A Further Study of Quadratic APN Permutations in Dimension Nine

Abstract

Recently, Beierle and Leander found two new sporadic quadratic APN permutations in dimension 9. Up to EA-equivalence, we present a single trivariate representation of those two permutations as $C_u \colon (\mathbb{F}_{2^m})^3 \rightarrow (\mathbb{F}_{2^m})^3, (x,y,z) \mapsto (x^3+uy^2z, y^3+uxz^2,z^3+ux^2y)$, where $m=3$ and $u \in \mathbb{F}_{2^3}\setminus\{0,1\}$ such that the two permutations correspond to different choices of $u$. We then analyze the differential uniformity and the nonlinearity of $C_u$ in a more general case. In particular, for $m \geq 3$ being a multiple of 3 and $u \in \mathbb{F}_{2^m}$ not being a 7-th power, we show that the differential uniformity of $C_u$ is bounded above by 8, and that the linearity of $C_u$ is bounded above by $8^{1+\lfloor \frac{m}{2} \rfloor}$. Based on numerical experiments, we conjecture that $C_u$ is not APN if $m$ is greater than $3$. We also analyze the CCZ-equivalence classes of the quadratic APN permutations in dimension 9 known so far and derive a lower bound on the number of their EA-equivalence classes. We further show that the two sporadic APN permutations share an interesting similarity with Gold APN permutations in odd dimension divisible by 3, namely that a permutation EA-inequivalent to those sporadic APN permutations and their inverses can be obtained by just applying EA transformations and inversion to the original permutations.