mathbb {F}_{2^{7}}$ (resp. $\mathbb {F}_{2^{9}}$ ) for which previously known invariants fail to do so.